# 3-periodic point implies positive topological entropy

When I learn some basic ergodic theory, I encounter an interesting exercise. As we all know, 3-periodic point often means chaos. Therefore, when a continuous map has a 3-periodic point, it may have positive topological entropy. The following question exactly talks about it.

Question: Let $$I=[0,1]$$, and $$f$$ be a continuous map from $$I$$ to itself. Suppose $$f$$ have a 3-periodic point, where 3 means minimal period. Then try to prove the topological entropy of $$f$$ is strictly positive.

Well, Sharkovsky's theorem tells us that $$f$$ has arbitrary periodic points, which means "chaos". But I do not know how to prove this exercise.

My idea is to construct a symbolic system as a factor or sub-system by dividing the interval with periodic points, for Bernoulli shift has positive topological entropy. But I fail. Please help me! Thank you!

• I'm not sure if this answer helps. Jan 9, 2022 at 7:51
• Side question, maybe I'm ignorant but is "3-periodic point often means chaos" really true for some types of systems, or are we just talking about the interval? Feb 12, 2022 at 5:56

## 2 Answers

The result you are looking for is contained as a particular case in Theorem 4.58, point ii), in the very nice reference work by Sylvie Ruette on interval topological dynamics.

I add some (hopefully) useful observations. You wrote: "Sharkovsky's theorem tells us that $$f$$ has arbitrary periodic points, which means "chaos" ". In fact, something more precise can be said, as period 3 (famously) implies chaos in the sense of Li-Yorke.

There are also some classical results showing that having a 3-periodic point is in fact a strictly stronger condition than Li-Yorke chaos, because there are interval maps which are Li-Yorke chaotic with zero topological entropy (Second example in Section 5.7 in the book by Ruette). Moreover, positive entropy implies Li-Yorke chaos (Section 5.3 there).

Update: Doing some research I just found out the following two facts, which can be of interest for the OP: the assumption of continuity can be in fact generalized in two ways:

1. "functions whose graph is a connected, $$G_\delta$$ set" as proved in Theorem 1.5 in Čiklová, M. (2005).

2. "Darboux functions" as proved in Natkaniec et al. (2010)

See Theorem 1 in The periodic points of maps of the disk and the interval. It states that if a continuous map $$f:[0,1]\rightarrow [0,1]$$ has a point of period $$n$$, and $$n$$ is not a power of two, then $$h_{\rm{top}}(f)>\frac{1}{n}\log 2$$. The proof establishes the lower bound via constructing a suitable $$(n,\epsilon)$$-separated subset for $$f$$. Also see Milnor and Thurston's paper on kneading theory which is concerned with interval maps with finitely many monotonic pieces. In sections 8-10, it relates the periodic points to the so called kneading invariant of $$f$$ which captures where in the interval the turning points (local extrema) go under iteration. Corollary 9.6 therein is the inverse of the result mentioned above: If $$s:={\rm{e}}^{h_{\rm{top}}(f)}>1$$, then $$f$$ has a periodic point whose period is not a power of two. Finally, the idea of constructing a Markov partition and studying the associated shift works if you further assume that the turning points of $$f$$ are eventually periodic. In that case, the turning points (local extrema) and their forward orbits cut the interval into finitely many subintervals. Each of the subintervals is mapped onto other via $$f$$. You can write down the transition matrix and compute the entropy as the logarithm of the leading eigenvalue of the matrix. For instance, if you have a unimodal map (a map with just one turning point), if the turning point is of period three, you can compute the entropy as $$\log\left(\frac{1+\sqrt{5}}{2}\right)$$.