On the local connectivity of attractors of Markov IFS

TL;DR

This paper extends Hata's theorem to planar Markov IFS, proving that connected attractors are also locally connected under certain conditions, and provides counterexamples to demonstrate the necessity of these conditions.

Contribution

It generalizes existing results by establishing local connectivity of attractors for Markov IFS under a strong Open Set Condition, with counterexamples showing the importance of hypotheses.

Findings

Connected attractors are locally connected under specified conditions

Counterexamples demonstrate the necessity of hypotheses

Extension of Hata's theorem to Markov IFS

Abstract

We prove an extension of M. Hata's theorem [4] for planar Markov Iterated Function Systems satisfying a strong version of the Open Set Condition. More precisely, if the attractor of such a system is connected, then it is locally connected. We construct counterexamples to show that all additional hypothesis are necessary.