Combinatorics of identities involving Meixner polynomials. (Combinatoire des identités sur les polynômes de Meixner.)

Foata, Dominique

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Let T be a geometrically finite rational map, p(T) its petal number and δ the Hausdorff dimension of its Julia set. We give a construction of the σ-finite and T-invariant measure equivalent to the δ-conformal measure. We prove that this measure is finite if and only if $\frac{p (T) + 1}{p (T)} δ > 2$ . Under this assumption and if T is parabolic, we prove that the only equilibrium states are convex combinations of the T-invariant probability and δ-masses at parabolic cycles.