On ‘observable’ Li–Yorke tuples for interval maps

Abstract

In this paper we study the set of Li–Yorke d-tuples and its d-dimensional Lebesgue measure for interval maps T : [0, 1] → [0, 1]. If a topologically mixing T preserves an absolutely continuous probability measure (with 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 an example, we show that for any d ⩾ 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.