Real Analyticity for random dynamics of transcendental functions

TL;DR

This paper proves the real analyticity of expected pressure and invariant densities in the random dynamics of transcendental functions, providing new insights into the Hausdorff dimension of Julia sets under mild assumptions.

Contribution

It introduces a simplified approach to establish analyticity results and extends Bowen's formula to random transcendental dynamics, with applications to hyperbolic systems.

Findings

Expected pressure and invariant densities are real analytic.

Hausdorff dimension of Julia sets varies analytically in hyperbolic systems.

Provides a new method under mild dynamical assumptions.

Abstract

Analyticity results of expected pressure and invariant densities in the context of random dynamics of transcendental functions are established. These are obtained by a refinement of work by Rugh leading to a simple approach to analyticity. We work under very mild dynamical assumptions. Just the iterates of the Perron-Frobenius operator are assumed to converge. We also provide a Bowen's formula expressing the almost sure Hausdorff dimension of the radial fiberwise Julia sets in terms of the zero of an expected pressure function. Our main application states real analyticity for the variation of this dimension for suitable hyperbolic random systems of entire or meromorphic functions.