On the Lebesgue measure of Li-Yorke pairs for interval maps

TL;DR

This paper studies the measure-theoretic properties of Li-Yorke pairs in multimodal interval maps, showing most such pairs are measure-zero, and characterizes conditions under which typical pairs are Li-Yorke or have dense orbits.

Contribution

It establishes measure-zero results for scrambled and wandering sets, and provides a trichotomy for Li-Yorke pairs in maps with Cantor attractors, using new techniques involving nice neighborhoods and Markov maps.

Findings

Measurable scrambled sets have zero Lebesgue measure.

Strongly wandering sets have zero Lebesgue measure.

In topologically mixing maps without Cantor attractors, typical pairs are Li-Yorke.

Abstract

We investigate the prevalence of Li-Yorke pairs for $C^2$ and $C^3$ multimodal maps $f$ with non-flat critical points. We show that every measurable scrambled set has zero Lebesgue measure and that all strongly wandering sets have zero Lebesgue measure, as does the set of pairs of asymptotic (but not asymptotically periodic) points. If $f$ is topologically mixing and has no Cantor attractor, then typical (w.r.t. two-dimensional Lebesgue measure) pairs are Li-Yorke; if additionally $f$ admits an absolutely continuous invariant probability measure (acip), then typical pairs have a dense orbit for $f \times f$. These results make use of so-called nice neighborhoods of the critical set of general multimodal maps, and hence uniformly expanding Markov induced maps, the existence of either is proved in this paper as well. For the setting where $f$ has a Cantor attractor, we present a trichotomy explaining when the set of Li-Yorke pairs and distal pairs have positive two-dimensional Lebesgue measure.