Perturbations of rational Misiurewicz maps

TL;DR

This paper studies how rational Misiurewicz maps behave under perturbations, showing that non-Lattes Misiurewicz maps can be approximated by hyperbolic maps, with implications for complex dynamics.

Contribution

It demonstrates that non-Lattes rational Misiurewicz maps with full Julia set can be approximated by hyperbolic maps, expanding understanding of their stability.

Findings

Non-Lattes Misiurewicz maps can be approximated by hyperbolic maps.

The Julia set being the entire sphere is a key condition.

Perturbation properties differ for Lattes and non-Lattes maps.

Abstract

In this paper we investigate the perturbation properties of rational Misiurewicz maps, when the Julia set is the whole sphere (the other case is treated in [1]). In particular, we show that if f is a Misiurewicz map and not a flexible Lattes map, then we can find a hyperbolic map arbitrarily close to f.