Omega-limit sets and invariant chaos in dimension one

TL;DR

This paper clarifies the equivalence of different definitions of basic sets in one-dimensional dynamics, extends results to graph maps, and shows that certain chaotic properties are mutually equivalent under invariance conditions.

Contribution

It fills the proof gap for the equivalence of definitions of basic sets and generalizes chaos results to continuous maps of finite graphs.

Findings

Equivalence of definitions of basic sets in one-dimensional dynamics.

Generalization of chaos properties to graph maps.

Mutual equivalence of several chaotic properties under invariance.

Abstract

Omega-limit sets play an important role in one-dimensional dynamics. During last fifty year at least three definitions of basic set has appeared. Authors often use results with different definition. Here we fill in the gap of missing proof of equivalency of these definitions. Using results on basic sets we generalize results in paper [P.~Oprocha, Invariant scrambled sets and distributional chaos, Dyn. Syst. 24 (2009), no. 1, 31--43.] to the case continuous maps of finite graphs. The Li-Yorke chaos is weaker than positive topological entropy. The equivalency arises when we add condition of invariance to Li-Yorke scrambled set. In this note we show that for a continuous graph map properties positive topological entropy; horseshoe; invariant Li-Yorke scrambled set; uniform invariant distributional chaotic scrambled set and distributionaly chaotic pair are mutually equivalent.