Suppose that f : S-2 -> 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.
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