Duke Mathematical Journal

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

Peter Haïssinsky and Kevin M. Pilgrim

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Suppose that f:S2S2 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.

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Duke Math. J., Volume 163, Number 13 (2014), 2517-2559.

First available in Project Euclid: 1 October 2014

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Primary: 37F30: Quasiconformal methods and Teichmüller theory; Fuchsian and Kleinian groups as dynamical systems 54E40: Special maps on metric spaces
Secondary: 20F67: Hyperbolic groups and nonpositively curved groups 30C65: Quasiconformal mappings in $R^n$ , other generalizations

conformal dimension modulus of curves word hyperbolic group weak tangent space rational map Lattès maps


Haïssinsky, Peter; Pilgrim, Kevin M. Minimal Ahlfors regular conformal dimension of coarse expanding conformal dynamics on the sphere. Duke Math. J. 163 (2014), no. 13, 2517--2559. doi:10.1215/00127094-2819408. https://projecteuclid.org/euclid.dmj/1412168850

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