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

Abstract

Suppose that $f:S^2\to S^2$ determines a dynamical system on the sphere which is topologically coarse expanding conformal in the sense of our previous work. We prove that if its Ahlfors regular conformal dimension $Q$ is realized by some metric $d$, then either (i) $Q=2$ and $f$ is topologically conjugate to a semihyperbolic rational map with Julia set equal to the whole sphere or (ii) $Q>2$ 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 analogue of a known result for Gromov hyperbolic groups with a two-sphere boundary.