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 $T$.
Proof:
- n=1 case: Trivial.
- 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:
- 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…