Recurrence determinism and Li-Yorke chaos for interval maps

TL;DR

This paper investigates recurrence determinism in interval dynamical systems, revealing how it characterizes different types of omega-limit sets and relates to Li-Yorke chaos, providing new insights into predictability and chaos.

Contribution

It offers a novel analysis of recurrence determinism in interval maps, linking it to chaos types and omega-limit set structures, and characterizing non-chaotic and Li-Yorke chaotic maps.

Findings

Recurrence determinism distinguishes three main types of omega-limit sets.

Strongly non-chaotic maps have recurrence determinism equal to one.

Li-Yorke non-chaotic maps have positive recurrence determinism.

Abstract

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.