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.

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