On "observable" Li-Yorke tuples for interval maps

TL;DR

This paper investigates the measure-theoretic properties of Li-Yorke tuples in interval maps, revealing how different invariant measures influence the prevalence of such tuples, especially in the context of Manneville-Pomeau maps.

Contribution

It demonstrates that for certain interval maps, the set of Li-Yorke d-tuples can have full measure while higher-order tuples may have zero measure, highlighting nuanced measure-theoretic behavior.

Findings

Li-Yorke d-tuples can have full Lebesgue measure under certain conditions.

The measure of Li-Yorke (d+1)-tuples can be zero even when d-tuples are prevalent.

Manneville-Pomeau maps exemplify complex measure properties of Li-Yorke tuples.

Abstract

In this paper we study the set of Li-Yorke $d$-tuples and its $d$-dimensional Lebesgue measure for interval maps $T\colon [0,1] \to [0,1]$. If a topologically mixing $T$ preserves an absolutely continuous probability measure 9with respect to Lebesgue), then the $d$-tuples have Lebesgue full measure, but if $T$ preserves an infinite absolutely continuous measure, the situation becomes more interesting. Taking the family of Manneville-Pomeau maps as example, we show that for any $d \ge 2$, it is possible that the set of Li-Yorke $d$-tuples has full Lebesgue measure, but the set of Li-Yorke $d+1$-tuples has zero Lebesgue measure.