The evolution and distribution of the periodic critical values of iterated differentiable functions

We consider the dynamical system (A,T)(A,T), where AA is a class of differentiable functions defined on some interval and T:A→AT:A→A is the operator Tϕ≔f∘ϕTϕ≔f∘ϕ, where ff is a differentiable mm-modal map. For the particular case of ff being a topologically exact map we study the growth rate of critical points of the iterated functions. Considering functions in AA whose critical values are periodic points for ff, 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 ff, we developed an algorithm for computing the itineraries of the critical values of these functions.