Interval map

1.period 3 induce chaos

theorem:if a interval map $latex T:I\to I$ have a period 3 point $latex x$,then $latex \forall n\in N^*$,there is a period n point for $latex T$.


n=1 case. trivial

$latex n>1,n\neq 3$ case:

the key point is to consider the structure of monotone interval contain previous one with fix length.

this will easy to lead a proof.


2.a work of J.Milnor and W.Thurston.

$latex N(T^n)$ defined as the number of monotone interval of the map $latex T^n$.

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


3.monotone Markov map

this structure have two property:

1.piesewise monotone and $C^1$,the derive has control!


there is a relative dynamic system with this a shift map with a relative $latex n\times n$ matrix $latex A$.

this two dynamic system have a lot of relation,the key one is:

the topological entropy of monotone Markov map is just the unique maximum eigenvalue of $latex A$.

and some byproduct…