On the attractor of piecewise expanding maps of the interval

TL;DR

This paper demonstrates that piecewise expanding maps of the interval are generally stable in their attractor structure under small changes, providing a topological understanding and elementary proof of periodic orbit density.

Contribution

It offers a topological framework for understanding the stability of attractors in piecewise expanding maps and proves the density of periodic orbits with elementary methods.

Findings

Number of ergodic attractors remains stable under perturbations

Periodic orbits are dense in the attractor

Provides a topological description of the attractor

Abstract

We consider piecewise expanding maps of the interval with finitely many branches of monotonicity and show that they are generically combinatorially stable, i.e., the number of ergodic attractors and their corresponding mixing periods do not change under small perturbations of the map. Our methods provide a topological description of the attractor and, in particular, give an elementary proof of the density of periodic orbits.