Persistence of order and structure in chaos

Abstract

In this paper some results are presented concerning one-dimensional chaotic maps with arbitrarily many critical points. Let f be a chaotic map belonging to some suitable class of C 1 maps from a nontrivial interval X into itself. Assuming that f is of class C 1+α for some α > 0, we have that the set of aperiodic points for f has Lebesgue measure zero; further, if f( X) is bounded then there exists a positive integer p such that almost every point in the interval is asymptotically periodic with period p. Moreover, it will turn out that this asymptotically periodic behaviour in the complicated dynamics of f is persistent under small smooth perturbations. The topological structure of the nonwandering set of f will be described, and this structure is invariant under small C 1 perturbations of the map f. Assuming that f is of class C 2, the map f is C 2 structurally stable provided that f satisfies some suitable conditions. Finally, it will turn out that maps with a negative Schwarzian derivative belong to the suitable class of maps mentioned above.