The omega-limit sets of quadratic Julia sets

TL;DR

This paper characterizes the omega-limit sets of dendritic Julia sets for quadratic maps, showing they are exactly the closed, internally chain transitive invariant sets, and establishes shadowing for these maps.

Contribution

It provides a complete characterization of omega-limit sets for quadratic dendritic Julia sets using symbolic dynamics and proves shadowing property for these maps.

Findings

Omega-limit sets are exactly the closed, internally chain transitive invariant sets.

Quadratic maps with dendritic Julia sets have shadowing.

A set is an omega-limit set if and only if it is internally chain transitive.

Abstract

In this paper we characterize $\w$-limit sets of dendritic Julia sets for quadratic maps. We use Baldwin's symbolic representation of these spaces as a non-Hausdorff itinerary space and prove that quadratic maps with dendritic Julia sets have shadowing, and also that for all such maps, a closed invariant set is an $\w$-limit set of a point if, and only if, it is internally chain transitive.