Julia sets and periodic kneading sequences

Abstract

We construct abstract Julia sets homeomorphic to Julia sets for complex polynomials of the form fc(z) = z2 + c, having an associated periodic kneading sequence of the form \({\overline{\alpha\ast}}\) which is not a period n-tupling. We show that there is a single simply-defined space of “itineraries” which contains homeomorphic copies of all such Julia sets in a natural combinatorial way, with dynamical properties which are derivable directly from the combinatorics. This also leads to a natural definition of abstract Julia sets even for those kneading sequences which are not realized by any polynomial fc, with similar dynamical properties.