Distance expanding random mappings, thermodynamical formalism, Gibbs measures and fractal geometry

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The theory of random dynamical systems, rooted in stochastic differential equations, provides a framework to analyze the evolution of systems with approximately known input and output data based on probability distributions. This manuscript introduces measurable expanding random dynamical systems and develops the thermodynamical formalism, establishing exponential decay of correlations and analyticity of expected pressure, despite the absence of the spectral gap property. We apply this theory to explore the fractal properties of conformal random systems, proving Bowen’s formula and developing the multifractal formalism of Gibbs states. By examining the behavior of Birkhoff sums of the pressure function, we classify systems into two categories: quasi-deterministic systems, which exhibit many deterministic properties, and essentially random systems, which are generic and not bi-Lipschitz equivalent to deterministic systems. Notably, we demonstrate that in the essentially random case, the Hausdorff measure vanishes, contradicting a conjecture by Bogenschutz and Ochs. Finally, we present applications of our findings to specific conformal random systems, providing a positive answer to a question raised by Bruck and Buger regarding the Hausdorff dimension of quadratic random Julia sets.