Symbolic dynamics for iterated smooth functions

Abstract

We consider the discrete dynamical system (A,T), where A is a class of smooth real functions defined on some interval and T:A→A is an operator Tφ:=foφ, where f is a function on the real line. Iteration of smooth maps appears naturally in the study of continuous difference equations and boundary value problems such as those in Romanenko et al [2], Severino et al [3], Sharkovsky et al [4]-[5], Vinagre et al [6]. In our work we develop some techniques of symbolic dynamics for the discrete dynamical system (A,T). We analyze in detail the case in which T arises from unimodal interval maps.