The behaviour of the nonwandering set of a piecewise monotonic interval map under small perturbations

Abstract

In this paper piecewise monotonic maps $T [0,1]\to[0,1]$ are considered. Let $Q$ be a finite union of open intervals, and consider the set $R(Q)$ of all points whose orbits omit $Q$. The influence of small perturbations of the endpoints of the intervals in $Q$ on the dynamical system $(R(Q),T)$ is investigated. The decomposition of the nonwandering set into maximal topologically transitive subsets behaves very unstably. Nonetheless, it is shown that a maximal topologically transitive subset cannot be completely destroyed by arbitrary small perturbations of $Q$. Furthermore it is shown that every sufficiently "big" maximal topologically transitive subset of a sufficiently small perturbation of $(R(Q),T)$ is "dominated" by a topologically transitive subset of $(R(Q),T)$.