Three Theorems of Poincare

In this talk, I looked at three theorems due to Poincaré concerning the long-term behavior of dynamical systems, namely, the Poincaré–Perron Theorem, the Poincaré–Bendixson Theorem and the rather counterintuitive Poincaré Recurrence Theorem, along with their proofs. Certain aspects of ergodic theory and topological dynamics, for instance, their applications in number theory and analysis (Green–Tao Theorem and Szemerédi’s Theorem), were also touched upon.

1. Poincaré: three theorems on the asymptotic behavior of dynamical systems Arpan Saha Engineering Physics with Nanoscience IIT Bombay October 3, 2010

2. Change alone is eternal , perpetual , immortal . – Arthur Schopenhauer

3. Henri Poincaré (1854 - 1912) A polymath , he is known in mathematical circles as the Last Universalist due to the large number of significant contributions he made to various fields of mathematics and physics especially the then nascent study of dynamical systems .

4. The Three Theorems Even in this single field , the list of his contributions is no small one. We’ll hence be looking at only three of his theorems : Poincaré-Perron Theorem Poincaré- Bendixson Theorem Poincaré Recurrence Theorem

5. Some problems /questions Hopefully , these results shall enable us to solve some interesting problems such as: Given any finite colouring C 1 , C 2 , … , C k of the set of integers , there are arbitrary large arithmetic progressions of the same colour. A man with irrational step  walks around a circle of length 1. The circle has a ditch of width . Show that sooner or later , he will step into the ditch no matter how small  will be .

6. The Poincaré-Perron Theorem This theorem describes the long- term behavior of iterated maps defined by linear recurrence relations as approximately geometric progressions with common factor being a root of the ‘ characteristic polynomial’ of the recurrence .

7. Statement Given a linear homogeneous recurrence relation in with constant coefficients a x(n) + a 1 x(n + 1) + … + a k x (n + k) = 0 with characteristic roots  i , such that distinct roots have distinct moduli , then x(n + 1)/x(n)   i for some i, as n  . The Poincaré-Perron Theorem

8. Proof of the Theorem First we find a general solution for the recurrence relation as a linear combination of basis functions of n, in a manner analogous to the case of differential equations . We then express the ratio x(n + 1)/x(n) in terms of these solutions and compute the limit for the various cases of arbitrary constants. The Poincaré-Perron Theorem

9. Q. E. D.

10. Comments on the Proof What if distinct roots don’t have distinct moduli ? Let’s consider the case where k = 2 and a 1 and a 2 are real. We see that in general the limit does not exist . But for certain particular solutions, it does . The Poincaré-Perron Theorem

11. Terminology and Clarifications To proceed with the next theorem , we will need to develop some terminology and clarify concepts which haven’t been so clearly defined such as: Dynamical systems and flows Orbits , semiorbits and invariant sets Limit , limit point, - and - limit point Sequential compactness Transversal and flow box Monotone on trajectory Monotone on transversal

12. The Poincaré- Bendixson Theorem This theorem establishes the sufficient conditions for the phase- space trajectories through a given point to approach a limit cycle.

13. Statement of the theorem Every  -limit set of a C 1 flow defined over a sequentially compact and simply connected subset of the plane that does not contain an equilibrium point is a ( nondegenerate ) periodic orbit. The Poincaré- Bendixson Theorem

14. Proof of the Theorem The result immediately follows from four lemmas that we are about to prove : Lemma 1 : If the intersections of the positive semiorbit of a point with a transversal are monotonic on the trajectory , they are also monotonic on the transversal. Lemma 2 : The - limit set of a point cannot intersect a transversal at more than one point. Lemma 3 : An - limit point of an - limit point of a point lies on a periodic orbit . Lemma 4 : If the - limit of a point contains a nondegenerate period orbit , then the - limit set is the periodic orbit . The Poincaré- Bendixson Theorem

15. Q. E. D.

16. Comments on the proof We have seen that the proof critically hinges upon the validity of the Jordan Curve Theorem and the fact that the transversal is 1D. Hence the theorem is not applicable to manifolds of dimension greater than 2. But amongst those of dimension 2, only subsets homeomorphic to compact, simply connected subsets of the plane or the 2- sphere respect the PB theorem . Hence , PB doesn’t work for surfaces of higher genus such as tori as they don’t satisfy the Jordan curve theorem . The Poincaré- Bendixson Theorem

17. A Corollary If a positively ( negatively ) invariant, closed and bounded subset of a plane contains no stable ( unstable ) fixed point then it must contain an -(-) limit cycle. The Poincaré- Bendixson Theorem

18. What about structure? We’ll now move into very different kinds of questions about dynamical systems such as: Hamiltonian systems Finite systems Group actions Cyclic groups Bernoulli systems More specifically, we’ll be briefly looking at some systems with nice structural aspects, while spending some time over a rather cool theorem called the Poincaré Recurrence Theorem.

19. The Notion of Measure For our last theorem today , we will need to acquaint ourselves with some measure theoretic ideas . The concept of measure was spearheaded by Henri Lebesgue in his attempt to generalize the Riemann- Stieltjes Integral to much more exotic functions . Informally , it refers to the ‘size’ of a set.

20. The Notion of Measure More formally , if X is a space , and X a - algebra of the subsets of X that we consider ‘ measurable ’, then the measure  is assignment of nonnegative reals and + to sets in X , such that the following hold : The measure of the null set, () = 0 The measure of a countable union of pairwise disjoint sets of X is the sum of the measures of the individual sets. The triple (X, X , ) is referred to as a measure space and any flow  such that ((t, S)) = (S) for all t in the time set and all S in X is said to be measure - preserving .

21. The Poincaré Recurrence Theorem This theorem asserts that a dynamical system with a measure - preserving flow revisits any measurable set infinitely many times for almost all initial points in the set.

22. Statement If  = (X, X , ) is a measure space with (X) <  and f:    is a measure - preserving bijection, then for any measurable set E  X , the set of points x in E such that x  f n (E) for only finitely many natural numbers n, has measure zero . The Poincaré Recurrence Theorem

23. Proof of the Theorem Let E be the given measurable set and A n be the countably infinite union of f –n (E), f –(n + 1) (E), f –(n +2) (E) and so on. The proof now essentially becomes five steps : Argue E  A , and A l  A m if m < l Argue f –n ( A ) = A n , hence ( A n ) = ( A ) for all integers n. Show ( E\ A n )  ( A \ A n ) = 0 Conclude that the measure of the countably infinite union of E\ A 1 , E\ A 2 , E\ A 3 and so on is zero as well . Argue that this union is precisely the set of points x in E such that f n (x)  E for only finitely many n. The Poincaré Recurrence Theorem

24. Q. E. D.

25. Terence Tao’s Version Terence Tao gave a somewhat stronger version of the PR theorem . Continuing with the notation introduced in the previous formulation, Tao’s statement is : lim sup n  + (E  f n E )  ((E)) 2 This follows from the Cauchy-Schwarz Inequality for integrals , and is a more explicit qualitative strengthening of the Pigeonhole Principle . The Poincaré Recurrence Theorem

26. The Recurrence Paradox We will briefly digress to remark on a curious paradox that has becomed inextricably linked with PR’s history . Liouville showed that Hamilton’s Equations of Motion preserve volume in phase space i.e. they give rise to a measure - preserving system. The Poincaré Recurrence Theorem must hence apply . But for large collections of particles such as those in a gas , it seemingly contradicts the Second Law of Thermodynamics . The Poincaré Recurrence Theorem

27. The Recurrence Paradox Ernst Zermelo, in his letters to Boltzmann, used this point to argue against the kinetic theory of gases . Boltzmann replied that it was permissible for a system of large number of particles to exhibit low - entropy fluctuations. What is your take on this ? The Poincaré Recurrence Theorem

28. Returning to PR, we can, in a similar spirit, prove the following theorem in ‘topological dynamics’: P Let (U  ),    ( being some indexing set), be an open cover of a topological dynamical system (X,  ), and let k > 0 be an integer. Then there exists an open set U in this cover and a shift n  1 such that U  f n U  …  f (k – 1) n U   (Equivalently, there exist U, n, and a point x such that x, f n x , … , f (k – 1) n x  U .)

29. Further Applications Ergodic theory is the only framework which attempts to understand ‘the structure and randomness of primes’. Topological Dynamics provides many insights in the areas of combinatorics and number theory .

30. Any set of positive integers with positive upper density contains arbitrary large arithmetic progressions. – Szemerédi’s theorem The sequence of primes has arbitrarily large arithmetic progressions. – Green – Tao theorem

31. References Robinson, Dynamical Systems , World Scientific Elaydi , An Introduction to Difference Equations , Springer Milne -Thomson, The Calculus of Finite Differences Shivamoggi , Nonlinear Dynamics and Chaotic Phenomena , Kluwer Academic Terence Tao’s Mathematical Blog: terrytao.wordpress.com Shepelyanski (2010), Poincaré Recurrences in Hamiltonian Systems with Few Degrees of Freedom Dutta (1966), On Poincaré’s Recurrence Theorem Schwartz (1963), A Generalization of Poincaré- Bendixson Theorem to Closed Two - Dimensional Manifolds , AJM, Vol. 85, No. 3 Barreira , Poincaré Recurrence : Old and New

32. Thank you !