Suppose that 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 is realized by some metric , then either (i) and is topologically conjugate to a semihyperbolic rational map with Julia set equal to the whole sphere or (ii) and 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.
Peter Haïssinsky. Kevin M. Pilgrim. "Minimal Ahlfors regular conformal dimension of coarse expanding conformal dynamics on the sphere." Duke Math. J. 163 (13) 2517 - 2559, 1 October 2014. https://doi.org/10.1215/00127094-2819408