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Conformal measures and matings between Kleinian groups and quadratic polynomials

Marianne Freiberger (2007)

Fundamenta Mathematicae

Following results of McMullen concerning rational maps, we show that the limit set of matings between a certain class of representations of C₂ ∗ C₃ and quadratic polynomials carries δ-conformal measures, and that if the correspondence is geometrically finite then the real number δ is equal to the Hausdorff dimension of the limit set. Moreover, when f is the limit of a pinching deformation f t 0 ≤ t < 1 we give sufficient conditions for the dynamical convergence of f t .

Contractive curves.

Bonifant, Araceli, Dabija, Marius (2002)

International Journal of Mathematics and Mathematical Sciences

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...

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