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$\displaystyle h_{top}(T) \leq \max(0, D \log L).$

The idea of this proof is simple. We can evidently assume ${L>1}$ (if this is not already the case, take ${L}$ very close to 1 and let ${L \rightarrow 1^+}$). Then we need to consider the metric ${d_n(x,y) \equiv \sup_{0 \leq i \leq n-1} d(T^ix,T^iy)}$ and consider minimal ${\epsilon}$-spanning sets with respect to this metric. However, ${d_n \leq L^{n-1} d \leq L^n d}$.

Now consider ${b(L^{-1}\epsilon)}$, the minimal cover of ${X}$ by ${L^{-n}\epsilon}$ balls (with respect to the original metric ${d}$). Then any point of ${X}$ is ${L^{-n}\epsilon}$ close to one of the centers of these balls (with metric ${d}$). The previous remark implies that any point of ${X}$ is ${\epsilon}$-close with the metric ${d_n}$. In particular, the minimal ${\epsilon}$-spanning set of ${(X, d_n)}$ has cardinality at most ${b(L^{-n}\epsilon)}$. Call this ${S(n, d , \epsilon)}$, as before; then

$\displaystyle \limsup \frac{1}{n} \log S(n, d, \epsilon) \leq \limsup \frac{1}{n}\log b(L^{-n} \epsilon).$

Now, by definition, ${\log b(L^{-n}\epsilon)}$ is of the order ${D| \log (L^{-n} \epsilon)|}$. Plugging this into the last expression gives that

$\displaystyle \limsup \frac{1}{n} \log S(n, d, \epsilon) \leq D \limsup \frac{1}{n}( n \log L - \epsilon) = D \log L .$

Letting ${\epsilon \rightarrow 0}$, we get on the left side the topological entropy. This establishes the bound.

Given the remarks made earlier about the ball dimension of a manifold and Lipschitzness of smooth maps, it follows that:

Corollary 2 Let ${M}$ be a compact Riemannian manifold of dimension ${n}$, ${f: M \rightarrow M}$ a smooth map. Then$\displaystyle h_{top}(f) \leq \max(0, n \sup||Df||).$

2. The topological entropy of maps of the circle

We now consider a special (but important) low-dimensional case, namely that where the ground space is the unit circle ${S^1}$. It is well-known from elementary algebraic topology that the homotopy classes of maps ${S^1 \rightarrow S^1}$ are in bijection with the integers, i.e. that the fundamental group of ${S^1}$ is the integers. In this way, we can associate to each continuous ${f: S^1 \rightarrow S^1}$ an invariant, called its degree, which turns multiplicativity to composition. For instance, the map ${z \rightarrow z^k}$ has degree ${k}$.

It is possible, as we now show, to connect this with topological entropy.

Theorem 3 Let ${f; S^1 \rightarrow S^1}$ be continuous and of degree ${k}$. Then ${h_{top}(f) \geq \log |k|}$.

The idea of the proof is simple–it’s basically to reverse the Lipschitz argument above. We shall use the following mini-lemma : Let ${g: S^1 \rightarrow S^1}$ be continuous and suppose we can decompose ${S^1}$ into arcs ${I_1, \dots, I_l}$ such that the image ${f(I_i)}$ of each arc ${I_i}$ has length at most ${\pi}$. Then ${g}$ has degree ${\leq l}$. The reason is as follows. Lift ${g}$ to a map ${G:[0,1] \rightarrow \mathbb{R}}$ such that ${ G(0)=0}$. Then on the pull-back of each arc ${I_i}$ in the interval ${[0,1]}$, ${G}$ can jump by at most 1 because ${g(I_i)}$ is contained in a semicircle.

Now, let’s prove the theorem. Suppose ${h_{top}(f) < \log |k|}$. Pick ${L}$ between the two numbers. Then for ${n}$ large, the smallest ${\pi}$-spanning subset of ${S^1}$ with respect to ${d_n}$ has cardinality at most ${L^n}$. Call this set ${x_1, \dots, x_Q}$. These points are contained in small arcs ${I_i, 1 \leq i \leq Q}$ that cover ${S^1}$ and such that ${f^n(I_i)}$ has length at most ${\pi}$. This is because we can take ${I_i}$ to be the ${\pi}$-neighborhood of ${x_i}$ in the metric ${d_n}$. It now follows by the previous mini-lemma that the degree of ${f^n}$ is at most ${Q \leq L^n}$. But the degree is multiplicative and must therefore be ${k^n >L^n}$. This contradiction establishes the result.

I don’t think I’m going to actually prove it, but I’ll mention another (significantly deeper) version of this idea. First, recall that if ${M}$ is an oriented ${n}$-manifold and ${f: M \rightarrow M}$ continuous, then the top singular cohomology group ${H^n(M, \mathbb{R})}$ is one-dimensional, and the associated map

$\displaystyle f^*: H^n(M, \mathbb{R}) \rightarrow H^n(M, \mathbb{R})$

is called the degree of ${f}$. In the smooth case, which is what we care about, we can construct this (in view of the isomorphism between de Rham and singular cohomology) by taking an ${n}$-form ${\Omega}$ whose integral is nonzero and considering

$\displaystyle \frac{ \int f^* \Omega }{\int \Omega } ;$

this is the degree.

The following is the theorem that I will state without proof:

Theorem 4 Let ${f: M \rightarrow M}$ be a smooth map of an oriented compact, ${n}$-dimensional manifold ${M}$. Then we have ${h_{top}(f) \geq \log(\deg f)}$.

For a proof, I refer you to Chapter 8 of Katok-Hassselblatt’s Introduction to the Modern Theory of Dynamical Systems, which is my primary source for this material.