Topological Entropy

There are several qualitative invariants of dynamical systems, such as density of periodic points, topological mixing, hyperbolicity. The answer whether a concrete dynamical system has this property or not would be “Yes” or “No”. There do exists quantitative invariants, for example the growth of period points. But perhaps the most important quantitative invariant is a topological entropy.

Definition. In some sense, the topological entropy describes the growth of orbits all at the same time. To be precise, for two points $x,y \in X$ one can define the metric

$\displaystyle d_n^f(x)=max_{0 \leq i \leq n-1} d(f^i(x),f^i(y))$

for a given positive integer $n$. Let $B(x,r,n)=\{y \in X: d_n^f(x,y), the ball of radius $r$. Let $S_d(f,r,n)$ be the minimal number of such ball, sufficient to cover the whole of $X$.

Then

$\displaystyle h_{top}(f)=\lim_{r \to 0} \lim_{n \to \infty} \frac{ \log S_d(f,r,n)}{n}$  .

There are several other definitions for the topological entropy. Sometimes it is not quite useful to use covering sets for computation reasons. For example, one can use the so-called separating sets, i.e. the sets such that the pairwise distance between any two points in the metric $d_n^f > r$  . The cardinality of the maximal separating set is denoted by $N_d(f,r,n)$ and can be used similarly for the definition of the topological entropy, substituting $S_d(f,r,n)$ by $N_d(f,r,n)$ in the formula above.

For example, for the expanding map $E_m(x)=mx \quad mod (1)$, the topological entropy $h_{top}(E_m)=\log|m|$.

Proof. In general, for expanding maps the distance between two points growth until it becomes larger than some constant ($1/2m$ for $E_m$). Let us choose two points $x,y$ with $d(x,y)<\frac{1}{2m^n}$.Then

$d_n^{E_m}(x,y)=d(E_m^{n-1}(x),E_m^{n-1}(x)) = m^{n-1}d(x,y)$.

If we want to have

$d_n^{E_m}(x,y)> m^{-k}$, then we must have $d(x,y)>m^{-k-n}$.

Hence if we choose $S=\{i\cdot m^{-k-n}: 0\leq i \leq m^{k+n}-1 \}$ all the points from $S$ will have pairwise distances between each other at least $m^{-k}$. And the set $S$ can serve as a separating set with $|S|=m^{k+n}$, we obtain that

$h_{top}(E_m)=\log |m|$.

Bowen has asked to find a topological entropy of the complex polynomial of degree $d$. More generally M.Lyubich gave the answer for any rational function in the complex plane. The lower bound $h_{top}(f) \geq \log (deg f)$ was known before due to Misiurewisz-Przytycki theorem.

Theorem. Let $f: S^2 \to S^2$ be a rational function, that is not a constant. Then the topological entropy
$h_{top}(f)=\log (deg f)$.