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Regular and limit sets for holomorphic correspondences

S. Bullett, C. Penrose (2001)

Fundamenta Mathematicae

Holomorphic correspondences are multivalued maps f = Q ̃ ₊ Q ̃ ₋ - 1 : Z → W between Riemann surfaces Z and W, where Q̃₋ and Q̃₊ are (single-valued) holomorphic maps from another Riemann surface X onto Z and W respectively. When Z = W one can iterate f forwards, backwards or globally (allowing arbitrarily many changes of direction from forwards to backwards and vice versa). Iterated holomorphic correspondences on the Riemann sphere display many of the features of the dynamics of Kleinian groups and rational maps, of which...

Rigidity of harmonic measure

I. Popovici, Alexander Volberg (1996)

Fundamenta Mathematicae

Let J be the Julia set of a conformal dynamics f. Provided that f is polynomial-like we prove that the harmonic measure on J is mutually absolutely continuous with the measure of maximal entropy if and only if f is conformally equivalent to a polynomial. This is no longer true for generalized polynomial-like maps. But for such dynamics the coincidence of classes of these two measures turns out to be equivalent to the existence of a conformal change of variable which reduces the dynamical system...

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