Interval map

Period 3 Induces Chaos

Theorem: If an interval map $T:I \to I$ has a period 3 point $x$, then for all $n \in \mathbb{N}^*$, there is a period $n$ point for $T$.


  1. n=1 case: Trivial.
  2. n>1, n ≠ 3 case: The key point is to consider the structure of the monotone interval containing the previous one with a fixed length. This will easily lead to the proof.

A Work of J. Milnor and W. Thurston

$N(T^n)$ is defined as the number of monotone intervals of the map $T^n$.

Theorem: $h(T) = \lim_{n \to \infty} \frac{1}{n} \log N(T^n)$.

Monotone Markov Map

This structure has two properties:

  1. Piecewise monotone and $C^1$, the derivative has control.

There is a relatively dynamic system with this map, which is a shift map with a relative $n \times n$ matrix $A$.

These two dynamic systems have a lot of relations, the key one being: the topological entropy of the monotone Markov map is just the unique maximum eigenvalue of $A, and some byproducts…