2011/03/21 by Haïssinsky, Peter, Pilgrim, Kevin M.
#Dynamical Systems (math.DS) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1103.4019
We prove that if the Ahlfors regular conformal dimension Q of a topologically cxc map on the sphere f: S2 → S2 is realized by some metric d on S2, 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.