Geometry of hyperbolic Julia-Lavaurs sets

Mariusz Urbanski (1) , Michel Zinsmeister (2)

1 Department of Mathematics and Statistics [Texas Tech]
2 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans

Let J_σ be the Julia-Lavaurs set of a hyperbolic Lavaurs map gσ be its Hausdorff dimension. We show that the upper ball-(box) counting dimension and the Hausdorff dimension of Jσ are equal, that the hσ-dimensional Hausdorff measure of J_σ vanishes and that the hσ-dimensional packing measure of Jσ is positive and finite. If gσ is derived from the parabolic quadratic polynomial f(z) = z2 + Image, then the Hausdorff dimension hσ is a real-analytic function of σ. As our tool we study analytic dependence of the Perron-Frobenius operator on the symbolic space with infinite alphabet.

Domaines: Systèmes dynamiques [math.DS]

Mariusz Urbanski, Michel Zinsmeister. Geometry of hyperbolic Julia-Lavaurs sets. Indagationes Mathematicae, 2001, Vol 12, pp.273-292. ⟨hal-00079633⟩