Let $f \colon X \to X$ be a dynamical system on a compact metric space. In this lecture we introduce the topological entropy $\mathsf{h}_{\operatorname{top}}(f)$ of $f$.

This is a non-negative real number (or $\infty$), which attempts to give a quantitative measure of how “ complex” the dynamics of $f$ are. As one might expect, trying to reduce the entire dynamics of $f$ to a single number is only partially successful.

As a first step we show that topological entropy is a dynamical invariant: if $f$ is conjugate to $g$ then $\mathsf{h}_{\operatorname{top}}(f) =\mathsf{h}_{\operatorname{top}}(g)$.

Next lecture we will compute the topological entropy of some of our standard “model” dynamical systems.

For now though, let us summarise the most important property of topological entropy in a single handy table:

If the entropy is: The dynamics are usually:
$\mathsf{h}_{ \operatorname{top}}(f)=0$ Simple
$\mathsf{h}_{ \operatorname{top}}(f) >0$ Complex
$\mathsf{h}_{ \operatorname{top}}(f) =\infty$ 🤯