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 $\omega$-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

Strong laws for recurrence quantification analysis

The recurrence rate and determinism are two of the basic complexity measures studied in the recurrence quantification analysis. In this paper, the recurrence rate and determinism are expressed in terms of the correlation sums, and strong laws of large numbers are given for them