Persistent Markov partitions and hyperbolic components of rational maps

TL;DR

This paper constructs persistent Markov partitions near hyperbolic components of rational maps, characterizes neighboring hyperbolic components via symbolic dynamics, and identifies conditions for matings, advancing understanding of complex dynamics.

Contribution

It introduces a method to analyze hyperbolic components using Markov partitions and symbolic dynamics, including conditions for matings, which are novel results in the field.

Findings

Constructed persistent Markov partitions near hyperbolic components.

Characterized neighboring hyperbolic components through symbolic dynamics.

Identified conditions under which hyperbolic components are matings.

Abstract

Markov partitions persisting in a neighbourhood of hyperbolic components of rational maps were constructed under the condition that closures of Fatou components are disjoint in \cite{R1}. Given such a partition, we characterize all nearby hyperbolic components in terms of the symbolic dynamics. This means we can count them, and also obtain topological information. We also determine extra conditions under which all nearby type IV hyperbolic components are given by matings. These are probably the first known results of this type.