Evolution and distribution of the periodic critical values of iterated differentiable functions

Abstract

We consider the dynamical system (A,T), where A is a class of differentiable functions defined on some interval and T:A→A is the operator Tφ:=foφ, where f is a differentiable m-modal map. For the particular case of f being a topologically exact map we study the growth rate of critical points of the iterated functions. Considering functions in A whose critical values are periodic points for f, we analyze the evolution as well as the distribution of the periodic critical values of the iterated functions. For this, using only the kneading invariant of f , we developed an algorithm for computing the itineraries of the critical values of these functions.