Shadowing and $\omega $-limit sets of circular Julia sets

By A. Barwell, Jonathan Meddaugh, Brian E. Raines. Ergodic Theory and Dynamical Systems, 2011

52 citations recorded by the source corpus.

Abstract In this paper we consider quadratic polynomials on the complex plane ${f}_{c} (z)= {z}^{2} + c$ and their associated Julia sets, ${J}_{c} $. Specifically, we consider the case that the kneading sequence is periodic and not an $n$-tupling. In this case ${J}_{c} $ contains subsets that are homeomorphic to the unit circle, usually infinitely many disjoint such subsets. We prove that ${f}_{c} : {J}_{c} \rightarrow {J}_{c} $ has shadowing, and we classify all $\omega $-limit sets for these maps by showing that a closed set $R\subseteq {J}_{c} $ is internally chain transitive if, and only if, there is some $z\in {J}_{c} $ with $\omega (z)= R$.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/293241fec9da7a3b2834bb520e8bfb20d19d1a51