## Duke Mathematical Journal

### 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) $Q\gt 2$ 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.

#### Article information

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

Dates
First available in Project Euclid: 1 October 2014

https://projecteuclid.org/euclid.dmj/1412168850

Digital Object Identifier
doi:10.1215/00127094-2819408

Mathematical Reviews number (MathSciNet)
MR3265557

Zentralblatt MATH identifier
1384.37056

#### Citation

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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