Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups
orchidealecian
9 views
67 slides
Oct 13, 2024
Slide 1 of 67
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
About This Presentation
Seminar PUC- Pontifical Catholic University of Rio de Janeiro,
Rio de Janeiro, Brazil.
26 January 2024.
Author: Orchidea Maria Lecian
Speaker: Orchidea Maria Lecian
Title: Some new theorems
on scarred waveforms
in desymmetrised PSL(2, Z) group
and in its congruence subgroups
Abstract: The scarred wa...
Slide Content
Some new theorems
on scarred waveforms
in desymmetrised PSL(2, Z) group
and in its congruence subgroups.
Orchidea Maria Lecian
Sapienza University of Rome,
Rome, Italy.
PUC- Pontifical Catholic University of Rio de Janeiro,
Rio de Janeiro, Brazil.
26 January 2024
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
on scarred waveforms
in desymmetrised PSL(2, Z) group
and in its congruence subgroups.
Orchidea Maria Lecian
Sapienza University of Rome,
Rome, Italy.
PUC- Pontifical Catholic University of Rio de Janeiro,
Rio de Janeiro, Brazil.
26 January 2024
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Abstract
The scarred wavefunctions of the desymmetrisedPSL(2,Z) group and
those of its congruence subgroups are newly studied.
The construction of irrationals after the (Farey)-Pell method is compared
with the qualities of the quadratic fields.
The action of automorphisms on trees is recalled to explain the action of
the Hecke operators on the Maass waveforms.
The Margulis measure (which acquires a multiplicative constant under
the action of certainUflows) on quadratic fields is used.
The opportune Birkhoff reduced surfaces of section are chosen.
Closed geodesics are newly proven to scar the waveforms according to the
quadratic field they are constructed after:
a) in the desymmetrisedPSL(2,Z) group, the scarred waveforms are
newly proven to be obtained under the action of theUflow on the
Margulis measure which acts on the quadratics fields which define the
(also, classes) of closed geodesics;
b) in the congruence subgroups of the desymmetrisedPSL(2,Z) domain,
the scarred wavefunctions are newly proven to occur under the effect of
the action of the Bogomolny transfer operators on the Margulis measure.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The scarred wavefunctions of the desymmetrisedPSL(2,Z) group and
those of its congruence subgroups are newly studied.
The construction of irrationals after the (Farey)-Pell method is compared
with the qualities of the quadratic fields.
The action of automorphisms on trees is recalled to explain the action of
the Hecke operators on the Maass waveforms.
The Margulis measure (which acquires a multiplicative constant under
the action of certainUflows) on quadratic fields is used.
The opportune Birkhoff reduced surfaces of section are chosen.
Closed geodesics are newly proven to scar the waveforms according to the
quadratic field they are constructed after:
a) in the desymmetrisedPSL(2,Z) group, the scarred waveforms are
newly proven to be obtained under the action of theUflow on the
Margulis measure which acts on the quadratics fields which define the
(also, classes) of closed geodesics;
b) in the congruence subgroups of the desymmetrisedPSL(2,Z) domain,
the scarred wavefunctions are newly proven to occur under the effect of
the action of the Bogomolny transfer operators on the Margulis measure.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Summary
•from automorphisms on trees to Hecke theory on Maass eigenforms;
•the (Farey-)Pell construction of irrationals from Farey trees: the
quadratic-fields formalism,
the graphs of the Gauss map;
•Self-adjoint-ness conditions of the Bogomolny transfer operators;
•Motivations:
the defintion(s) of the Perron-Frobenius operator(s) of the Gauss map
are still under study;
D.H. Mayer, On the thermodynamic formalism for the Gauss map, Commun. Math. Phys. 130, 311 (1990).
available only ’suitable operator-valued power series’ on opportune
’dynamicalζfunctions’;
S. Isola, On the spectrum of Farey and Gauss maps, Nonlinearity 15, 1521 (2002).
their application for Maass waveforms was not ensured yet.
•phase space of the Anosov systems and the Gauss map;
•Anosov systems and geodesics;
•the Hitchin systems and the Hecke operators
•The Hitchin elements and the tiling,
Weyl chambers;
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
•from automorphisms on trees to Hecke theory on Maass eigenforms;
•the (Farey-)Pell construction of irrationals from Farey trees: the
quadratic-fields formalism,
the graphs of the Gauss map;
•Self-adjoint-ness conditions of the Bogomolny transfer operators;
•Motivations:
the defintion(s) of the Perron-Frobenius operator(s) of the Gauss map
are still under study;
D.H. Mayer, On the thermodynamic formalism for the Gauss map, Commun. Math. Phys. 130, 311 (1990).
available only ’suitable operator-valued power series’ on opportune
’dynamicalζfunctions’;
S. Isola, On the spectrum of Farey and Gauss maps, Nonlinearity 15, 1521 (2002).
their application for Maass waveforms was not ensured yet.
•phase space of the Anosov systems and the Gauss map;
•Anosov systems and geodesics;
•the Hitchin systems and the Hecke operators
•The Hitchin elements and the tiling,
Weyl chambers;
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
•The volume-preserving property (with Anosov flow);
•More general results from the proof of the Verjovsky Conjecture.
•the Bogomolny transfer operators;
•About the solution of the Bogomolny-Fredholm integral equation;
•The choice of the one-dimensional (Poincar´e) surface;
•more about the Bogomolny Transfer Operators;
•the Margulis measure;
•The need for the Bogmolny transfer operators (with Hecke
theory)
Motivation: for application on Maass eigenforms of the Hamiltonians;
•Asymptotic estimates on lengths of closed geodesics;
•Some theorems about measure spaces of the desymmetrised
systems fromPSL(2,Z)and of those from its congruence
subgroups;
•Remarks;
•Some theorems to prove the existence of scars of the
desymmetrised systems fromPSL(2,Z)and from its congruence
subgroups (also with algebra);
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
•More general results from the proof of the Verjovsky Conjecture.
•the Bogomolny transfer operators;
•About the solution of the Bogomolny-Fredholm integral equation;
•The choice of the one-dimensional (Poincar´e) surface;
•more about the Bogomolny Transfer Operators;
•the Margulis measure;
•The need for the Bogmolny transfer operators (with Hecke
theory)
Motivation: for application on Maass eigenforms of the Hamiltonians;
•Asymptotic estimates on lengths of closed geodesics;
•Some theorems about measure spaces of the desymmetrised
systems fromPSL(2,Z)and of those from its congruence
subgroups;
•Remarks;
•Some theorems to prove the existence of scars of the
desymmetrised systems fromPSL(2,Z)and from its congruence
subgroups (also with algebra);
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
•Prospective studies:
1 The Ljapunov exponent of scarred systems;
2 Homology classes and topological entropy;
3 Homology of closed geodesics: need for cohomology;
4 More about automorphic forms;
5 Topological entropy of geodesics flows;
6 Cohomology techniques;
7 Farrell-Ontaneda(-related) methods;
8 Markov approximation;
9 The Isola map.
•Theorem.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
1 The Ljapunov exponent of scarred systems;
2 Homology classes and topological entropy;
3 Homology of closed geodesics: need for cohomology;
4 More about automorphic forms;
5 Topological entropy of geodesics flows;
6 Cohomology techniques;
7 Farrell-Ontaneda(-related) methods;
8 Markov approximation;
9 The Isola map.
•Theorem.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Introduction
Regular trees are defined from the Cayley graphs of free groups; they can
be also derived from Tits buildings constructions. Due to the rigidity
properties of the Laplace-Beltrami operator on the UPHP, the free groups
to be considered is the diffeomorphism group.
Periodic orbits of the desymmetrisedPSL(2,Z) group and those of its
congruence subgroups Γn(PSL(2,Z)) are irrationals which can be
represented after continued-fraction decomposition, which contains an
infinite succession of Farey sequences, which, on their turn, correspond to
closed geodesics.
In the present work, the main steps of the derivation of the action of
automorphisms for trees is followed after the definition of the Ichar zeta
function by means of the Hecke operators; the pertinent considerations
are newly developped about the desymmetrizedPSL(2,Z). The
derivation of the properties of the eigenfunctions of the Ichar zeta
function is obtained after following the properties of the Hecke operators.
The new analysis is aimed at analysing the action of the Hecke operators
on the Maass waveforms and the polynomial behaviour of the Maass
waveforms.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Regular trees are defined from the Cayley graphs of free groups; they can
be also derived from Tits buildings constructions. Due to the rigidity
properties of the Laplace-Beltrami operator on the UPHP, the free groups
to be considered is the diffeomorphism group.
Periodic orbits of the desymmetrisedPSL(2,Z) group and those of its
congruence subgroups Γn(PSL(2,Z)) are irrationals which can be
represented after continued-fraction decomposition, which contains an
infinite succession of Farey sequences, which, on their turn, correspond to
closed geodesics.
In the present work, the main steps of the derivation of the action of
automorphisms for trees is followed after the definition of the Ichar zeta
function by means of the Hecke operators; the pertinent considerations
are newly developped about the desymmetrizedPSL(2,Z). The
derivation of the properties of the eigenfunctions of the Ichar zeta
function is obtained after following the properties of the Hecke operators.
The new analysis is aimed at analysing the action of the Hecke operators
on the Maass waveforms and the polynomial behaviour of the Maass
waveforms.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The definition of Pell-Farey approximation is here newly adapted to the
new Farey-Pell trees in connection with their continued-fraction Gauss
decomposition; accordingly, it is possible to define the periodic orbits of
the billiard systems.
The construction of the continued-fraction decompositions of the periodic
orbits from the quadratic fields is therefore obtained, for the purpose of
making the trees automorphisms act on them. The action of the Hecke
operators on the Maass waveforms is compared the action of the U flows
on the probability space defined after the Anosov flow in the space whose
measure is defined from that of the probability space induced after the
Anosov flow. The Margulis measure acquires a multiplicative constant
under the action of U flows on quadratic fields. The space on which the
eigenfunctions live in one defined after the measure of the probability
space, of which the measure is one induced after eh Anosov flow.
The phenomenon of scars is therefore explained as due to the action of
the U flows on the quadratic fields form which the periodic orbits are
constructed after the Farey-Pell construction.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
new Farey-Pell trees in connection with their continued-fraction Gauss
decomposition; accordingly, it is possible to define the periodic orbits of
the billiard systems.
The construction of the continued-fraction decompositions of the periodic
orbits from the quadratic fields is therefore obtained, for the purpose of
making the trees automorphisms act on them. The action of the Hecke
operators on the Maass waveforms is compared the action of the U flows
on the probability space defined after the Anosov flow in the space whose
measure is defined from that of the probability space induced after the
Anosov flow. The Margulis measure acquires a multiplicative constant
under the action of U flows on quadratic fields. The space on which the
eigenfunctions live in one defined after the measure of the probability
space, of which the measure is one induced after eh Anosov flow.
The phenomenon of scars is therefore explained as due to the action of
the U flows on the quadratic fields form which the periodic orbits are
constructed after the Farey-Pell construction.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Complements of the Hecke theory: starting from
automorphisms on trees
The main steps in the derivation of the procedures are here followed to
study the action of automorphisms on trees and therefore the action of
Hekce operators on Mass waveforms.
The UPHP (h) is considered.
Letχbe a finite-dimension unitary representation of a group Γ, i.e.
χ: Γ→Vn(C); (1)
letVχbe the representation space, and letnbe s.t.dimC=n.
The space is defined as
Hχ(h) =L
2
Γ(h,Vχ), (2)
consisting of (Γ, χ).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
automorphisms on trees
The main steps in the derivation of the procedures are here followed to
study the action of automorphisms on trees and therefore the action of
Hekce operators on Mass waveforms.
The UPHP (h) is considered.
Letχbe a finite-dimension unitary representation of a group Γ, i.e.
χ: Γ→Vn(C); (1)
letVχbe the representation space, and letnbe s.t.dimC=n.
The space is defined as
Hχ(h) =L
2
Γ(h,Vχ), (2)
consisting of (Γ, χ).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The automorphic functionfsuch thatf:h→Vχbe one of finite norm,
where the latter is defined as
<f,g>=
Z
Γ/h
<f(z),g(z)>Vχ
dµ(z) (3)
∀(Γ, χ) automorphic functionsfandg∈L
2
Γ
(h,Vχ), with
dµ=
dxdy
y
2
; (4)
the scalar product inVχis thus defined.
The following equality holds
f(γz) =χ(γ)f(z) (5)
∀γ∈Γ,∀z∈h.
The Laplace-Beltrami operator acquires the meaning
∆χf+λf= 0, (6)
withf∈Hχ(h).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
where the latter is defined as
<f,g>=
Z
Γ/h
<f(z),g(z)>Vχ
dµ(z) (3)
∀(Γ, χ) automorphic functionsfandg∈L
2
Γ
(h,Vχ), with
dµ=
dxdy
y
2
; (4)
the scalar product inVχis thus defined.
The following equality holds
f(γz) =χ(γ)f(z) (5)
∀γ∈Γ,∀z∈h.
The Laplace-Beltrami operator acquires the meaning
∆χf+λf= 0, (6)
withf∈Hχ(h).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
LetVΓthe set of conjugacy classes which are primitive in a group Γ, with
Γ̸=I.
For any arbitraryg∈PSL(2,R), let
N(g) =e
l(g)
, (7)
withl(g) =minz∈hd(z,g(z)), withd(z,z
′
) the hyperbolic distance inh.
The Sel’berg zeta functionZSis defined as
ZS(Γ,s;χ) = Π
{P}∈VΓ
Π
k=∞
k=0det[IVχ
−χ(P)N(P)
−s−k
], (8)
withRe(s)>1,s∈C.Pis the conjugacy subclasses of prime geodesics
andN(p) the length.
One looks for the meromorphic continuation
d
ds
logZS(Γ,s;χ). (9)
One takes the Riemann hypothesis.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Γ̸=I.
For any arbitraryg∈PSL(2,R), let
N(g) =e
l(g)
, (7)
withl(g) =minz∈hd(z,g(z)), withd(z,z
′
) the hyperbolic distance inh.
The Sel’berg zeta functionZSis defined as
ZS(Γ,s;χ) = Π
{P}∈VΓ
Π
k=∞
k=0det[IVχ
−χ(P)N(P)
−s−k
], (8)
withRe(s)>1,s∈C.Pis the conjugacy subclasses of prime geodesics
andN(p) the length.
One looks for the meromorphic continuation
d
ds
logZS(Γ,s;χ). (9)
One takes the Riemann hypothesis.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The group of automorphisms on a tree
BeXa tree a set of its verticesSomX(’sommets’) and its edgesArX
(are’tes), i.e. such that
X= (SomX,ArX); (10)
it defines an infinite tree, locally finite, with a non-empty graph without
cycles.
Be Γ the group of automorphisms on the treeX, which acts freely on the
treeXwith a finite factorizable graph Γ/X: according to the Shraier
Theorem, it is the free group.
Beχ: Γ→un(C) finite unitary representation of the group Γ.
BeVχthe pertinent space.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
BeXa tree a set of its verticesSomX(’sommets’) and its edgesArX
(are’tes), i.e. such that
X= (SomX,ArX); (10)
it defines an infinite tree, locally finite, with a non-empty graph without
cycles.
Be Γ the group of automorphisms on the treeX, which acts freely on the
treeXwith a finite factorizable graph Γ/X: according to the Shraier
Theorem, it is the free group.
Beχ: Γ→un(C) finite unitary representation of the group Γ.
BeVχthe pertinent space.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The Sel’berg trace formula for (Γ, χ)
The Sel’berg trace formula for (Γ, χ) consists of functions of the automorphisms on
the treeX.
Be the algebraH(X,C) the algebra of the Hecke operators on the treeX.
By definition,k∈ H(X,C) is
K=
m=∞
X
m=0
k(m)Tm, (11)
withk(m)∈C,m= 0,1,2, ...the roots.
Then, one has that
m=∞
X
m=0
|k(m)|(q+ 1)q
m−1
<∞, (12)
withTmthe invariant operators automorphisms onX
(Tmϕ)(v)
X
v
′
∈SomX,d(v,v
′
)=m
ϕ(v
′
) (13)
∀q∈ L∞(X,V), being a normed vectorial space, where the norm is defined as
||ϕ||∞=supv∈SomX|ϕ(v)|. (14)
The spaceL∞(X,V) is the natural space for the operators that act fromH(X,C).
From the spectral theory, one has the action of the operators fromH(X,C) from the
Hilbert spaceL2(X,V). By definition, the Hilbert spaceL2(X,V) consists of all the
functionsϕ:SomX→V.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The Sel’berg trace formula for (Γ, χ) consists of functions of the automorphisms on
the treeX.
Be the algebraH(X,C) the algebra of the Hecke operators on the treeX.
By definition,k∈ H(X,C) is
K=
m=∞
X
m=0
k(m)Tm, (11)
withk(m)∈C,m= 0,1,2, ...the roots.
Then, one has that
m=∞
X
m=0
|k(m)|(q+ 1)q
m−1
<∞, (12)
withTmthe invariant operators automorphisms onX
(Tmϕ)(v)
X
v
′
∈SomX,d(v,v
′
)=m
ϕ(v
′
) (13)
∀q∈ L∞(X,V), being a normed vectorial space, where the norm is defined as
||ϕ||∞=supv∈SomX|ϕ(v)|. (14)
The spaceL∞(X,V) is the natural space for the operators that act fromH(X,C).
From the spectral theory, one has the action of the operators fromH(X,C) from the
Hilbert spaceL2(X,V). By definition, the Hilbert spaceL2(X,V) consists of all the
functionsϕ:SomX→V.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The functions enjoy the property
||ϕ2||={
X
v∈SomX
< ϕ(v), ϕ(u)>V}
1/2
<∞, (15)
where< , >Vis the Hermitian scalar product in the spaceV.
The scalar product is written also as
(ϕ, ψ) =
X
v∈SomX
< ϕ(v), ψ(v)>V. (16)
The norm ofTmfrom Eq. (11) is bounded as
||Tm||≤(q+ 1)q
m−1
,m= 1,2, ... (17)
This means that one considers only operators bounded inL∞(X,V).
It is verified that the multiplication (i.e. composition) rule of the operators{Tm}
m=∞
m=0
is written as
T
2
1
=T2+ (q+ 1)T0, (18a)
T1Tm=Tm+1+ (q)Tm−1,m≥2 : (18b)
the algebraH(X,C) is commutative.
Any element fromH(X,C)is a polynomial.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
||ϕ2||={
X
v∈SomX
< ϕ(v), ϕ(u)>V}
1/2
<∞, (15)
where< , >Vis the Hermitian scalar product in the spaceV.
The scalar product is written also as
(ϕ, ψ) =
X
v∈SomX
< ϕ(v), ψ(v)>V. (16)
The norm ofTmfrom Eq. (11) is bounded as
||Tm||≤(q+ 1)q
m−1
,m= 1,2, ... (17)
This means that one considers only operators bounded inL∞(X,V).
It is verified that the multiplication (i.e. composition) rule of the operators{Tm}
m=∞
m=0
is written as
T
2
1
=T2+ (q+ 1)T0, (18a)
T1Tm=Tm+1+ (q)Tm−1,m≥2 : (18b)
the algebraH(X,C) is commutative.
Any element fromH(X,C)is a polynomial.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The Ihara zeta function
The Ihara zeta function writes
ZX(u) = Π
[C](1−u
ν(C)
)
−1
, (19)
for which the following specifications are needed. BeCa closed path; BeCthe
sequence of verticesC=υ1, υ2, ..., υη=υ1; beν(C) the length ofC=η−1 number
of edges inC.
It is interesting to notice that, from the number of vertices and from that of edges, it
is possible to calculate the Euler characteristic ofX. The Ihara zeta function Eq. (19)
is the product over eigenvalue classes of primitive closed back-track-less geodesics
tail-less cycles of positive lengthηinX, withν(C) the length ofC, where the latter is
equivalent to the number of edges inC.
Thefundamental group for graphsGdefines homotopy classes of cycles inX
beginning and ending at the pointρunder the product obtained from the composition
(of the generators) of paths. The fundamental group for graphs is
G=π1(X, ρ) (20)
ofXfor a pointρinX.
A tree is defined as a connected graph without cycles.
A spanning tree is here defined as a tree which is a subgraph ofXand which includes
every vertex ofX.
To a closed pathCstarting and ending atρthere corresponds a conjugacy class{C}
inG;Cis the homotopy class (Chas no backtracking).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The Ihara zeta function writes
ZX(u) = Π
[C](1−u
ν(C)
)
−1
, (19)
for which the following specifications are needed. BeCa closed path; BeCthe
sequence of verticesC=υ1, υ2, ..., υη=υ1; beν(C) the length ofC=η−1 number
of edges inC.
It is interesting to notice that, from the number of vertices and from that of edges, it
is possible to calculate the Euler characteristic ofX. The Ihara zeta function Eq. (19)
is the product over eigenvalue classes of primitive closed back-track-less geodesics
tail-less cycles of positive lengthηinX, withν(C) the length ofC, where the latter is
equivalent to the number of edges inC.
Thefundamental group for graphsGdefines homotopy classes of cycles inX
beginning and ending at the pointρunder the product obtained from the composition
(of the generators) of paths. The fundamental group for graphs is
G=π1(X, ρ) (20)
ofXfor a pointρinX.
A tree is defined as a connected graph without cycles.
A spanning tree is here defined as a tree which is a subgraph ofXand which includes
every vertex ofX.
To a closed pathCstarting and ending atρthere corresponds a conjugacy class{C}
inG;Cis the homotopy class (Chas no backtracking).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
There exists a one-to-one correspondence between conjugacy subclasses{C}inGand
equivalence classes of backtrack-less tail-less cycles [C] inX. The one-to-one
correspondence is explained as the fact that the elements of the equivalence class of
C
∗
are the closed cycles of minimal length freely homotopic toC(i.e. for which the
base point is not fixed).
There exists a universal covering tree˜X: it is regular iffXis regular; there exists an
action of the fundamental groupGon˜Xsuch that˜X/G
∼
=X.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
equivalence classes of backtrack-less tail-less cycles [C] inX. The one-to-one
correspondence is explained as the fact that the elements of the equivalence class of
C
∗
are the closed cycles of minimal length freely homotopic toC(i.e. for which the
base point is not fixed).
There exists a universal covering tree˜X: it is regular iffXis regular; there exists an
action of the fundamental groupGon˜Xsuch that˜X/G
∼
=X.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
It is now possible to further comment on Eq. (19). It is the product over equivalence
classes of primitive closed backtrack-less tail-less cyclesC= (υ1, υ2, ..., υη=υ1) of
positive lengthη∈ X;ν(C) =η−1 which equals the length ofC, and also the
number of edges inC.
It is therefore possible to state that
ZX(u) =
X
{C}primitiveconjugacyclassesinG=π1(X,ρ);C̸=
ˆ
I
(1−u
ν(C)
)
−1
, (21)
whereν(C) is now the length of an elementC
∗
ib the equivalence class of
backtrack-less tail-less paths corresponding to{C}.
In other words,ν(C) is the minimal length of all cycles freely homotopic toC.
From
Y. Ihara, On discrete subgroups of the two by two projective linear group over p-adic fields, Journal of the
Mathematical Society of Japan, 18, 219-235 (1966), Theorem 1.
Theorem
letXbe a connected (ρ+ 1) regular graph with adjacent matrixA, andrthe rank of
the fundamental group:the Ihara zeta function after Eq. (19) is the reciprocal of a
polynomialas
Z
−1
X
(u)
r−1
det[
ˆ
I−Au+ρu
2ˆ
I]. (22)
The analogy with the Riemann hypothesis is taken.
A. B. Venkov, A. M. Nikitin, [Formula of the trace of Sel’berg’s, graphs of Ramanujan’s, and some problems in
mathematical Physics, Algebra and Analysis 5, 3 (1993)] St. Petersburg Math. J. 5, 419 (1994).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
classes of primitive closed backtrack-less tail-less cyclesC= (υ1, υ2, ..., υη=υ1) of
positive lengthη∈ X;ν(C) =η−1 which equals the length ofC, and also the
number of edges inC.
It is therefore possible to state that
ZX(u) =
X
{C}primitiveconjugacyclassesinG=π1(X,ρ);C̸=
ˆ
I
(1−u
ν(C)
)
−1
, (21)
whereν(C) is now the length of an elementC
∗
ib the equivalence class of
backtrack-less tail-less paths corresponding to{C}.
In other words,ν(C) is the minimal length of all cycles freely homotopic toC.
From
Y. Ihara, On discrete subgroups of the two by two projective linear group over p-adic fields, Journal of the
Mathematical Society of Japan, 18, 219-235 (1966), Theorem 1.
Theorem
letXbe a connected (ρ+ 1) regular graph with adjacent matrixA, andrthe rank of
the fundamental group:the Ihara zeta function after Eq. (19) is the reciprocal of a
polynomialas
Z
−1
X
(u)
r−1
det[
ˆ
I−Au+ρu
2ˆ
I]. (22)
The analogy with the Riemann hypothesis is taken.
A. B. Venkov, A. M. Nikitin, [Formula of the trace of Sel’berg’s, graphs of Ramanujan’s, and some problems in
mathematical Physics, Algebra and Analysis 5, 3 (1993)] St. Petersburg Math. J. 5, 419 (1994).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Construction of periodic orbits from (Farey) sequences
•Periodic orbits are intended to be studied after the approximation of
irrationals:
H. Cohen, X. F. Roblot, Computing the Hilbert Class Field of Real Quadratic Fields, Mathematics of Computation
69, 1229 (2000).
•the irrationals to be approximated are those constituted of an infinite
repetition of Farey sequences, i.e. as envisaged from
I. Akkus, Nurettin Irmak, G. Kizilaslan, Farey-Pell squence, approximation to inrrationals and Hurwitz’s inequality,
Bulletin of Mathematical Analysis and Applications 8, 11 (2016);
J. L. Sikorav, Best rational approximations of an irrational number, arXiv:1807.06284 [math.NT].
•Ideals for continued fractions are defined
M. J. Jacobson Jr., H. C. Williams, Ideals and Continued Fractions, in: Solving the Pell Equation, CMS Books in
Mathematics, Springer, New York, USA (2008).
•The relations between the Gauss map and continued fractions was
studied:
M. Bunder, K. Tognettti, Conitnued fractions and the Gauss map, Acta Mathematica Academiae Paedagogicae
Nyi’regyhi’aziensis 21, 113 (2005).
the main passages of the derivation are here followed. In particular, the
graph of an iterate over [0,1/2] is symmetric to the graph of the next
higher iterate over 1/2,1.
TheProofis given into two parts.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
•Periodic orbits are intended to be studied after the approximation of
irrationals:
H. Cohen, X. F. Roblot, Computing the Hilbert Class Field of Real Quadratic Fields, Mathematics of Computation
69, 1229 (2000).
•the irrationals to be approximated are those constituted of an infinite
repetition of Farey sequences, i.e. as envisaged from
I. Akkus, Nurettin Irmak, G. Kizilaslan, Farey-Pell squence, approximation to inrrationals and Hurwitz’s inequality,
Bulletin of Mathematical Analysis and Applications 8, 11 (2016);
J. L. Sikorav, Best rational approximations of an irrational number, arXiv:1807.06284 [math.NT].
•Ideals for continued fractions are defined
M. J. Jacobson Jr., H. C. Williams, Ideals and Continued Fractions, in: Solving the Pell Equation, CMS Books in
Mathematics, Springer, New York, USA (2008).
•The relations between the Gauss map and continued fractions was
studied:
M. Bunder, K. Tognettti, Conitnued fractions and the Gauss map, Acta Mathematica Academiae Paedagogicae
Nyi’regyhi’aziensis 21, 113 (2005).
the main passages of the derivation are here followed. In particular, the
graph of an iterate over [0,1/2] is symmetric to the graph of the next
higher iterate over 1/2,1.
TheProofis given into two parts.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
As afirst step, one studies the discontinuities of the Gauss map on a continued
fraction, as one can outline after posing the Gauss mapGonxasG(x), and the
continued fraction ofx, wherexis defined as
x∈(0,1] ={0,a1,a2, ...} (23)
, and noticing that, for
1
a1+ 1
<x≤
1
a1
, (24)
one has that
lim
x→
1
a
1
+1
−G(x) = 0 (25)
and
lim
x→
1
a
1
+1
+G(x) = 1, (26)
from which all the discontinuities are outlined.
Forx, then−thtotal convergentC
n
is defined as
Cn≡
pn
qn
≡ {0,a1,a2, ...,an} (27)
where
pn
qn
is an irreducible fraction.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
fraction, as one can outline after posing the Gauss mapGonxasG(x), and the
continued fraction ofx, wherexis defined as
x∈(0,1] ={0,a1,a2, ...} (23)
, and noticing that, for
1
a1+ 1
<x≤
1
a1
, (24)
one has that
lim
x→
1
a
1
+1
−G(x) = 0 (25)
and
lim
x→
1
a
1
+1
+G(x) = 1, (26)
from which all the discontinuities are outlined.
Forx, then−thtotal convergentC
n
is defined as
Cn≡
pn
qn
≡ {0,a1,a2, ...,an} (27)
where
pn
qn
is an irreducible fraction.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The study from
B.Bates, M. Bunder, K. Tognettti, Conitnued fractions and the Gauss map, Acta Mathematica Academiae
Paedagogicae Nyi’regyhi’aziensis 21, 113 (2005).
accords with the study of
J. L. Sikorav, Best rational approximations of an irrational number, arXiv:1807.06284 [math.NT].
The graphs of the Gauss map have therefore to be studied.
The transfer operators of the Gauss map are studied in
S. Bettin, S. Drappeau, L. Spiegelhofer, Statistical distribution of the Stern sequence, Comment. Math. Helv. 94
(2019), no. 2, pp. 241–271.
The study of ’Best rational approximations of an irrational number’ is focused on the
research of the best approximations of an irrational number by rational numbers. Such
study has to be then specified with respect to the wanted definition of irrational
consisting of an infinite repetition of Farey sequences.
The degree of approximation of an irrational to a rational can be chosen within an
accuracy at leisure. By means of fractions, the best approximation of an irrational to a
rational is obtained from a fraction with the absolute value of the denominator smaller
than a given quantity. The approximation is completed after the concept of nearest
integer, and after the Dirichlet theorem.
In
H. Cohen, X. F. Roblot, Computing the Hilbert Class Field of Real Quadratic Fields, Mathematics of Computation
69, 1229 (2000).
the computation of the Hilbert Class Field of Real Quadratic Fields is recalled to be
based on theStark’s conjectures,which have not been proven yet; the problem is
usefully complemented after
J.T. Tate, Les Conjectures de Stark sur les Fonctions L d’Artin en s = 0, Progress in Math. 47, Birkhaeuser,
Boston, 1984;
R. Terras, The Determination of Incomplete Gamma Functions through Analytic Integration, J. Comput. Phys. 31,
146 (1979).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
B.Bates, M. Bunder, K. Tognettti, Conitnued fractions and the Gauss map, Acta Mathematica Academiae
Paedagogicae Nyi’regyhi’aziensis 21, 113 (2005).
accords with the study of
J. L. Sikorav, Best rational approximations of an irrational number, arXiv:1807.06284 [math.NT].
The graphs of the Gauss map have therefore to be studied.
The transfer operators of the Gauss map are studied in
S. Bettin, S. Drappeau, L. Spiegelhofer, Statistical distribution of the Stern sequence, Comment. Math. Helv. 94
(2019), no. 2, pp. 241–271.
The study of ’Best rational approximations of an irrational number’ is focused on the
research of the best approximations of an irrational number by rational numbers. Such
study has to be then specified with respect to the wanted definition of irrational
consisting of an infinite repetition of Farey sequences.
The degree of approximation of an irrational to a rational can be chosen within an
accuracy at leisure. By means of fractions, the best approximation of an irrational to a
rational is obtained from a fraction with the absolute value of the denominator smaller
than a given quantity. The approximation is completed after the concept of nearest
integer, and after the Dirichlet theorem.
In
H. Cohen, X. F. Roblot, Computing the Hilbert Class Field of Real Quadratic Fields, Mathematics of Computation
69, 1229 (2000).
the computation of the Hilbert Class Field of Real Quadratic Fields is recalled to be
based on theStark’s conjectures,which have not been proven yet; the problem is
usefully complemented after
J.T. Tate, Les Conjectures de Stark sur les Fonctions L d’Artin en s = 0, Progress in Math. 47, Birkhaeuser,
Boston, 1984;
R. Terras, The Determination of Incomplete Gamma Functions through Analytic Integration, J. Comput. Phys. 31,
146 (1979).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
As asecond step, it is therefore necessary to approach the issue after reviewing the
relationships amongreal quadratic fields, the class numbers of quadratic fields, and
the continued fraction expansion of the related ideals
S. Louboutin, R.A. Mollin, H.C. Williams, Class numbers of real quadratic fields, continued fractions, reduced
ideals, prime-producing quadratic polynomials and quadratic residue covers, Canadian Journal of Mathematics 44,
824 (1992).
Indeed, the ideals of the quadratic fields have to be analysed with respect to thePell
primes.
In
I. Akkus, N. Irmak, G. Kizilaslan, Farey-Pell sequence, approximation to irrationals and Hurwitz’s inequality,
Bulletin of Mathematical Analysis and Applications, 8, 11 (2016).
Farey-Pell sequences and Pell numbers are defined; furthermore, the approximation to
irrationals is given.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
relationships amongreal quadratic fields, the class numbers of quadratic fields, and
the continued fraction expansion of the related ideals
S. Louboutin, R.A. Mollin, H.C. Williams, Class numbers of real quadratic fields, continued fractions, reduced
ideals, prime-producing quadratic polynomials and quadratic residue covers, Canadian Journal of Mathematics 44,
824 (1992).
Indeed, the ideals of the quadratic fields have to be analysed with respect to thePell
primes.
In
I. Akkus, N. Irmak, G. Kizilaslan, Farey-Pell sequence, approximation to irrationals and Hurwitz’s inequality,
Bulletin of Mathematical Analysis and Applications, 8, 11 (2016).
Farey-Pell sequences and Pell numbers are defined; furthermore, the approximation to
irrationals is given.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
ThePell sequenceis defined after the relation
Pn= 2Pn−1+Pn−2,n≥2, (28)
withP0= 0,P1= 1; the corresponding Binet formula reads
Pn=
α
n
−β
n
α−β
, (29)
withαandβbeing the solutions of the characteristic equationx
2
−2x−1 = 0.
A Farey-Pell sequence of orderPnis defined as the set of all possible fractionsPi/Pj,
order as 0<Pi<Pj<Pn.
A Farey-Pell sequence of orderPnis a set of functionsPi/Pj: the sequence is denoted
asFPn. Ther−thofFPnis denoted asFP
(r)n.
FP
(r)nis defined as a point of symmetry ifFP
(r−1)nandFP
(r+2)nshare the same
denominator.
Therefore,FP
(r+k+1)nandFP
(r−k)nshare the same denominatoriffone of them is a
point of symmetry.
Theapproximation of a rational to an irrationalis performed as follows.
LetFPn,1be the ordered set consistent of all rationals inFPnwith the mediant of
consecutive rationals inFPn.
FPn,s+1consists of all the rationals inFPn,swhere the latter have mediant of
consecutive rationals; thus
PPn=
s=∞
[
s=1
FPn,s (30)
is dense in the interval [0,∞).
Therefore,every irrational can be approximated by a sequence of rationals inPPn,1.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Pn= 2Pn−1+Pn−2,n≥2, (28)
withP0= 0,P1= 1; the corresponding Binet formula reads
Pn=
α
n
−β
n
α−β
, (29)
withαandβbeing the solutions of the characteristic equationx
2
−2x−1 = 0.
A Farey-Pell sequence of orderPnis defined as the set of all possible fractionsPi/Pj,
order as 0<Pi<Pj<Pn.
A Farey-Pell sequence of orderPnis a set of functionsPi/Pj: the sequence is denoted
asFPn. Ther−thofFPnis denoted asFP
(r)n.
FP
(r)nis defined as a point of symmetry ifFP
(r−1)nandFP
(r+2)nshare the same
denominator.
Therefore,FP
(r+k+1)nandFP
(r−k)nshare the same denominatoriffone of them is a
point of symmetry.
Theapproximation of a rational to an irrationalis performed as follows.
LetFPn,1be the ordered set consistent of all rationals inFPnwith the mediant of
consecutive rationals inFPn.
FPn,s+1consists of all the rationals inFPn,swhere the latter have mediant of
consecutive rationals; thus
PPn=
s=∞
[
s=1
FPn,s (30)
is dense in the interval [0,∞).
Therefore,every irrational can be approximated by a sequence of rationals inPPn,1.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Self-adjoint-ness conditions of the Bogomolny
transfer operators
The statistical properties in the complete phase space can be
recovered from the return map defined by the Anosov-flow
properties on the reduced Birkhoff surface of section.
OML, Reduced 2-dimensional Birkhoff surfaces of section of the desymmetrized
PSL(2, Z) group: the Anosov characterization, International Journal Of Mathematics
And Computer Research- IJMCR 11, 3255 (2023).
TheoremIn particular, the paradigm is obtained by reconstructing
the complete phase space and extended Bogomolny map on the
reduced 2-dimensional Birkhoff surface, where the Bogomolny
transfer operators are straightforward proved to be self-adjoint.
ProofThe Bogomolny transfer operators commute with the
Laplacian, which has rigidity properties on the Upper Poincar´e half
Plane.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
transfer operators
The statistical properties in the complete phase space can be
recovered from the return map defined by the Anosov-flow
properties on the reduced Birkhoff surface of section.
OML, Reduced 2-dimensional Birkhoff surfaces of section of the desymmetrized
PSL(2, Z) group: the Anosov characterization, International Journal Of Mathematics
And Computer Research- IJMCR 11, 3255 (2023).
TheoremIn particular, the paradigm is obtained by reconstructing
the complete phase space and extended Bogomolny map on the
reduced 2-dimensional Birkhoff surface, where the Bogomolny
transfer operators are straightforward proved to be self-adjoint.
ProofThe Bogomolny transfer operators commute with the
Laplacian, which has rigidity properties on the Upper Poincar´e half
Plane.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Phase space of the Anosov systems
In
G. A. Noskov, Bounded shortening in Coxeter complexes and buildings. Mathematical structures and modelling 8,
10 (2001).
the symbolic dynamics is codified as the ’word ordering’ (where the latter
is the allowed compostions of generators) of the geometrically finite
hyperbolic group, as explained in
E. Bogomolny, N. Cairoli, Giannoni, Schmit, Arithmetical chaos, Phys.Rept. 291, 219 (1997).
for complexes (i.e. ’sharp sides); the desymmetrizedPSL(2,Z) group can
be considered for these purposes: the conjugacy subclasses can be
considered also for the congruence subgroups, provided that the proper
hyperbolic reflections (Kasner transformations) are inserted within the
composition of generators.
Theorem:The properties of the Gauss map in the desymmetrised
PSL(2,Z) allow one to further classify the (i.e. also closed) geodesics.
Proof:By applying the Gauss map to the continued-fraction
decomposition of the geodesics on the UPHP.
Discussion:The role of the entries of the continued-fraction
decompositions on the Birkhoff surface should be understood after the
further study of the Anosov properties.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
In
G. A. Noskov, Bounded shortening in Coxeter complexes and buildings. Mathematical structures and modelling 8,
10 (2001).
the symbolic dynamics is codified as the ’word ordering’ (where the latter
is the allowed compostions of generators) of the geometrically finite
hyperbolic group, as explained in
E. Bogomolny, N. Cairoli, Giannoni, Schmit, Arithmetical chaos, Phys.Rept. 291, 219 (1997).
for complexes (i.e. ’sharp sides); the desymmetrizedPSL(2,Z) group can
be considered for these purposes: the conjugacy subclasses can be
considered also for the congruence subgroups, provided that the proper
hyperbolic reflections (Kasner transformations) are inserted within the
composition of generators.
Theorem:The properties of the Gauss map in the desymmetrised
PSL(2,Z) allow one to further classify the (i.e. also closed) geodesics.
Proof:By applying the Gauss map to the continued-fraction
decomposition of the geodesics on the UPHP.
Discussion:The role of the entries of the continued-fraction
decompositions on the Birkhoff surface should be understood after the
further study of the Anosov properties.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Anosov systems and geodesics
The properties of the (normalised-velocity) geodesics flows which
define particular surfaces in (some definitions of) phase space (or
’reduced phase space’) are connected with those of the Anosov
systems.
A. Zeghib, Subsystems of Anosov Systems, American Journal of Mathematics 117,
1431 (1995).
The definition of Anosov representation the Lie groupsSL(n,R) is
provided in
R. Canary, K. Tsouvalas, Topological restrictions on Anosov representations, e-print
math arXiv:1904.02002.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The properties of the (normalised-velocity) geodesics flows which
define particular surfaces in (some definitions of) phase space (or
’reduced phase space’) are connected with those of the Anosov
systems.
A. Zeghib, Subsystems of Anosov Systems, American Journal of Mathematics 117,
1431 (1995).
The definition of Anosov representation the Lie groupsSL(n,R) is
provided in
R. Canary, K. Tsouvalas, Topological restrictions on Anosov representations, e-print
math arXiv:1904.02002.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The symbolic dynamics is codified as the ’word ordering’ (where
the latter is the allowed compositions of generators) of the
geometrically finite hyperbolic group;
G. A. Noskov, Bounded shortening in Coxeter complexes and buildings.
Matematicheskie Struktury i Modelirovanie 8, 10 (2001).
for complexes (i.e. ’sharp sides’), see:
E. Bogomolny, N. Cairoli, Quantum maps for transfer operators, Physica D: Nonlinear
Phenomena 67, 88 (1993).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
the latter is the allowed compositions of generators) of the
geometrically finite hyperbolic group;
G. A. Noskov, Bounded shortening in Coxeter complexes and buildings.
Matematicheskie Struktury i Modelirovanie 8, 10 (2001).
for complexes (i.e. ’sharp sides’), see:
E. Bogomolny, N. Cairoli, Quantum maps for transfer operators, Physica D: Nonlinear
Phenomena 67, 88 (1993).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Theorem
The desymmetrized modular group can be considered for these
purposes: the conjugacy subclasses can be considered also for the
congruence subgroups, provided that the proper hyperbolic
reflections are inserted within the composition of generators.
Proof
By construction.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The desymmetrized modular group can be considered for these
purposes: the conjugacy subclasses can be considered also for the
congruence subgroups, provided that the proper hyperbolic
reflections are inserted within the composition of generators.
Proof
By construction.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The Hitching systems and the Hecke operators
∃symplectic map between the Hitchin systems (correlated with some
holomorphic bundles).
These maps define a ’symplectic Hecke correspondence’.
The maps are proven to be constructed by means of the Hecke
correspondence of the underlying holomorphic bundles.
The symplectic Hecke correspondence SHC provides one with the
construction of the Baecklund transformations in the Hitchin systems
wchich are set over ’Riemann curves with marked points’.
A.M. Levin, M.A. Olshanetsky, A. Zotov, Hitchin Systems - Symplectic Hecke Correspondence and
Two-dimensional Version, Commun. Math. Phys. 236, 93 (2003).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
∃symplectic map between the Hitchin systems (correlated with some
holomorphic bundles).
These maps define a ’symplectic Hecke correspondence’.
The maps are proven to be constructed by means of the Hecke
correspondence of the underlying holomorphic bundles.
The symplectic Hecke correspondence SHC provides one with the
construction of the Baecklund transformations in the Hitchin systems
wchich are set over ’Riemann curves with marked points’.
A.M. Levin, M.A. Olshanetsky, A. Zotov, Hitchin Systems - Symplectic Hecke Correspondence and
Two-dimensional Version, Commun. Math. Phys. 236, 93 (2003).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The Hitchin elements and the tiling
The Hitchin elements and the Coxeter functors
More in general, for polygon groups, the Hitchin scheme is spelled out in
G.-S. Lee, L. Marquis, Discrete Coxeter groups, available on
https://perso.univrennes1.fr/ludovic.marquis/pdf/SurveyDiscreteCoxGp.pdf.
Periodic orbits can be understood as generated form an algebra of
idempotent operators (where the definition extends to the phase space
trivially as the geodesics velocity is constant and normalised,i.e. such
that the definition of the reduced Birkhoff surface directly extends from
that of the Poincar´e surface for the return maps of the periodic orbits
generated after the idempotent elements): the corresponding Coxeter
functors are defined in
S. A. Kruglyak, On Coxeter functors for some classes of algebra generated by
idempotents, Ukrainian Mathematical Journal 56, 1189 (2004).
Non-stationarity of compoisiton of mapsThe non-stationary
composition of chaotic maps is related to non-stationary compositions of
Anosov diffeomorphisms in
M. Stenlund, Non-stationary compositions of Anosov diffeomorphisms, Nonlinearity 24, 2991 (2011).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The Hitchin elements and the Coxeter functors
More in general, for polygon groups, the Hitchin scheme is spelled out in
G.-S. Lee, L. Marquis, Discrete Coxeter groups, available on
https://perso.univrennes1.fr/ludovic.marquis/pdf/SurveyDiscreteCoxGp.pdf.
Periodic orbits can be understood as generated form an algebra of
idempotent operators (where the definition extends to the phase space
trivially as the geodesics velocity is constant and normalised,i.e. such
that the definition of the reduced Birkhoff surface directly extends from
that of the Poincar´e surface for the return maps of the periodic orbits
generated after the idempotent elements): the corresponding Coxeter
functors are defined in
S. A. Kruglyak, On Coxeter functors for some classes of algebra generated by
idempotents, Ukrainian Mathematical Journal 56, 1189 (2004).
Non-stationarity of compoisiton of mapsThe non-stationary
composition of chaotic maps is related to non-stationary compositions of
Anosov diffeomorphisms in
M. Stenlund, Non-stationary compositions of Anosov diffeomorphisms, Nonlinearity 24, 2991 (2011).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Anosov flow for Weyl chambers
The codification of the geodesics flow of billiards on the UPHP (as we
define for the billiards) is codified into the properties of an Anosov
flow for the Weyl chambers in
Felipe A. Ram´ırez, Invariant distributions and cohomology for geodesic flows and higher cohomology of higher-rank
Anosov actions, Journal of Functional Analysis 265, 1002 (2013).
where the Weyl chambers are represented on the UPHP as the copies of
the desymmetrised-modular-group domain
(needed to unfold the trajectories of the congruence subgroup(s)); the
related cohomology tools allow one to to evaluate the definitions available
for the topic under investigation after facilities of the results of the
GL groups for which the application to reflection groups (which construct
the Weyl chambers) is provided
in
Jeffrey Danciger, Fran cois Gu´eritaud, Fanny Kassel, Gye-Seon Lee, Ludovic Marquis, Convex cocompactness for
Coxeter groups, arXiv:2102.02757.
(as summoned in the very recent discoveries of the references therein).
Cohomology to be studied.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The codification of the geodesics flow of billiards on the UPHP (as we
define for the billiards) is codified into the properties of an Anosov
flow for the Weyl chambers in
Felipe A. Ram´ırez, Invariant distributions and cohomology for geodesic flows and higher cohomology of higher-rank
Anosov actions, Journal of Functional Analysis 265, 1002 (2013).
where the Weyl chambers are represented on the UPHP as the copies of
the desymmetrised-modular-group domain
(needed to unfold the trajectories of the congruence subgroup(s)); the
related cohomology tools allow one to to evaluate the definitions available
for the topic under investigation after facilities of the results of the
GL groups for which the application to reflection groups (which construct
the Weyl chambers) is provided
in
Jeffrey Danciger, Fran cois Gu´eritaud, Fanny Kassel, Gye-Seon Lee, Ludovic Marquis, Convex cocompactness for
Coxeter groups, arXiv:2102.02757.
(as summoned in the very recent discoveries of the references therein).
Cohomology to be studied.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The non-stationary composition of chaotic maps is related to
non-stationary
compositions of Anosov diffeomorphisms in
Mikko Stenlund, Non-stationary compositions of Anosov diffeomorphisms, Nonlinearity
24 (2011) 2991–3018.
in order to consider the statistical properties and the composition
of maps for flows.
In
N.I. Chernov, Markov approximations and decay of correlations for Anosov flows, Annals of Mathematics 147, 269
(1998).
, the Markov properties of the Poincar´e surfaces are studied as far
as far as the Anosov representation is concerned;the definition
should be moulded for the Birkhoff surfaces thereafter.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
non-stationary
compositions of Anosov diffeomorphisms in
Mikko Stenlund, Non-stationary compositions of Anosov diffeomorphisms, Nonlinearity
24 (2011) 2991–3018.
in order to consider the statistical properties and the composition
of maps for flows.
In
N.I. Chernov, Markov approximations and decay of correlations for Anosov flows, Annals of Mathematics 147, 269
(1998).
, the Markov properties of the Poincar´e surfaces are studied as far
as far as the Anosov representation is concerned;the definition
should be moulded for the Birkhoff surfaces thereafter.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
In
A. Zeghib, Subsystems of Anosov Systems, American Journal of Mathematics 117,
1431 (1995).
, the properties of the (normalised-velocity) geodesics flows which
define particular surfaces in (some definitions of) phase space (or
’reduced phase space’) are connected with those of the Anosov
systems:for which the latter definition of reduction of the
phase space should be considered.
Theorem
The surfaces connected with those of the Anosov systems are the
reduced Birkhhoff surfaces.
Proof
By construction after the rigidity of the Laplacian operators on the
UPHP.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A. Zeghib, Subsystems of Anosov Systems, American Journal of Mathematics 117,
1431 (1995).
, the properties of the (normalised-velocity) geodesics flows which
define particular surfaces in (some definitions of) phase space (or
’reduced phase space’) are connected with those of the Anosov
systems:for which the latter definition of reduction of the
phase space should be considered.
Theorem
The surfaces connected with those of the Anosov systems are the
reduced Birkhhoff surfaces.
Proof
By construction after the rigidity of the Laplacian operators on the
UPHP.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The volume-preserving property
The use of the formalism of the Anosov flow is found in the properties of
volume-preserving actions under the hypotheses that it be transitive;the
volume-preserving property is essential in the analysis of the
PSL(2,Z)billiards and to those of its congruence subgroups
M. Asaoka, On invariant volumes of codimension-one Anosov flows and the Verjovsky conjecture, Inventiones
mathematicae 178 , 1 (2009).
Theorem
The maps here considered are volume-preserving.
ProofThe Hamiltonian here considered is one with vanishing potential
(everywhere but on the boundaries of the grouppal structure) and allows
one to normalise the velocities such that the phase space is
three-dimensional and there exists a reduced conserved form on it (by
construction).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The use of the formalism of the Anosov flow is found in the properties of
volume-preserving actions under the hypotheses that it be transitive;the
volume-preserving property is essential in the analysis of the
PSL(2,Z)billiards and to those of its congruence subgroups
M. Asaoka, On invariant volumes of codimension-one Anosov flows and the Verjovsky conjecture, Inventiones
mathematicae 178 , 1 (2009).
Theorem
The maps here considered are volume-preserving.
ProofThe Hamiltonian here considered is one with vanishing potential
(everywhere but on the boundaries of the grouppal structure) and allows
one to normalise the velocities such that the phase space is
three-dimensional and there exists a reduced conserved form on it (by
construction).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
More general results
from the proof of the Verjovsky Conjecture
To every codimension-one Anosov flow on a manifold of dimension
greater than three there corresponds the suspension of a hyperbolic toral
automorphism (i.e. which is topologically equivalent).
K. War, Proof of the Verjovsky Conjecture, math arXiv:2309.10944.
Furthermore, ’possible more general result that says that for every
codimension-one volume-preserving Anosov flow on a manifold of
dimension greater than three, a suitable time change guarantees that the
stable and unstable sub-bundles are then jointly integrable’.
TheoremThe manifold with dimension greater than three here
considered is that of the complete phase space of the billiards systems.
Proof
The definition of billiards here adopted is one for which the time changes
are traced within the properties of the Gauss map on continued fractions
(by construction).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
from the proof of the Verjovsky Conjecture
To every codimension-one Anosov flow on a manifold of dimension
greater than three there corresponds the suspension of a hyperbolic toral
automorphism (i.e. which is topologically equivalent).
K. War, Proof of the Verjovsky Conjecture, math arXiv:2309.10944.
Furthermore, ’possible more general result that says that for every
codimension-one volume-preserving Anosov flow on a manifold of
dimension greater than three, a suitable time change guarantees that the
stable and unstable sub-bundles are then jointly integrable’.
TheoremThe manifold with dimension greater than three here
considered is that of the complete phase space of the billiards systems.
Proof
The definition of billiards here adopted is one for which the time changes
are traced within the properties of the Gauss map on continued fractions
(by construction).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The Bogomolny Transfer Operators
The trace formulas are validated after the verification of the existence of
transfer operators
M. Tabor, Chaos and Integrability in Non-linear Dynamics: an Introduction, Wiley, NY, 1989.
The topic is developped and applied to the Selberg trace formula in
R.E. Prange, Oleg Zaitsev, R. Narevich, Trace formulas and Bogomolny’s transfer operator, Physica E 9, 564
(2001).
The definition of the trace formulas is propedeutically studied after the
study of the definition of th existence of a kernelK(q,q
′
,E) which
defines a Fredholm integral equation
ψ(q) =
Z
PSS
dq
′
K(q,q
′
,E)ψ(q
′
) (31)
which is defined on a one-dimensional Poincar´e surface of sectionPSS.
Poincar´e surfaces of sections for the dynamics of the flow associated with
the desymmetrisedPSL(2,Z) domain are studied in
OML, Reduced 2-dimensional Birkhoff surfaces of section of the desymmetrized PSL(2, Z) group: the Anosov
characterization, Int.Journ. Math. and Computer research 11, 3261 (2023).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The trace formulas are validated after the verification of the existence of
transfer operators
M. Tabor, Chaos and Integrability in Non-linear Dynamics: an Introduction, Wiley, NY, 1989.
The topic is developped and applied to the Selberg trace formula in
R.E. Prange, Oleg Zaitsev, R. Narevich, Trace formulas and Bogomolny’s transfer operator, Physica E 9, 564
(2001).
The definition of the trace formulas is propedeutically studied after the
study of the definition of th existence of a kernelK(q,q
′
,E) which
defines a Fredholm integral equation
ψ(q) =
Z
PSS
dq
′
K(q,q
′
,E)ψ(q
′
) (31)
which is defined on a one-dimensional Poincar´e surface of sectionPSS.
Poincar´e surfaces of sections for the dynamics of the flow associated with
the desymmetrisedPSL(2,Z) domain are studied in
OML, Reduced 2-dimensional Birkhoff surfaces of section of the desymmetrized PSL(2, Z) group: the Anosov
characterization, Int.Journ. Math. and Computer research 11, 3261 (2023).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Solution of the Bogomolny-Fredholm integral equation
Eq. (31) is demonstrated to have non-trivial solution iffEis within the spectrum; it is
integrated after the boundary integral method
E.B. Bogomolny, Semiclassical quantization of multidimensional systems, Nonlinearity 5 (1992) 805.
From
M. Tabor, Chaos and Integrability in Non-linear Dynamics: an Introduction, Wiley, NY, 1989.
Theorem
The semiclassical expression of Eq. (31) is obtained after substituting the kernelK
with the opportune (Bognomolny transfer) operatorTon the suitably-chosen orbits.
Proof
The surface of section Σ can be chosen after the definitions from
OML, Reduced 2-dimensional Birkhoff surfaces of section of the desymmetrized PSL(2, Z) group: the Anosov
characterization, Int. Journ. Math. and Computer research 11, 3261 (2023).
(which are suitably for the requests of a four-dimensional phase space).
Theorem
the trajectory on which the integration is considered is defined through the trajectory
the actionSE(q,q
′
) follows in the phase space, for which the surface of section is
defined
OML, Reduced 2-dimensional Birkhoff surfaces of section of the desymmetrized P SL(2, Z) group: the Anosov
characterization, Int. Journ. Math. and Computer research 11, 3261 (2023).
with a total energyE, and which connects the pointqwith the pointq
′
.
Proof
The Anosov characterisation induces the definition of the corresponding Poincar´e
surface of section.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Eq. (31) is demonstrated to have non-trivial solution iffEis within the spectrum; it is
integrated after the boundary integral method
E.B. Bogomolny, Semiclassical quantization of multidimensional systems, Nonlinearity 5 (1992) 805.
From
M. Tabor, Chaos and Integrability in Non-linear Dynamics: an Introduction, Wiley, NY, 1989.
Theorem
The semiclassical expression of Eq. (31) is obtained after substituting the kernelK
with the opportune (Bognomolny transfer) operatorTon the suitably-chosen orbits.
Proof
The surface of section Σ can be chosen after the definitions from
OML, Reduced 2-dimensional Birkhoff surfaces of section of the desymmetrized PSL(2, Z) group: the Anosov
characterization, Int. Journ. Math. and Computer research 11, 3261 (2023).
(which are suitably for the requests of a four-dimensional phase space).
Theorem
the trajectory on which the integration is considered is defined through the trajectory
the actionSE(q,q
′
) follows in the phase space, for which the surface of section is
defined
OML, Reduced 2-dimensional Birkhoff surfaces of section of the desymmetrized P SL(2, Z) group: the Anosov
characterization, Int. Journ. Math. and Computer research 11, 3261 (2023).
with a total energyE, and which connects the pointqwith the pointq
′
.
Proof
The Anosov characterisation induces the definition of the corresponding Poincar´e
surface of section.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The choice of the one-dimensional
(Poincar´e) surface
The choice of the one-dimensional surface section is motivated after the
analysis of the Markov partition induced after rules of the
operators-ordering in the conjugacy classes of the groups considered.
Theorem
It also coincides with one from the definition of the Birkhoff surface of
section from
OML, Reduced 2-dimensional Birkhoff surfaces of section of the desymmetrized P SL(2, Z) group: the Anosov
characterization, Int. Journ. Math. and Computer research 11, 3261 (2023).
Proof
By construction after the requirements of the Anosov flow of the system.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
(Poincar´e) surface
The choice of the one-dimensional surface section is motivated after the
analysis of the Markov partition induced after rules of the
operators-ordering in the conjugacy classes of the groups considered.
Theorem
It also coincides with one from the definition of the Birkhoff surface of
section from
OML, Reduced 2-dimensional Birkhoff surfaces of section of the desymmetrized P SL(2, Z) group: the Anosov
characterization, Int. Journ. Math. and Computer research 11, 3261 (2023).
Proof
By construction after the requirements of the Anosov flow of the system.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The Bogomolny Transfer Operators
The semiclassical expression of the Bogomolny transfer operator is
rewritten
TE(q,q
′
) =
1
2iπℏ
s
∂
2
Se(q,q
′
)
∂q∂q
′
·e
i
ℏ
S(q,q
′
,E)+i
π
2
ν
(32)
The semiclassical Poincar´e map is
transfer operatorTE(q,q
′
) as
TE(q,q
′
)
Z
Σ
TE(q,q
′
)ψ(q)d
N
q (33)
being Σ the chosen surface of section.
TheoremThe semiclassical limitℏ→0 is demonstrated to exist iff the
spectrum can contain the eigenvalue 1.
Proof
After grand Riemann hypothesis to all automorphic zeta functions, such
as Mellin transforms of the Hecke eigenforms.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The semiclassical expression of the Bogomolny transfer operator is
rewritten
TE(q,q
′
) =
1
2iπℏ
s
∂
2
Se(q,q
′
)
∂q∂q
′
·e
i
ℏ
S(q,q
′
,E)+i
π
2
ν
(32)
The semiclassical Poincar´e map is
transfer operatorTE(q,q
′
) as
TE(q,q
′
)
Z
Σ
TE(q,q
′
)ψ(q)d
N
q (33)
being Σ the chosen surface of section.
TheoremThe semiclassical limitℏ→0 is demonstrated to exist iff the
spectrum can contain the eigenvalue 1.
Proof
After grand Riemann hypothesis to all automorphic zeta functions, such
as Mellin transforms of the Hecke eigenforms.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The (k)-dim Schoredinger equation is reduced to a (k−1)-dim quantum
map.
E.B. Bogomolny, Semiclassical quantization of multidimensional systems, Comm. At. Mol. Phys. 25 (1990) 67;
E.B. Bogomolny, Semiclassical quantization of multidimensional systems, Nonlinearity 5 (1992) 805.
Theoremthe operatorTadmits an invariant function
˜
ψ. The invariant
function
˜
ψis written as
˜ψ(q
′
)≡
Z
Σ
Te(q
′
,q)˜ψ(q)d
N
q. (34)
ProofThe invariant function
˜
ψis
condition of the considered Fredholm determinant as
det(ˆ1−TE) = 0. (35)
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
map.
E.B. Bogomolny, Semiclassical quantization of multidimensional systems, Comm. At. Mol. Phys. 25 (1990) 67;
E.B. Bogomolny, Semiclassical quantization of multidimensional systems, Nonlinearity 5 (1992) 805.
Theoremthe operatorTadmits an invariant function
˜
ψ. The invariant
function
˜
ψis written as
˜ψ(q
′
)≡
Z
Σ
Te(q
′
,q)˜ψ(q)d
N
q. (34)
ProofThe invariant function
˜
ψis
condition of the considered Fredholm determinant as
det(ˆ1−TE) = 0. (35)
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
TheoremThe solutions of Eq. (35) are proven to coincide with the
dynamical zeta function as infinite product over all the periodic orbits
E.B. Bogomolny, Semiclassical quantization of multidimensional systems, Comm. At. Mol. Phys. 25 (1990) 67;
E.B. Bogomolny, Semiclassical quantization of multidimensional systems, Nonlinearity 5 (1992) 805.
Proof
under the suitable conditions:
- Poisson summation formula,
A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications
to Dirichlet series, J. Indian Math. Soc. 20 (1956) 47;
D. Hejhal, The Selberg trace formula for PSL(2, R), Lectures Notes in Mathematics, Vol. 548, A. Dold, B.
Eckmann, (Springer, New York, 1979)- Vol. 1001 (Springer, New York, 1983);
- separation of variables, energy levels from the Green functions,...
M.C. Gutzwiller, Periodic Orbits and Classical Quantization Conditions, J. Math. Phys. 12, 343 (1971);
M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, New York, 1990).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
dynamical zeta function as infinite product over all the periodic orbits
E.B. Bogomolny, Semiclassical quantization of multidimensional systems, Comm. At. Mol. Phys. 25 (1990) 67;
E.B. Bogomolny, Semiclassical quantization of multidimensional systems, Nonlinearity 5 (1992) 805.
Proof
under the suitable conditions:
- Poisson summation formula,
A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications
to Dirichlet series, J. Indian Math. Soc. 20 (1956) 47;
D. Hejhal, The Selberg trace formula for PSL(2, R), Lectures Notes in Mathematics, Vol. 548, A. Dold, B.
Eckmann, (Springer, New York, 1979)- Vol. 1001 (Springer, New York, 1983);
- separation of variables, energy levels from the Green functions,...
M.C. Gutzwiller, Periodic Orbits and Classical Quantization Conditions, J. Math. Phys. 12, 343 (1971);
M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, New York, 1990).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The quantum mapsTsis
operatorU.
Motivations:the defintion(s) of the Perron-Frobenius operator(s) of the
Gauss map are still under study.
D.H. Mayer, On the thermodynamic formalism for the Gauss map, Commun. Math. Phys. 130, 311 (1990).
In
R.E. Prange, Oleg Zaitsev, R. Narevich, Trace formulas and Bogomolny’s transfer operator, , Physica E 9, 564
(2001).
the existence of an ergodic measure on the Banach spaces on which these
operators acts is ensured.
The qualities of the defined operators to be applied on irrational are
discussed still in
D.H. Mayer, On the thermodynamic formalism for the Gauss map, Commun. Math. Phys. 130, 311 (1990).
furthermore, selected topics about the meromorphic continuation of the
objects are calculated.
TheoremThe existence of an ergodic measure onL
2
spaces is
calculated in the following.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
operatorU.
Motivations:the defintion(s) of the Perron-Frobenius operator(s) of the
Gauss map are still under study.
D.H. Mayer, On the thermodynamic formalism for the Gauss map, Commun. Math. Phys. 130, 311 (1990).
In
R.E. Prange, Oleg Zaitsev, R. Narevich, Trace formulas and Bogomolny’s transfer operator, , Physica E 9, 564
(2001).
the existence of an ergodic measure on the Banach spaces on which these
operators acts is ensured.
The qualities of the defined operators to be applied on irrational are
discussed still in
D.H. Mayer, On the thermodynamic formalism for the Gauss map, Commun. Math. Phys. 130, 311 (1990).
furthermore, selected topics about the meromorphic continuation of the
objects are calculated.
TheoremThe existence of an ergodic measure onL
2
spaces is
calculated in the following.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The Margulis measure
Hypotheses
Be
•U-flow (AnosovCflow) on a compact Riemannian manifoldW
n
;
•onW
n
∃two foliation with layers:
−contracting leaves, and
−expanding leaves;
•each one of the foliations is invariant under theU-flow.
Proposition
It is possible on all expanding leaves to induce aσ-finite
countably additive measure s.t.
1) under the action of theU-flow the measure is multiplied times a
constant; and
2) the measures of the canonically-isomorphic sets coincide.
Proposition
3) Analogous propositions are proven for other foliations.
G.A. Margulis, Certain measures associated withU-flows on compact manifolds,
Functional Analysis and Its Applications 4, 55 (1970).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Hypotheses
Be
•U-flow (AnosovCflow) on a compact Riemannian manifoldW
n
;
•onW
n
∃two foliation with layers:
−contracting leaves, and
−expanding leaves;
•each one of the foliations is invariant under theU-flow.
Proposition
It is possible on all expanding leaves to induce aσ-finite
countably additive measure s.t.
1) under the action of theU-flow the measure is multiplied times a
constant; and
2) the measures of the canonically-isomorphic sets coincide.
Proposition
3) Analogous propositions are proven for other foliations.
G.A. Margulis, Certain measures associated withU-flows on compact manifolds,
Functional Analysis and Its Applications 4, 55 (1970).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The need for the Bogmolny transfer operators
•The quantum uniqueness conjectures has not been proven yet for the
dymmetrizedPSL(2,) billiards and for its congruence subgrouppal
structures;
P. Sarnak, Recent progress on the quantum ergodicity conjecture, Bull. (New series) of the Americal mahtmatical
Society 48, 211 (2011).
•It is therefore necessary to apply the Bogomolny theory after the
qualities induced after the specifications allowed after the
Hecke-operators theory.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
•The quantum uniqueness conjectures has not been proven yet for the
dymmetrizedPSL(2,) billiards and for its congruence subgrouppal
structures;
P. Sarnak, Recent progress on the quantum ergodicity conjecture, Bull. (New series) of the Americal mahtmatical
Society 48, 211 (2011).
•It is therefore necessary to apply the Bogomolny theory after the
qualities induced after the specifications allowed after the
Hecke-operators theory.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Aymptotic estimates on lengths of closed geodesics
•surfaces of constant negative curvature−1:
∃a countable infinity of closed geodesics within a specific homology
class, with asymptotic formulas for their length
R. Phillips, P. Sarnak, Closed geodesics in homology classes, Duke Math. J. 55, 287 (1987);
A. Katsuda, T. Sunada, Homology and closed geodesics in a compact Riemann surface, Amer. J. Math. 110, 145
(1988).
∃C>0 :
π(t,α)
Ce
t
/t
g+1→1,t→ ∞
l(γ) length,
l(γ)≤t
αhomology class
π(t, α)number of closed geodesics
ggenus
M. Pollicott, American Journal of Mathematics 113, 379 (1991).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
•surfaces of constant negative curvature−1:
∃a countable infinity of closed geodesics within a specific homology
class, with asymptotic formulas for their length
R. Phillips, P. Sarnak, Closed geodesics in homology classes, Duke Math. J. 55, 287 (1987);
A. Katsuda, T. Sunada, Homology and closed geodesics in a compact Riemann surface, Amer. J. Math. 110, 145
(1988).
∃C>0 :
π(t,α)
Ce
t
/t
g+1→1,t→ ∞
l(γ) length,
l(γ)≤t
αhomology class
π(t, α)number of closed geodesics
ggenus
M. Pollicott, American Journal of Mathematics 113, 379 (1991).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Some theorems about Measure spaces
of the desymmetrised systems from PSL(2, Z) and
of those if its congruence subgroups
Theorem
In billiards of thePSL(2,Z) desymmetrised group, and in its congruence
subgroups, the results recalled from [Pollicott, 1991] hold.
Proof
The metric is one induced after the Maass waveforms onPSL(2,Z).
Theorem
The space obtained is thereforeL
2
(dµ),dµbeing the measure on the
UPHP.
Proof
The measuredµon the UPHP is one obtained after probability space
defined after the action of the Anosov flow on the reduced Birkhoff
surface of section.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
of the desymmetrised systems from PSL(2, Z) and
of those if its congruence subgroups
Theorem
In billiards of thePSL(2,Z) desymmetrised group, and in its congruence
subgroups, the results recalled from [Pollicott, 1991] hold.
Proof
The metric is one induced after the Maass waveforms onPSL(2,Z).
Theorem
The space obtained is thereforeL
2
(dµ),dµbeing the measure on the
UPHP.
Proof
The measuredµon the UPHP is one obtained after probability space
defined after the action of the Anosov flow on the reduced Birkhoff
surface of section.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Remarks
•The measure presented in
V. Baladi, M. Demers, On the Measure of Maximal Entropy for Finite Horizon Sinai Billiard Maps, J. Amer. Math.
Soc. 33, 381 (2020).
is not of use in the present analysis, as the points at infinity are in the
present analysis excluded form the billiard representation.
•The presence of Hecke operators on arithmetical manifolds acting on
L
2
functions has been analysed in
D. Jakobson, N. S. Nadirashvili, J. Toth, Geometric properties of eigenfunctions, Russian Mathematical Surveys,
56, 1085 (2001) , and in the references 138-139-140 ibidem.
•The results can be compared with those form the Severini-Egorov
Theorem.
•From
N. Hashiguchi, PL-representations of Anosov foliations, Ann. Inst. Fourier, Grenoble 42, 937 (1992).
the Anosov flow is defined after the choice of an opportune Birkhoff
surface of section; the features of the so-defined harmonic maps are
exposed in
F. T. Farrell, P. Ontaneda, Branched Covers of Hyperbolic Manifolds and Harmonic Maps, arXiv math/0403140.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
•The measure presented in
V. Baladi, M. Demers, On the Measure of Maximal Entropy for Finite Horizon Sinai Billiard Maps, J. Amer. Math.
Soc. 33, 381 (2020).
is not of use in the present analysis, as the points at infinity are in the
present analysis excluded form the billiard representation.
•The presence of Hecke operators on arithmetical manifolds acting on
L
2
functions has been analysed in
D. Jakobson, N. S. Nadirashvili, J. Toth, Geometric properties of eigenfunctions, Russian Mathematical Surveys,
56, 1085 (2001) , and in the references 138-139-140 ibidem.
•The results can be compared with those form the Severini-Egorov
Theorem.
•From
N. Hashiguchi, PL-representations of Anosov foliations, Ann. Inst. Fourier, Grenoble 42, 937 (1992).
the Anosov flow is defined after the choice of an opportune Birkhoff
surface of section; the features of the so-defined harmonic maps are
exposed in
F. T. Farrell, P. Ontaneda, Branched Covers of Hyperbolic Manifolds and Harmonic Maps, arXiv math/0403140.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Some theorems to prove the existence of Scars of
the desymmetrised systems from PSL(2, Z) and
from its congruence subgroups
•The Bogomolny transfer operators are indeed applied on reflections.
•The use of Hecke operators outlines why the Maass waveforms exhibit
at most polynomial growth.
•The construction of Farey trees arrives from the approximation of
rational to irrationals after the method of Farey-Pell.
•The trace functions on Farey trees (graphs) is therefore defined.
• •The construction achieved is extended to the congruence subgroups
of the desymmetrisedPSL(2,Z) group after the application of the
dynamical transfer operators.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
the desymmetrised systems from PSL(2, Z) and
from its congruence subgroups
•The Bogomolny transfer operators are indeed applied on reflections.
•The use of Hecke operators outlines why the Maass waveforms exhibit
at most polynomial growth.
•The construction of Farey trees arrives from the approximation of
rational to irrationals after the method of Farey-Pell.
•The trace functions on Farey trees (graphs) is therefore defined.
• •The construction achieved is extended to the congruence subgroups
of the desymmetrisedPSL(2,Z) group after the application of the
dynamical transfer operators.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The appearance ofscarsat the semiclassical regime ishere provento
arise as a phenomena due to the measure associated to the considered
(classical, scarring) trajectories and the properties they obey under the
action of theUflow.
In
A.C. Nordentoft, Concentration of closed geodesics in the homology of modular curves, Forum of Mathematics
Sigma 11, 1 (2023).
the behaviour of the geodesics of reals quadratic fields as concentrating
around the Eisenstein line are studied.
The results of Margulis
ibidem
can therefore be applied as far as the existence of the Margulis measure
which acquires the Margulis multiplicative constant under the action of
theUflow.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
arise as a phenomena due to the measure associated to the considered
(classical, scarring) trajectories and the properties they obey under the
action of theUflow.
In
A.C. Nordentoft, Concentration of closed geodesics in the homology of modular curves, Forum of Mathematics
Sigma 11, 1 (2023).
the behaviour of the geodesics of reals quadratic fields as concentrating
around the Eisenstein line are studied.
The results of Margulis
ibidem
can therefore be applied as far as the existence of the Margulis measure
which acquires the Margulis multiplicative constant under the action of
theUflow.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The multiplicative constant that the Margulis measure acquires can
compare a special case of the multiplicative character of the group (in
comparison with formula Eq. (2.11) of
E. Bogomolny, N. Cairoli, M.-J. Giannoni, C. Schmit, Arithmetical chaos, Phys. Rept. 291, 219 (1997) Eq. (2.11).
in the Sel’berg trace formula expression.
In particular, it is crucial to note from now on that the investigated
geodesics concentrate according to the features of the quadratic field and
not according to the qualities of the (classes of) closed geodesics, thus
the following theorems hold
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
compare a special case of the multiplicative character of the group (in
comparison with formula Eq. (2.11) of
E. Bogomolny, N. Cairoli, M.-J. Giannoni, C. Schmit, Arithmetical chaos, Phys. Rept. 291, 219 (1997) Eq. (2.11).
in the Sel’berg trace formula expression.
In particular, it is crucial to note from now on that the investigated
geodesics concentrate according to the features of the quadratic field and
not according to the qualities of the (classes of) closed geodesics, thus
the following theorems hold
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Theorem
In the desymmetrisedPSL(2,Z) group, the scarred waveforms are those
obtained after the action of theUflow on the Margulis measure which
acts on the quadratics fields which define the (also, classes) of closed
geodesics.
ProofBy construction of the closed geodesics after applying the billiard
map on the quadratic fields, from which quadratic fields the closed
geodesics are obtained to be formed after the Pell method.
Theorem
The action of the free-diffeomeorphisms groups is used in the action of
the U flow.
Proof
Due to the rigidity of the Laplace operator on surfaces of constant
negative curvature.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
In the desymmetrisedPSL(2,Z) group, the scarred waveforms are those
obtained after the action of theUflow on the Margulis measure which
acts on the quadratics fields which define the (also, classes) of closed
geodesics.
ProofBy construction of the closed geodesics after applying the billiard
map on the quadratic fields, from which quadratic fields the closed
geodesics are obtained to be formed after the Pell method.
Theorem
The action of the free-diffeomeorphisms groups is used in the action of
the U flow.
Proof
Due to the rigidity of the Laplace operator on surfaces of constant
negative curvature.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Theorem
In the congruence subgroups of the desymmetrisedPSL(2,Z) domain,
the scarred wavefunctions occur under the effect of the action of the
Bogomolny transfer operators on the Margulis measure.
Proof
By construction.
Corollary
The Margulis measure and the Bogomolny transfer operators commute.
Proof
By construction.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
In the congruence subgroups of the desymmetrisedPSL(2,Z) domain,
the scarred wavefunctions occur under the effect of the action of the
Bogomolny transfer operators on the Margulis measure.
Proof
By construction.
Corollary
The Margulis measure and the Bogomolny transfer operators commute.
Proof
By construction.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The Kasner transformations are those defined in
O.M. Lecian, Reflections on the hyperbolic plane, International Journal of Modern Physics D 22, 1350085 (2013).
which are needed in order to tile the UPHP according to the patterns of
the Γncongruence subgroups of the desymmetrisedPSL(2,Z) domain
starting from the desymmetrisedPSL(2,Z) group.
Theorem
The overall action of the U flows is decomposed for the congruence
subgroups Γnof the desymmetrised domain ofpsl(2,Z) as consisting of
the billiard transformations, the Bogomolny transfer operators, and the
Kasner transformation needed to apply the Bogomolny transfer operators
only on the boundary of the billiard tables (or on the chosen Birkhoff
surface of section).
Proof
The Kasner transformations are those needed to tile the wanted Γn
portions of the UPHP according to the desymmetrised domain of
PSL(2,Z), and therefore the let the Bogomolny transfer operator act
only on the chosen Birkhoff surface of section.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
O.M. Lecian, Reflections on the hyperbolic plane, International Journal of Modern Physics D 22, 1350085 (2013).
which are needed in order to tile the UPHP according to the patterns of
the Γncongruence subgroups of the desymmetrisedPSL(2,Z) domain
starting from the desymmetrisedPSL(2,Z) group.
Theorem
The overall action of the U flows is decomposed for the congruence
subgroups Γnof the desymmetrised domain ofpsl(2,Z) as consisting of
the billiard transformations, the Bogomolny transfer operators, and the
Kasner transformation needed to apply the Bogomolny transfer operators
only on the boundary of the billiard tables (or on the chosen Birkhoff
surface of section).
Proof
The Kasner transformations are those needed to tile the wanted Γn
portions of the UPHP according to the desymmetrised domain of
PSL(2,Z), and therefore the let the Bogomolny transfer operator act
only on the chosen Birkhoff surface of section.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
RemarkIt is important to remark that the properties of the
desymmetrisedPSL(2,Z) group and those of its congruence subgroups
were deduced after the properties ofpsl(2,Z) alone, according to the
newly-proposed analysis of the U flow.
RemarkThe quantum properties and the semiclassical ones of the
billiard are ascribed to the properties of the Bogomolny transfer
operators, according to which these operators act on the phase space (or
on the restricted phase space selected after the reduced Birkhoff surface
of section chosen after the definition of the Anosov flow).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
desymmetrisedPSL(2,Z) group and those of its congruence subgroups
were deduced after the properties ofpsl(2,Z) alone, according to the
newly-proposed analysis of the U flow.
RemarkThe quantum properties and the semiclassical ones of the
billiard are ascribed to the properties of the Bogomolny transfer
operators, according to which these operators act on the phase space (or
on the restricted phase space selected after the reduced Birkhoff surface
of section chosen after the definition of the Anosov flow).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Prospective studies 1:
The Ljapunov exponent of scarred systems
Intorductory material about billiard systems of the Gauss map are
proposed in
C. Manchein, M.W. Beims, Gauss map and Lyapunov exponents of interacting particles in a billiard, Chaos,
Solitons & Fractals 39, 2041 (2009).
with study of ’pseudo-integrable systems with more complicated invariant
surfaces of the flow’.
The behaviour of the Ljapunov exponent related to the reduced
Birkhoff usrface of section has to be explored.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The Ljapunov exponent of scarred systems
Intorductory material about billiard systems of the Gauss map are
proposed in
C. Manchein, M.W. Beims, Gauss map and Lyapunov exponents of interacting particles in a billiard, Chaos,
Solitons & Fractals 39, 2041 (2009).
with study of ’pseudo-integrable systems with more complicated invariant
surfaces of the flow’.
The behaviour of the Ljapunov exponent related to the reduced
Birkhoff usrface of section has to be explored.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Prospective studies 2:
Homology classes and topological entropy
In
V. Baladi, Periodic orbits and dynamical spectra, Ergod. Th. & Dynam. Sys. 18, 255 (1998).
, the spectrum of dynamical zeta functions connected with generalised zeta functions
is studied to be defined after dynamical transfer operators on Banach spaces.
The composition of these transfer operators to the hyperbolic reflections allows one to
reconstruct the conjugacy classes of the congruence subgroups of the desymmetrised
PSL(2,Z) group
OML, arXiv:1507.07085 [gr-qc]version2.
in particular, the construction applies also to the conjugacy classes of the Γn
subgroups which define the periodic orbits, which are expressed as (infinite) ’Farey
trees’ from the Farey-Pell paradigm.
The specification of the Sel’berg trace formula of the desymmetrisedPSL(2,Z) groups
and that of its Γncongruence subgroups will be addressed elsewhere.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Homology classes and topological entropy
In
V. Baladi, Periodic orbits and dynamical spectra, Ergod. Th. & Dynam. Sys. 18, 255 (1998).
, the spectrum of dynamical zeta functions connected with generalised zeta functions
is studied to be defined after dynamical transfer operators on Banach spaces.
The composition of these transfer operators to the hyperbolic reflections allows one to
reconstruct the conjugacy classes of the congruence subgroups of the desymmetrised
PSL(2,Z) group
OML, arXiv:1507.07085 [gr-qc]version2.
in particular, the construction applies also to the conjugacy classes of the Γn
subgroups which define the periodic orbits, which are expressed as (infinite) ’Farey
trees’ from the Farey-Pell paradigm.
The specification of the Sel’berg trace formula of the desymmetrisedPSL(2,Z) groups
and that of its Γncongruence subgroups will be addressed elsewhere.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The latest results gathered in
V. Baladi, M. Demers, On the Measure of Maximal Entropy for Finite Horizon Sinai Billiard Maps, J. Amer. Math.
Soc. 33, 381 (2020).
are not applied in the present work, because the billiard there-analysed is one for
which the points on the absolute are considered as part of the billiard, i.e. differently
from the definition from
I.P. Cornfeld , S.V. Fomin , Ya.G. Sinai, Ergodic Theory, Springer New York, USA (1982). [?].
The analysis of
Zh.-P Serr, Trees, amalgams andSL2, Matematika 18, 03 (1974) Section 3.1.
is not in the present work considered, as the problem of inversions is raised in the
definition of the action on trees.
Indeed, the hyperbolic transformations which define the desymmetrised group and its
congruence subgroups are reflections and not inversions.
Similarly, for the congruence subgroups, a different number of (hyperbolic) reflections
is needed (to tile the UPHP), even though the length of the (closed) geodesics is
unchanged, which is proven after the introduction of the (opportune) transfer
operators. In the present work, a different analysis is pursued, form which the use of
the trees in the study of the scars is used.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
V. Baladi, M. Demers, On the Measure of Maximal Entropy for Finite Horizon Sinai Billiard Maps, J. Amer. Math.
Soc. 33, 381 (2020).
are not applied in the present work, because the billiard there-analysed is one for
which the points on the absolute are considered as part of the billiard, i.e. differently
from the definition from
I.P. Cornfeld , S.V. Fomin , Ya.G. Sinai, Ergodic Theory, Springer New York, USA (1982). [?].
The analysis of
Zh.-P Serr, Trees, amalgams andSL2, Matematika 18, 03 (1974) Section 3.1.
is not in the present work considered, as the problem of inversions is raised in the
definition of the action on trees.
Indeed, the hyperbolic transformations which define the desymmetrised group and its
congruence subgroups are reflections and not inversions.
Similarly, for the congruence subgroups, a different number of (hyperbolic) reflections
is needed (to tile the UPHP), even though the length of the (closed) geodesics is
unchanged, which is proven after the introduction of the (opportune) transfer
operators. In the present work, a different analysis is pursued, form which the use of
the trees in the study of the scars is used.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The Hamiltonian flow is studied as follows.
As from
M. Pollicot, Homology and Closed Geodesics in a Compact Negatively Curved Surface, American Journal of
Mathematics, 113, 379 (1991).
on a compact manifoldMof constant negative curvature−1, the torus is constructed
from the closed geodesics, where the metric is a flat one, after the 1-forms onM. In
this case, the equidistribution property holds, given two different homology classesα
andβof geodesics of lengthxas
limx→∞
π(α,x)
π(β,x)
= 1 (36)
beingπthe number of prime closed geodesics.
L. Phillips, P. Sarnak, Closed geodesics in homology classes, Duke Mathematical Journal 55, 287 (1987).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
As from
M. Pollicot, Homology and Closed Geodesics in a Compact Negatively Curved Surface, American Journal of
Mathematics, 113, 379 (1991).
on a compact manifoldMof constant negative curvature−1, the torus is constructed
from the closed geodesics, where the metric is a flat one, after the 1-forms onM. In
this case, the equidistribution property holds, given two different homology classesα
andβof geodesics of lengthxas
limx→∞
π(α,x)
π(β,x)
= 1 (36)
beingπthe number of prime closed geodesics.
L. Phillips, P. Sarnak, Closed geodesics in homology classes, Duke Mathematical Journal 55, 287 (1987).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Prospective studies 3:
Homology of closed geodesics
If the fundamental group of the manifold has infinitely many conjugacy
classes, the closed curve representations with minimal length are the
infinitely many geometrically distinct periodic geodesics.
Theorem
For finite fundamental group, there are infinitely many geometrically
distinct periodic geodesics if the real cohomology ring of the manifold or
any of its covers requires at least two generators
M. Vigue-Poirrer, D. Sullivan, The homology theory of the closed geodesic problem, J. Differential Geometry, 11,
633 (1976);
after the algebra in
D. Gromoll, W. Meyer, Periodic geodesics on compact Riemannian manifolds, J. Differential Geometry 3, 493
(1969).
Proof
ForPSL(2,Z) and for its congruence subgroups, yes, by construction.
Need of cohomology
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Homology of closed geodesics
If the fundamental group of the manifold has infinitely many conjugacy
classes, the closed curve representations with minimal length are the
infinitely many geometrically distinct periodic geodesics.
Theorem
For finite fundamental group, there are infinitely many geometrically
distinct periodic geodesics if the real cohomology ring of the manifold or
any of its covers requires at least two generators
M. Vigue-Poirrer, D. Sullivan, The homology theory of the closed geodesic problem, J. Differential Geometry, 11,
633 (1976);
after the algebra in
D. Gromoll, W. Meyer, Periodic geodesics on compact Riemannian manifolds, J. Differential Geometry 3, 493
(1969).
Proof
ForPSL(2,Z) and for its congruence subgroups, yes, by construction.
Need of cohomology
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Propsective studies 4:
More about automorphic forms
Xcompact hyperbolic surface
- the closed geodesics onXare the periodic orbits of the geodesics flow:
counting the geodesics of the homology classes ofPSL(2,R)
Γ⊂PSL(2,R)
Method:
- trace formula,
- convolution operator,
- Fourier series of the trace.
S. Zelditch, Geodesics in homology classes and periods of automorphic forms, Advances in Mathematics 88, 113
(1991).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
More about automorphic forms
Xcompact hyperbolic surface
- the closed geodesics onXare the periodic orbits of the geodesics flow:
counting the geodesics of the homology classes ofPSL(2,R)
Γ⊂PSL(2,R)
Method:
- trace formula,
- convolution operator,
- Fourier series of the trace.
S. Zelditch, Geodesics in homology classes and periods of automorphic forms, Advances in Mathematics 88, 113
(1991).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Prospective studies 5:
Topological entropy of geodesic flows
Method:use the complete inverse images of the points under the
mapping.
G.A. Margulis, Applications of ergodic theory to the investigation of manifolds of negative curvature, Functional
analysis and its applications, 3, pages 335–336, (1969).
On a compact Riemann surfaceMof negative curvature without
boundariesthe topological entropy is defined from the entropy of the
map of a geodesic flow, and it equals the exponential rate growth.
A. Manning, Topological Entropy for Geodesic Flows, Annals of Mathematics Second Series 110, 567 (1979).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Topological entropy of geodesic flows
Method:use the complete inverse images of the points under the
mapping.
G.A. Margulis, Applications of ergodic theory to the investigation of manifolds of negative curvature, Functional
analysis and its applications, 3, pages 335–336, (1969).
On a compact Riemann surfaceMof negative curvature without
boundariesthe topological entropy is defined from the entropy of the
map of a geodesic flow, and it equals the exponential rate growth.
A. Manning, Topological Entropy for Geodesic Flows, Annals of Mathematics Second Series 110, 567 (1979).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Prospective studies 6:
Cohomology techniques
Apply the results of
M.H. Lee, Hecke operators on cohomology, Revista de la union matematica argentina 50, 99 (2009);
M. Furusawa, M. Tezuka, N. Yagita, Note on Hecke operators and cohomoogy ofPSL(2,‡), Kodai Math. J. 14,
173 (1991);
F. Williams, R.J. Wisner, Cohomology of certain congruence subgroups of the modular group, Proceedings of the
American Mathematical Society 126, 1331 (1998).
to the desymmetrisedPSL(2,Z) group and to its congruence subgroups.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Cohomology techniques
Apply the results of
M.H. Lee, Hecke operators on cohomology, Revista de la union matematica argentina 50, 99 (2009);
M. Furusawa, M. Tezuka, N. Yagita, Note on Hecke operators and cohomoogy ofPSL(2,‡), Kodai Math. J. 14,
173 (1991);
F. Williams, R.J. Wisner, Cohomology of certain congruence subgroups of the modular group, Proceedings of the
American Mathematical Society 126, 1331 (1998).
to the desymmetrisedPSL(2,Z) group and to its congruence subgroups.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Prospective studies 7:
Farrel-Ontaneda(-related) methods
LetM,Nbe closed negatively curved manifolds;
∃homeomorphismf:M→Ns.t.
fis not homeotopic to a diffeomorpihsm
F.T. Farrell, P. Ontaneda, Branched covers of hyperbolic manifolds and harmonic maps, arXiv:math/0403140
[math.DG].
what about piecewise homomorphisms
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Farrel-Ontaneda(-related) methods
LetM,Nbe closed negatively curved manifolds;
∃homeomorphismf:M→Ns.t.
fis not homeotopic to a diffeomorpihsm
F.T. Farrell, P. Ontaneda, Branched covers of hyperbolic manifolds and harmonic maps, arXiv:math/0403140
[math.DG].
what about piecewise homomorphisms
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Prospective studies 8:
Markov approximation
More research directions can be envisaged after Markov approximations
of Sinai billiards.
D. Sza’sz, Markov Approximations and Statistical Properties of Billiards. In: Holden, H., Piene, R. (eds) The Abel
Prize 2013-2017. The Abel Prize. Springer, Cham (2019).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Markov approximation
More research directions can be envisaged after Markov approximations
of Sinai billiards.
D. Sza’sz, Markov Approximations and Statistical Properties of Billiards. In: Holden, H., Piene, R. (eds) The Abel
Prize 2013-2017. The Abel Prize. Springer, Cham (2019).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Prospective studies 9:
The Isola map
Gauss map as one induced after the Farey map:
the corresponding holomorphic functions
- live on Hilbert spaces - are obtained after generalized Borel transform
and Laplace transform,
- invariant under the action of the transfer operators associated to the
maps.
Suitable ’operator-valued power series’ defined which act on the
associated zerta function.
S. Isola, On the spectrum of Farey and Gauss maps, Nonlinearity 15, 1521 (2002).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
The Isola map
Gauss map as one induced after the Farey map:
the corresponding holomorphic functions
- live on Hilbert spaces - are obtained after generalized Borel transform
and Laplace transform,
- invariant under the action of the transfer operators associated to the
maps.
Suitable ’operator-valued power series’ defined which act on the
associated zerta function.
S. Isola, On the spectrum of Farey and Gauss maps, Nonlinearity 15, 1521 (2002).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Further perspectives
1) the representation of the Hitchin components is given a geometric
interpretation in the projective spaces in
F. Labourie, Anosov Flows, Surface Groups and Curves in Projective Space, Inventiones Mathematicae 165, 51
(2006).
2) the action of the Anosov flow on the Weyl chambers is explained in
F. Gu´eritaud, O. Guichard, F. Kassel, A. Wienhard, Anosov representations and proper actions, Geom. Topol. 21,
485 (2017).
and adapted to those of the Coxeter group ibidem, pag. 17, as far as the
GLgroups are concerned;
3) the new characterization of the basic properties of the Anosov
subgroups on some Lie groups is given in
M. Kapovich, B. Leeb, J. Porti, Anosov subgroups: Dynamical and geometric characterizations, European Journal
of Mathematics 3, 808 (2017).
where the notion of geodesics flow is avoided; nevertheless, the notion of
expansion along the geodesics in the group is kept, which allows one to
recover the notion of unfolding;
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
1) the representation of the Hitchin components is given a geometric
interpretation in the projective spaces in
F. Labourie, Anosov Flows, Surface Groups and Curves in Projective Space, Inventiones Mathematicae 165, 51
(2006).
2) the action of the Anosov flow on the Weyl chambers is explained in
F. Gu´eritaud, O. Guichard, F. Kassel, A. Wienhard, Anosov representations and proper actions, Geom. Topol. 21,
485 (2017).
and adapted to those of the Coxeter group ibidem, pag. 17, as far as the
GLgroups are concerned;
3) the new characterization of the basic properties of the Anosov
subgroups on some Lie groups is given in
M. Kapovich, B. Leeb, J. Porti, Anosov subgroups: Dynamical and geometric characterizations, European Journal
of Mathematics 3, 808 (2017).
where the notion of geodesics flow is avoided; nevertheless, the notion of
expansion along the geodesics in the group is kept, which allows one to
recover the notion of unfolding;
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
4) the maps of the Anosov representation of hyperbolic groups are
written in
S. Dowdall, Anosov-representations: Basic definitions and properties, e-print
https://lukyanenko.net/conferences/htt2013/Dowdall.pdf and the references therein.
the topic is based on the fact that the representations of the Hitchin
components are Anosov;
5) the rigidity of the Weyl chambers flow is studied in
M. Kanai, Rigidity of the Weyl chamber flow, and vanishing theorems of Matsushima and Weil, Ergod. Th.
Dynam. Sys. 29, 1273 (2009).
Theorem: The rigidity of the Weyl chambers flow is due to the rigidity
of the Laplace-Beltrami operator.
Proof: The Weyl chambers can be constructed after the opportune tiling
of the UPHP after the apportune hyperbolic reflections on the
desymmetrsedPSL(2,Z) domain.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
written in
S. Dowdall, Anosov-representations: Basic definitions and properties, e-print
https://lukyanenko.net/conferences/htt2013/Dowdall.pdf and the references therein.
the topic is based on the fact that the representations of the Hitchin
components are Anosov;
5) the rigidity of the Weyl chambers flow is studied in
M. Kanai, Rigidity of the Weyl chamber flow, and vanishing theorems of Matsushima and Weil, Ergod. Th.
Dynam. Sys. 29, 1273 (2009).
Theorem: The rigidity of the Weyl chambers flow is due to the rigidity
of the Laplace-Beltrami operator.
Proof: The Weyl chambers can be constructed after the opportune tiling
of the UPHP after the apportune hyperbolic reflections on the
desymmetrsedPSL(2,Z) domain.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Further references
G.A. Noskov, Bounded shortening in Coxeter complexes and buildings,
Mathematical structures and modelling, 8, 10 (2001);
A. Levin, M. Olshanetsky, A. Smirnov, A. Zotov, Characteristic Classes
and Hitchin Systems. General Construction, Commun. Math. Phys. 316,
1 (2012).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
G.A. Noskov, Bounded shortening in Coxeter complexes and buildings,
Mathematical structures and modelling, 8, 10 (2001);
A. Levin, M. Olshanetsky, A. Smirnov, A. Zotov, Characteristic Classes
and Hitchin Systems. General Construction, Commun. Math. Phys. 316,
1 (2012).
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Thank You for Your attention.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Some new theorems on scarred waveforms in desymmetrised PSL(2, Z) group and in its congruence subgroups.Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
Download
Download Slideshow Get the original presentation fileQuick Actions
Statistics
- Views 9
- Slides 67
- Age 415 days
Related Slideshows
23
Earthquakes_Type of Faults_Science G8.pptx
OctabellFabila1
31 views
15
Quiz #1 Science 10 in the first quarter for jhs
HendrixAntonniAmante
30 views
9
Astronomy history from long ago till doday
ssuserbd9abe
29 views
9
Great history of astronomy from long ago till today
ssuserbd9abe
27 views
20
EARTHQUAKE-DRILL.powerpoint.............
chalobrido8
30 views
9
History of astronomy from old times to the present times
ssuserbd9abe
30 views