Minimal Ahlfors regular conformal dimension of coarse conformal dynamics on the sphere

TL;DR

This paper characterizes the minimal Ahlfors regular conformal dimension for certain sphere maps, showing it is either 2 with rational maps or greater than 2 with torus affine maps, extending known results for hyperbolic groups.

Contribution

It establishes a dichotomy for the conformal dimension of topologically cxc maps on the sphere, linking it to specific classes of dynamical systems and providing a new proof of related results.

Findings

If Q=2, the map is conjugate to a semihyperbolic rational map with Julia set equal to the sphere.

If Q>2, the map lifts to an affine expanding map of a torus with distinct real eigenvalues.

The methods offer a new proof of a known result for Gromov hyperbolic groups with sphere boundary.

Abstract

We prove that if the Ahlfors regular conformal dimension $Q$ of a topologically cxc map on the sphere $f: S^2 \to S^2$ is realized by some metric $d$ on $S^2$, then either Q=2 and $f$ is topologically conjugate to a semihyperbolic rational map with Julia set equal to the whole Riemann sphere, or $Q>2$ and $f$ is topologically conjugate to a map which lifts to an affine expanding map of a torus whose differential has distinct real eigenvalues. This is an analog of a known result for Gromov hyperbolic groups with two-sphere boundary, and our methods apply to give a new proof.