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On the dimensions of conformal repellers. Randomness and parameter dependency. (English) Zbl 1181.37066
The author gives s simple proof of a Bowen’s formula relating the Hausdorff dimension of a conformal repeller to the zero of the corresponding pressure function. A generalization of Bowen’s theorem is presented for time-dependent repellers arising as invariant subsets of sequences of a class of conformally expanding maps. It is proved that if the sequence is randomly chosen then the Hausdorff dimension of the repeller coincides with its upper and lower box dimensions and is given by a generalization of Bowen’s formula. In the case of a random hyperbolic Julia set on the Riemann sphere, the author shows that if the family of maps and their probability law depend in a real-analytic fashion on parameters, then its almost sure Hausdorff dimension exhibits similar dependence.

##### MSC:
 37F35 Conformal densities and Hausdorff dimension 37D35 Thermodynamic formalism, variational principles, equilibrium states 37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations 37H99 Random dynamical systems