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
Suryateja Gavva
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 C1, C2, … , Ck
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
a0x(n) + a1x(n + 1) + … + akx(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  .
ThePoincaré-PerronTheorem

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.
ThePoincaré-PerronTheorem

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 a1 and a2 are real.
We see that in general the limit does
not exist.
But for certain particular solutions, it
does.
ThePoincaré-PerronTheorem

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 C1 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.
ThePoincaré-BendixsonTheorem

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.
ThePoincaré-BendixsonTheorem

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.
ThePoincaré-BendixsonTheorem

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.
ThePoincaré-BendixsonTheorem

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  fn(E) for
only finitely many natural numbers
n, has measure zero.
ThePoincaréRecurrenceTheorem

23. Proof of the Theorem
Let E be the given measurable set and An 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  A0, and Al  Am if m < l
• Argue f–n(A0) = An, hence (An) = (A0) for all
integers n.
• Show (EAn)  (A0An) = 0
• Conclude that the measure of the countably
infinite union of EA1, EA2, EA3 and so on is zero
as well.
• Argue that this union is precisely the set of points x
in E such that fn(x)  E for only finitely many n.
ThePoincaréRecurrenceTheorem

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 supn  +(E  fnE)  ((E))2
This follows from the Cauchy-Schwarz Inequality for
integrals, and is a more explicit qualitative
strengthening of the Pigeonhole Principle.
ThePoincaréRecurrenceTheorem

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.
ThePoincaréRecurrenceTheorem

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?
ThePoincaréRecurrenceTheorem

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  fn U  …  f(k – 1)nU  
(Equivalently, there exist U, n, and a
point x such that x, fnx, … ,f(k – 1)nx  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!