We continue the discussion of the cohomological equation started yesterday and explain a situation where it can be solved.﻿

Given a compact space ${X}$, a continuous map ${T: X \rightarrow X}$, and a continuous ${g: X \rightarrow \mathbb{R}}$ with vanishing periodic data as in yesterday’s post, there are not many ways to construct a solution ${f}$ of the cohomological equation

$\displaystyle g = f \circ T - f.$

The basic thing to note that if ${f(x)}$ is known, then recursively we can determine ${f}$ on the entire orbit of ${x}$ in terms of ${g}$. In case the map ${T}$ is topologically transitive, say with a dense orbit generated by ${x_0}$, then by continuity the entire map ${f}$ is determined by its value on ${x_0}$.

This also provides the method for obtaining ${f}$ in the topologically transitive case. Namely, one picks ${f(x_0)}$ aribtrarily, defines ${f(T^ix_0)}$ in the only way possible by the cohomological equation. In this way one has ${f}$ defined on the entire orbit ${T^{\mathbb{Z}}(x_0)}$ such that on this orbit, the equation is satisfied. If one can show that ${f}$ is uinformly continuous on ${T^{\mathbb{Z}}(x_0)}$, then it extends to the whole space and must by continuity satisfy the cohomological equation there too.

This is the strategy behind the proof of the theorem of Livsic from the seventies, whose proof we shall sketch:

Theorem 1 (Livsic) Let ${M}$ be a compact Riemannian manifold, ${T: M \rightarrow M}$ a topologically transitive Anosov diffeomorphism. If ${g: M \rightarrow \mathbb{R}}$ is an ${\alpha}$-Holder function such that ${T^n p =p}$ implies ${\sum_{i=0}^{n-1} g(T^i p) = 0}$, then there exists an ${\alpha}$-Holder ${f: M \rightarrow \mathbb{R}}$ such that

$\displaystyle g = f \circ T -f .$

We start by considering a very simple problem. Let ${X}$ be a set, ${T: X \rightarrow X}$ be a bijection function, and ${g: X \rightarrow \mathbb{R}}$ a function. We want to know when the cohomological equation

$\displaystyle g = f \circ T - f$

can be solved for some ${f: X \rightarrow \mathbb{R}}$.

It turns out that this very simple question has an equally simple answer. The answer is that the equation can be solved if and only if for every finite (i.e., periodic) orbit ${O \subset X}$, we have ${\sum_{x \in O} g(x) = 0}$. The necessity of this is evident, because if we have such a solution, then

$\displaystyle \sum_O g(x) = \sum_0 f \circ T(x) - f(x) = \sum_O f(x) - \sum_O f(x) = 0$

because ${T}$ induces a bijection of ${O}$ with itself. This condition is called the vanishing of the periodic obstruction.

We further continue the discussion of topological entropy. Here, we discuss various results that bound above and below the entropy of a given map.

1. The topological entropy of Lipschitz maps

Many of the dynamical systems of interest are actually given by compact manifolds ${M}$ and smooth maps ${T: M \rightarrow M}$. These are always Lipschitz with respect to appropriate metrics. Indeed, choose a Riemannian metric on ${M}$ and let ${||\cdot||}$ denote the induced norm on the tangent spaces. Then ${\sup ||DT||}$ is a Lipschitz constant for ${T}$ with respect to the metric on ${M}$ induced by the Riemannian metric. In this case, the entropy is always finite. We shall prove this in a more general context.

Let ${X}$ be a compact metric space, and for ${\epsilon>0}$, let ${b(\epsilon)}$ denote the the number of ${\epsilon}$-balls necessary to cover ${X}$ (which is always finite). Then we call

$\displaystyle \limsup_{\epsilon \rightarrow 0} \frac{ \log b(\epsilon) }{ |\log \epsilon |}$

the ball dimension of ${X}$. For instance, an ${n}$-cube has ball dimension ${n}$. It follows more generally that a Riemannian ${n}$-manifold has ball dimension ${n}$.

The reason we shall use this concept below is that ${b(\epsilon)}$ gives an upper bound for a minimal ${\epsilon}$-spanning set of the space ${X}$. (Recall that an ${\epsilon}$-spanning set means that every point is ${\epsilon}$-close to it.) In fact, if the ${\epsilon}$-balls ${B_1, \dots, B_q}$ cover ${X}$, then the centers of these form an ${\epsilon}$-spanning set.

Theorem 1 Let ${X}$ be a compact metric space with finite ball dimension ${D}$. Suppose ${T:X \rightarrow X}$ is a Lipschitz continuous mapping with constant ${L}$, i.e. ${d(Tx,Ty) \leq Ld(x,y)}$ for all ${x,y \in X}$. Then the entropy of ${T}$ is finite and$\displaystyle h_{top}(T) \leq \max(0, D \log L).$ (more…)

We continue the discussion of topological entropy started yesterday.

-1. Basic properties-

So, recall that we attached an invariant ${h_{top}(T) \in [0, \infty]}$ to a transformation ${T: X \rightarrow X}$ of a compact metric space ${X}$. We showed that it was purely topological, i.e. invariant under semiconjugacies. However, we have yet to establish its basic properties and compute some examples.

In general, we can’t expect topological entropy to be additive, i.e. ${h_{top}(T \circ S) = h_{top}(T) + h_{top}(S)}$, even if ${T}$ and ${S}$ commute. The reason is that the identity—or any isometry—has zero entropy, while there are homeomorphisms with nonzero entropy.

However, we do have:

Theorem 1 If ${m \in \mathbb{Z}}$, then ${h_{top}(T^m) = |m| h_{top}(T)}$.

(Here if ${m \geq 0}$, this makes sense even for ${T}$ noninvertible.)

We handle the two cases ${m>0}$ and ${m=-1}$ (which together imply the result). In each, we will use the second definition of entropy that we gave in terms of coverings (which historically actually came first). Namely, the definition was

$\displaystyle h_{top}(T) = \sup_{\mathfrak{A}} \lim_{n} \frac{1}{n} \log \mathcal{N}( \mathfrak{A} \vee T^{-1}\mathfrak{A} \vee \dots \vee T^{-n+1}\mathfrak{A}).$ (more…)

In the theory of dynamical systems, it is of interest to have invariants to tell us when two dynamical systems are qualitatively “different.” Today, I want to talk about one particularly important one: topological entropy.

We will be in the setting of discrete dynamical systems: here a discrete dynamical system is just a pair ${(T,X)}$ for ${X}$ a compact metric space and ${T: X \rightarrow X}$ a continuous map.

Recall that two such pairs ${(T,X), (S,Y)}$ are called topologically conjugate if there is a homeomorphism ${h: X \rightarrow Y}$ such that ${T = h^{-1}Sh}$. This is a natural enough definition, and it is clearly an equivalence relation. For instance, it follows that there is a one-to-one correspondence between the orbits of ${T}$ and those of ${S}$. In particular, if ${T}$ has a fixed point, so does ${S}$. Admittedly this necessary criterion for determining whether ${T,S}$ are topologically conjugate is rather trivial.

Note incidentally that topological conjugacy needs to be considered even when one is studying smooth dynamical systems—in many cases, one can construct a homeomorphism ${h}$ as above but not a diffeomorphism. This is the case in the Hartman-Grobman theorem, which states that if ${f: M \rightarrow M}$ is a smooth map with a fixed point where the derivative is a hyperbolic endomorphism of the tangent space, then it is locally conjugate to the derivative (that is, the corresponding linear map). (more…)

Today, we will apply the technical lemma proved yesterday to proving a few special properties of Anosov diffeomorphisms. The first one states that if you have an approximate orbit, then you can approximate it by a real orbit. This may not sound like much, but it is false for isometries—and in fact, it gives another way of proving the structural stability result.

As usual, start with a compact manifold ${M}$ and an Anosov diffeomorphism ${f: M \rightarrow M}$. We can put a metric ${d}$ on ${M}$ (e.g. by imbedding ${M}$ in euclidean space, or using a Riemannian metric, etc.). To formalize this, fix ${\delta>0}$. We introduce the notion of a ${\delta}$-orbit. This is a two sided sequence ${\{x_n\}_{n \in \mathbb{Z}}}$ such that ${d(x_{n+1}, f(x_n)) \leq \delta}$.

Theorem 1 (Anosov shadowing lemma) Fix ${\epsilon>0}$ sufficiently small. There is ${\delta>0}$ such that any ${\delta}$-orbit ${\{x_n\}}$ can be shadowed by a unique real orbit ${\{y_n\}}$, i.e. ${y_{n+1} = f(y_n)}$ and ${d(x_n, y_n) < \epsilon}$ for all ${n \in \mathbb{Z}}$. (more…)

We now state and prove the ugly technical theorem invoked yesterday, that you can refine certain “approximate” solutions of conjugacy-like equations involving Anosov diffeomorphisms (and maps close to them—though actually one can prove that Anosov diffeomorphisms are open in the $C^1$ topology). The proof is rather complicated, but it will basically rely on familiar techniques: hyperbolic linearization (in Banach spaces!), the contraction principle, and simple algebraic manipulation.

Theorem 1 Let ${f}$ be an Anosov diffeomorphism of the compact manifold ${M}$. Then if ${\delta>0}$ is sufficiently small, there is ${\epsilon>0}$ satisfying the following condition. Suppose ${d_{C^1}(f,g)<\epsilon}$, and one has an “approximately commutative diagram” for a map $\phi: X \to M$:

with ${X}$ a topological space and ${h: X \rightarrow X}$ a homeomorphism: i.e. ${d(g \circ \phi, \phi \circ h)< \epsilon}$. Then there is a unique continuous ${\psi: X \rightarrow U}$ close to ${\phi}$ (namely ${d(\psi, \phi)<\delta}$) such that the modified diagram

commutes exactly.

So, how are we going to prove this? First, we want some sort of linearity, but we can’t add two elements of a manifold. Thus, we use the Whitney embedding theorem to assume without loss of generality that ${M}$ is a closed submanifold of ${\mathbb{R}^N}$. (more…)

I now know what I’m working on for my REU project; I’ll be studying (with two other undergraduates) a type of cohomology for dynamical systems. Misha Guysinsky, our mentor, has not explained the specific problem yet—perhaps that’ll come when we meet with him on Thursday. So I’ve spent the last weekend trying to learn a few basic facts about (especially hyperbolic) dynamical systems, which I will try to explain here.

1. Why do we care about hyperbolicity?

So, first a definition: let ${f: M \rightarrow M}$ be a ${C^1}$-morphism of a smooth manifold ${M}$. Suppose ${p \in M}$ is a fixed point. Then ${p}$ is called hyperbolic if the derivative ${Df_p: T_p(M) \rightarrow T_p(M)}$ has no eigenvalues on the unit circle. This comes from linear algebra: an endomorphism of a vector space is called hyperbolic if its eigenvalues are off the unit circle. Hyperbolicity is an important condition in dynamics, and I want to illustrate this with a few examples. (more…)

So, let’s fix a compact metric space ${X}$ and a transformation ${T: X \rightarrow X}$ which is continuous. We defined the space ${M(X,T)}$ of probability Borel measures which are ${T}$-invariant, showed it was nonempty, and proved that the extreme points correspond to ergodic measures (i.e. measures with respect to which ${T}$ is ergodic). We are interested in knowing what ${M(X,T)}$ looks like, based solely on the topological properties of ${T}$. Here are some techniques we can use:

1) If ${T}$ has no fixed points, then ${\mu \in M(X,T)}$ cannot have any atoms (i.e. ${\mu(\{x\})=0, x \in X}$). Otherwise ${\{x, Tx , T^2x, \dots \}}$ would have infinite measure.

2) The set of recurrent points in ${X}$ (i.e. ${x \in X}$ such that there exists a sequence ${n_i \rightarrow \infty}$ with ${T^{n_i}x \rightarrow x}$) has ${\mu}$-measure one. We proved this earlier.

3) The set of non-wandering points has measure one. We define this notion now. Say that ${x \in X}$ is wandering if there is a neighborhood ${U}$ of ${X}$ such that ${T^{-n}(U) \cap U = \emptyset, \forall n \in \mathbb{N}}$. In other words, the family of sets ${T^{i}(U), i \in \mathbb{Z}_{\geq 0}}$ is disjoint. If not, say that ${x}$ is non-wandering. Any recurrent point, for instance, is non-wandering, which implies that the set of non-wandering points has measure one.

Here is an example. (more…)

So, now it’s time to connect the topological notions of dynamical systems with ergodic theory (which makes use of measures).  Our first example will use the notion of topological transitivity, which we now introduce.  The next example will return to the story about recurrent points, which I talked a bit about yesterday.

Say that a homeomorphism ${T: X \rightarrow X}$ of a compact metric space ${X}$ is topologically transitive if there exists ${x \in X}$ with ${T^{\mathbb{Z}}x}$ dense in ${X}$.  (For instance, a minimal homeomorphism is obviously topologically transitive.)  Let ${\{ U_n \}}$ be a countable basis for the topology of ${X}$. Then the set of all such ${x}$ (with ${T^{\mathbb{Z}}x}$ dense) is given by

$\displaystyle \bigcap_n \bigcup_{i \in \mathbb{Z}} T^i U_n.$

In particular, if it is nonempty, then each ${\bigcup_{i \in \mathbb{Z}} T^i U_n}$ is dense—being ${T}$-invariant and containing ${U_n}$—and this set is a dense ${G_{\delta}}$ by Baire’s theorem.

Proposition 1 Let ${X}$ have a Borel probability measure ${\mu}$ positive on every nonempty open set, and let ${T: X \rightarrow X}$ be measure-preserving and ergodic. Then the set of ${x \in X}$ with ${\overline{T^{\mathbb{Z}}x}=X}$ is of measure 1, so ${T}$ is topologically transitive.

Indeed, each ${\bigcup_{i \in \mathbb{Z}} T^i U_n}$ must have measure zero or one by ergodicity, so measure 1 by hypothesis. Then take the countable intersection.

Poincaré recurrence

We now move to the abstract measure-theoretic framework, not topological.

Theorem 2 (Poincaré recurrence) Let ${T: X \rightarrow X}$ be a measure-preserving transformation on a probability space ${X}$. If ${E \subset X}$ is measurable, then there exists ${F \subset E}$ with ${\mu(E-F)=0}$ such that for each ${x \in F}$, there is a sequence ${n_i \rightarrow \infty}$ with ${T^{n_i} x \in E}$.

In other words, points of ${F}$ are ${T}$-frequently in ${E}$. (more…)