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

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

It follows that

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

If we choose an arbitrary cover ${\mathfrak{B}}$ and take ${\mathfrak{A}}$ to be the refined cover ${\mathfrak{B} \vee T^{-1}\mathfrak{B} \vee \dots T^{-m+1}\mathfrak{B}}$, it follows that the terms ${\frac{1}{n} \log \mathcal{N}( \mathfrak{A} \vee T^{-1}\mathfrak{A} \vee \dots \vee T^{-n+1}\mathfrak{A})}$ are just ${ \frac{m}{mn} \log \mathcal{N}( \mathfrak{B} \vee T^{-m}\mathfrak{B} \vee \dots \vee T^{-nm+1}\mathfrak{B})}$. Since ${\mathfrak{B}}$ was arbitrary, it follows that ${mh_{top}(T) \leq h_{top}(T^m)}$. But since when considering covers ${\mathfrak{A}}$ to take the sup over, we can always pass to refinements—and in particular, we can assume that ${\mathfrak{A}}$ is of the form ${\mathfrak{B} \vee T^{-1} \mathfrak{B} \vee \dots T^{-m+1}\mathfrak{B}}$ for some ${\mathfrak{B}}$, it follows equally that ${h_{top}(T^m) \leq mh_{top}(T)}$.

For ${m=-1}$, it is proved similarly; the key idea is that

$\displaystyle \mathcal{N}( \mathfrak{A} \vee T^{-1} \mathfrak{A} \vee \dots T^{-n+1 } \mathfrak{A}) = \mathcal{N}( T^{n-1}\mathfrak{A} \vee T^{n-2}\mathfrak{A} \vee \dots \vee \mathfrak{A})$

by applying the homeomorphism ${T^{n-1}}$ to the whole space.

There is another useful result when we have a decomposition of ${X}$ into finitely many invariant subspaces:

Theorem 2 Suppose ${T: X \rightarrow X}$ and ${X = \bigcup X_i}$ for the ${X_i}$ a finite collection of invariant, closed subsets. Then ${h_{top}(T) = \max_i h_{top}(T|{X_i})}$.

One direction is easy to see. Namely, if we have an open cover for some ${X_i}$, then we extend it (via ${X-X_i}$) to the whole space. It is then easy to see from this argument that for each ${i}$, ${h_{top}(T|_{X_i}) \leq h_{top}(T)}$. The other direction can be proved simliarly (an open cover covers ${X}$ iff it covers each ${X_i}$), and I won’t spend much more time on the straightforward details.

-2. Example-

We’d now like to actually compute the entropy of something. A canonical example of a dynamical system is given by the rotations of the circle. But these are isometries (with respect to a natural metric), so they all have entropy zero. We will try a more interesting example.

Instead, we will consider an example that begins the field of symbolic dynamics. To do this, fix a finite set ${\mathcal{F}}$ which we endow with the discrete topology. Then the space ${\Sigma = \mathcal{F}^{\mathbb{Z}}}$ with the product topology is compact and admits a homeomorphism ${T: \Sigma \rightarrow \Sigma}$ such that ${T}$ “shifts” the double-sided sequence ${\{x_n\}_{n \in \mathbb{Z}} \subset \mathcal{F}}$ by one (in whichever direction you want, but we’ll say to the right).

When computing the topological entropy of ${T}$, we will need to fix open covers of ${\Sigma}$. A good way to get open covers is as follows. Fix ${l and for ${a_{l}, \dots, a_{k} \in \mathcal{F}}$, consider the open sets ${O_{l,k}^{a_{l}\dots a_{k}}}$ defined such that a bisequence belongs to ${O_{l,k}^{a_{l}\dots a_{k}}}$ if and only if the value at ${i}$ is ${a_i}$ for ${l \leq i \leq k}$. These sets are open by definition of the product topology, and for ${l,k}$ fixed, the ${|\mathcal{F}|^{l-k}}$ open sets cover ${\Sigma}$.

Now these sets form a basis, so any open cover of ${\Sigma}$ can be refined to one of the form ${\{O_{l,k}^{a_l \dots a_k}, a_l \dots a_k \in \mathcal{F} \}}$, if ${l}$ is chosen sufficiently large negative and ${k}$ large positive. Call this covering ${\mathfrak{A}_{l,k}}$. It is easy to see that ${T^{-1} \mathfrak{A}_{l,k} = \mathfrak{A}_{l-1,k-1}}$. In particular,

$\displaystyle \mathfrak{A}_{l,k} \vee T^{-1} \mathfrak{A}_{l,k} \vee \dots \vee T^{-n+1} \mathfrak{A}_{l,k} = \mathfrak{A}_{l-n+1, k}.$

(Certain sets will come up more htan once, but we only include each once.) The covering ${\mathfrak{A}_{a,b}}$ is always disjoint, and it contains precisely ${|\mathcal{F}|^{b-a+1}}$ elements. So ${\log \mathcal{N}(\mathfrak{A}_{a,b}) = (b-a+1) \log |\mathcal{F}|}$.

Now, it is easy to see from this computation that the entropy of ${T}$ is ${\log |\mathcal{F}|}$.

-3. The periodic points-

Consider now an expansive homeomorphism ${T: X \rightarrow X}$. This means that there exists ${\epsilon>0}$ such that if ${x,y}$ are distinct, then there exists ${k \in \mathbb{Z}}$ with ${d(T^kx, T^ky) \geq \epsilon}$. In other words, no matter how precisely we know the value of ${x}$, we can’t know the value of the entire orbit with rpecision better than ${\epsilon}$—this is a form of chaotic behavior.

An example of an expansive homeomorphism is given by any Anosov diffeoorphism, as we discussed—this is a corollary of the uniqueness in the shadowing lemma. There are actually no expansive homeomorphisms of the unit circle ${S^1}$; there are expansive maps, but they are all topologically conjugate to some ${z \rightarrow z^{k}}$ for ${|k|>1}$, hence not homeomorphisms.

In this case, we shall connect the topological entropy of ${T}$ with the distribution of periodic points. We define

$\displaystyle p(T) = \limsup_{n \rightarrow \infty} \frac{1}{n} \log \# \{ p \ s.t. \ T^n p = p\}.$

Then ${p(T)}$ can be connected with the radius of convergence of the “zeta function” ${\sum P_n z^n}$, where ${P_n}$ is the number of points of order ${n}$.

The theorem we want to prove:

Theorem 3 $\displaystyle p(T) \leq h_{top}(T)$

for expansive ${T}$.

Indeed, we shall show that

$\displaystyle p(T) \leq \limsup \frac{1}{n} \log S'(n, \epsilon, T)$

where ${\epsilon}$ is an expansive constant for ${T}$, and ${S'(n, \epsilon,T)}$ is the cardinality of the maximal ${\epsilon}$-separated set for ${X}$ under the metric ${d_n}$ as defined in the definition of topological entropy.

To do this, we show

$\displaystyle P_n(T) \leq S'(n, \epsilon, T).$

Suppose ${x,y}$ both satisfy ${T^nx = x, T^ny = y}$ and are distinct; I claim that ${x,y}$ are ${\epsilon}$-separated with respect to the metric ${d_n}$. Indeed, this states that ${d(T^i(x), T^i(y)) \geq \epsilon}$ for some ${i = 0, 1, \dots, n-1}$. But by expansiveness, we have ${d(T^i(X), T^i(y)) \geq \epsilon}$ for some ${i \in \mathbb{Z}}$; by periodicity, we can assume ${i \in [0, n-1]}$ and this establishes the claim and the result.