Recurrence determinism and Li–Yorke chaos for interval maps

Recurrence determinism, one of the fundamental characteristics of recurrence quantification analysis, measures predictability of a trajectory of a dynamical system. It is tightly connected with the conditional probability that, given a recurrence, following states of the trajectory will be recurrences. In this paper, we study recurrence determinism of interval dynamical systems. We show that recurrence determinism distinguishes three main types of ω-limit sets of zero entropy maps: finite, solenoidal without non-separable points, and solenoidal with non-separable points. As a corollary, we obtain characterizations of strongly non-chaotic and Li–Yorke (non-)chaotic interval maps via recurrence determinism. For strongly non-chaotic maps, recurrence determinism is always equal to one. Li–Yorke non-chaotic interval maps are those for which recurrence determinism is always positive. Finally, Li–Yorke chaos implies the existence of a Cantor set of points with zero determinism.