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

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 $\omega (z)= R$.