On the local connectivity of attractors of Markov IFS

We prove an extension of M. Hata’s theorem [Japan J. Appl. Math. 2 (1985), 381–414] 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 the additional hypotheses are necessary.