Prime ends dynamics on invariant Peano continua

Abstract

The dynamics of a planar homeomorphism h is simple on any non-separating Peano continuum K that is invariant under h. This means that all limit sets on K are either fixed points or periodic orbits. The map h induces a homeomorphism h⁎ on the space of prime ends associated to K. The goal of this paper is to show that in some cases the dynamics on prime ends can have a certain complexity. We construct a dissipative homeomorphism with attractor K and h⁎ a Denjoy map.