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arXiv · 2607.04900

Thermodynamic formalism for hyperbolic random dynamical systems

Lucas Amorim·Matheus M Castro·Benoit Saussol·Sandro Vaienti
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Abstract

We develop thermodynamic formalism for random Anosov maps and uniformly H\"older random potentials. We assume uniform fibre hyperbolicity given by deterministic invariant cone fields, a one-dimensional stable direction, and a fibrewise mixing condition whose mixing time may depend on the base point. To do so, we construct adapted projective cones for the random Perron--Frobenius cocycle and prove that the cocycle contracts the associated Hilbert projective metrics. This allows us to construct a $\mathbb P$-relative equilibrium state, prove its uniqueness, and establish quenched exponential decay of correlations.

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Lucas Amorim, Matheus M Castro, Benoit Saussol, Sandro Vaienti. 2026-07-06. Thermodynamic formalism for hyperbolic random dynamical systems. https://arxiv.org/abs/2607.04900

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