The dynamics of hyperbolic rational maps with Cantor Julia sets

TL;DR

This paper explores conditions under which hyperbolic rational maps have Cantor Julia sets, showing that generically the minimal iterate for a certain inclusion is one, and establishing the connectedness of the shift locus.

Contribution

It proves that for generic hyperbolic rational maps, the minimal iterate is one, and demonstrates the connectedness of the shift locus of rational maps.

Findings

For generic maps, n_f=1, meaning the minimal iterate is one.

The shift locus S_d is connected for all degrees d.

Existence of a degree 4 map with n_f=2.

Abstract

Let $f:\hat{\mathbb C}\to\hat{\mathbb C}$ be a hyperbolic rational map of degree $d\ge2$ on the Riemann sphere. We give several conditions which are equivalent to the condition for the Julia set $J_f$ to be a Cantor set. It has been known that $J_f$ is a Cantor sets if and only if there exists a positive integer $n>0$ such that $\bar{f^{-n}(U)}\subset U$ for some open topological disc $U$ containing no critical values. Let $n_f$ denote the minimal positive integer satisfying the above. The problem is whether $n_f=1$ or not. Let $S_d$ denote the shift locus of rational maps of degree $d$. We show that $n_f=1$ for generic $f\in S_d$ and that there is a rational map $\bar f\in S_4$ with $n_{\bar f}=2$. We also prove that $S_d$ is connected using the generic case result. In particular, generic hyperbolic rational maps of degree $d$ with Cantor Julia sets are qc-conjugate to each other.