Cantor sets of circles of Sierpiński curve Julia sets

Abstract

AbstractOur goal in this paper is to give an example of a one-parameter family of rational maps for which, in the parameter plane, there is a Cantor set of simple closed curves consisting of parameters for which the corresponding Julia set is a Sierpiński curve. Hence, the Julia sets for each of these parameters are homeomorphic. However, each of the maps in this set is dynamically distinct from (i.e. not topologically conjugate to) any other map in this set (with only finitely many exceptions). We also show that, in the dynamical plane for any map drawn from a large open set in the connectedness locus in this family, there is a Cantor set of invariant simple closed curves on which the map is conjugate to the product of certain subshifts of finite type with the maps $z \mapsto \pm z^n$ on the unit circle.