# 12. Chaos Versus Positive Entropy

Chaos and positive entropy are two different ways of measuring the amount of complexity (or instability) in a dynamical system. Naively one might infer that the two concepts are related.

However in fact they are quite different: chaos is a *global* property, in the sense that it makes an assumption on the dynamics of $f$ across the entire space $X$. Meanwhile positive topological entropy is a *local* property, in the sense that if $f$ has positive entropy on a (possibly small) invariant set, then it has positive entropy on the entire space. This means that on many spaces $X$, it is possible to construct non-chaotic systems with arbitrarily large topological entropy.

Going the other way, on the space $ \Sigma_2$ of $(0,1)$-sequences, there exist chaotic systems with zero entropy. Meanwhile on the circle $S^1$, it is known that every chaotic system has positive entropy, but that there exist chaotic systems with arbitrarily small entropy. However for $S^2$ or $ \mathbb{T}^2$ it is an open problem whether there exist chaotic systems with zero entropy!

Today (and next week) we concern ourselves solely with the interval. Here things are much simpler. Our main result is that if $f \colon [0,1] \to [0,1]$ is a chaotic dynamical system then $\mathsf{h}_{\operatorname{top}}(f) \ge \log \sqrt{2}$. Conversely on the interval a system with positive entropy has a closed invariant subset on which it is chaotic (although we won't prove this).

Thus on the interval positive entropy *is* (up to restricting to an invariant set) equivalent to chaos.

Comments and questions?