---
tag: M44J4ZJ6Z8
kind: definition
name: Topological entropy
msc: [37B40]
defines: BDZYBXBFDE
uses: [0TVJZYB6SH, 0VKD1TP7F3, 0VGFHTM75Y, 04DXAFN2ES, 049PXYHW1T, A1XE59ZHNM]
sources:
  - url: https://en.wikipedia.org/wiki/Topological_entropy
---

## Definition

Let $(X, d)$ be a [[compact|0TVJZYB6SH]] [[metric space|0VKD1TP7F3]] and let $f \colon X \to X$ be a [[continuous map|0VGFHTM75Y]].
For an integer $n \ge 1$, define the metric $d_n(x, y) = \max_{0 \le i < n} d(f^i(x), f^i(y))$ on $X$.
For $\varepsilon > 0$, a set $E \subseteq X$ is *$(n, \varepsilon)$-separated* if $d_n(x, y) > \varepsilon$ for all distinct $x, y \in E$.
Let $s(n, \varepsilon)$ be the largest size of an $(n, \varepsilon)$-separated set, which is finite by compactness.
The *topological entropy* of $f$ is
$$h_{\mathrm{top}}(f) = \lim_{\varepsilon \to 0} \limsup_{n \to \infty} \frac{1}{n} \log s(n, \varepsilon) \in [0, \infty],$$
where $\log$ is the [[natural logarithm|04DXAFN2ES]].

## Remarks

The limit in $\varepsilon$ exists because the $\limsup$ grows as $\varepsilon$ decreases.
The value depends only on the topology, not on the metric.
Adler, Konheim and McAndrew defined it on any compact space through the growth of joins of [[open covers|049PXYHW1T]], and this gives the same number.
Topologically conjugate maps have equal entropy, and $h_{\mathrm{top}}(f^k) = k \, h_{\mathrm{top}}(f)$ for $k \ge 1$.
Example: the full one-sided [[shift|A1XE59ZHNM]] $\sigma$ on $\set{0, \ldots, k-1}^{\N}$ with $d(x, y) = 2^{-\min\set{i : x_i \neq y_i}}$.
Then $d_n(x, y) > 2^{-m}$ exactly when $x$ and $y$ differ at some coordinate $i < n + m - 1$, so $s(n, 2^{-m}) = k^{n + m - 1}$ and $h_{\mathrm{top}}(\sigma) = \log k$.
A map with zero entropy: an isometry has $d_n = d$, so $s(n, \varepsilon)$ does not depend on $n$ and $h_{\mathrm{top}} = 0$.

## Explanations

### L1 human

Topological entropy is the exponential growth rate of the number of orbit segments that can be told apart at a fixed small resolution.