ITERATION OF DIFFERENTIABLE FUNCTIONS UNDER m-MODAL MAPS

Abstract

We consider the dynamical system (A,T), where A is a class of differentiable real functions defined on some interval and T:A→A is an operator Tφ:= foφ, where f is a function on the real line. In this work we introduce and develop some techniques of symbolic dynamics for the dynamical system (A,T). We analyze in detail the case in which T arises from a differentiable m-modal map f. In this case we obtain a combinatorial description of the orbits of (A,T) which depends on the combinatorial description of the orbits of one dimensional dynamical system induced by f on the interval.