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

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.

41 pages, 3 figures

References in corpus (1)

• Ergodic theory for smooth one-dimensional dynamical systems

Cited by in corpus (4)

• The Hausdorff dimension of multiply Xiong chaotic sets
• Hausdorff measure of sets of distributional chaotic pairs for shift maps
• On "observable" Li-Yorke tuples for interval maps
• Chaotic and Topological Properties of Continued Fractions