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 $latex T$.
proof:
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 map.is 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…