Persistence of order and structure in chaos

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 C¹ maps from a nontrivial interval X into itself.

Assuming that f is of class C¹⁺ for some > 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 C¹ perturbations of the map f.

Assuming that f is of class C², the map f is C² 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.