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 topolog-ically 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 ≥ 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. 1.