On the Finiteness of Attractors for piecewise $C^2$ Maps of the Interval

TL;DR

This paper proves that for almost all points in certain smooth, piecewise maps of the interval, the long-term behavior is limited to a finite set of attractors, including periodic orbits and critical point cycles.

Contribution

It establishes the finiteness of non-periodic attractors for piecewise $C^2$ non-flat interval maps, extending understanding of their long-term dynamics.

Findings

Almost every point's omega-limit set is a periodic orbit, cycle of intervals, or critical point orbit closure.

Every such map has only finitely many non-periodic attractors.

The results apply to Lebesgue almost every initial point in the interval.

Abstract

We consider piecewise $C^2$ non-flat maps of the interval and show that, for Lebesgue almost every point, its omega-limit set is either a periodic orbit, a cycle of intervals or the closure of the orbits of a subset of the critical points. In particular, every piecewise $C^2$ non-flat map of the interval displays only a finite number of non-periodic attractors.