Thermodynamic formalism for hyperbolic random dynamical systems

Abstract: We develop thermodynamic formalism for random Anosov maps and uniformly Hölder 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.

• The paper introduces a novel thermodynamic formalism leveraging projective cone techniques to derive a precise spectral decomposition for random transfer operators.
• It establishes existence and uniqueness of relative equilibrium states for uniformly Hölder continuous potentials on hyperbolic random systems.
• The approach proves quenched exponential decay of correlations, quantifying fiberwise mixing and providing robust statistical stability insights.

Thermodynamic Formalism for Hyperbolic Random Dynamical Systems: A Technical Analysis

Introduction and Context

The paper develops a comprehensive thermodynamic formalism for random Anosov maps with uniformly Hölder continuous random potentials. The authors consider skew products $F: \Omega \times M \to \Omega \times M$, where the base dynamical system $(\theta, \Omega, \mathbb{P})$ is ergodic and $T_\omega$ are $\mathcal{C}^2$ fibrewise diffeomorphisms over a compact Riemannian manifold $M$. Uniform fibre hyperbolicity is ensured by deterministic invariant cone fields with a one-dimensional stable direction, accompanied by a fibrewise mixing condition with potentially base-point-dependent mixing times.

Their approach is grounded in the adaptation and extension of projective cone techniques for the random Perron–Frobenius cocycle, providing a spectral decomposition based on Hilbert projective metrics. This framework enables construction and uniqueness proofs of $\mathbb{P}$-relative equilibrium states, as well as establishment of quenched exponential decay of correlations.

Technical Contributions

Setting and Main Hypotheses

Random dynamical systems are treated as skew products with regularity and hyperbolicity assumptions, specifically:

• Uniform Hyperbolicity: Standing deterministic cone fields, invariant under the random sequence of diffeomorphisms, with $\dim E^s = 1$ and contraction/expansion estimates uniform over $\omega$.
• Fibrewise Mixing: A mixing time $N(\omega)$ guarantees every local unstable manifold achieves $\delta$-density after $(\theta, \Omega, \mathbb{P})$0 iterates, with either positive probability or an exponential tail (strengthened hypothesis).

The thermodynamic formalism is relativised: equilibrium states maximize $(\theta, \Omega, \mathbb{P})$1 over $(\theta, \Omega, \mathbb{P})$2-invariant measures projecting to $(\theta, \Omega, \mathbb{P})$3, with $(\theta, \Omega, \mathbb{P})$4 denoting relative entropy.

Construction of Adapted Cones

Building on earlier deterministic and non-invertible random systems (notably [viana1997stochastic], [liu2024exponential]), the authors construct adapted projective cones in spaces of leafwise log-Hölder functions and state-dependent observables. These cones encode stable-leaf averages and their compatibility with unstable holonomies, leading to a contraction property for the Perron–Frobenius cocycle in the associated Hilbert projective metric. Regularity assumptions ensure invariance under the transfer operator, with precise control of the cones' projective diameters after a return to sufficiently mixing blocks.

Quenched Spectral Decomposition

A key technical achievement is the proof of a fiberwise spectral gap for the random transfer operator cocycle at the level of the Hilbert projective metric. For almost every $(\theta, \Omega, \mathbb{P})$5 there exist positive random eigenfunctions $(\theta, \Omega, \mathbb{P})$6 and dual functionals $(\theta, \Omega, \mathbb{P})$7 such that iterations of the transfer operator decompose as

$(\theta, \Omega, \mathbb{P})$8

where the remainder decays at a rate controlled by block returns to sufficiently mixing fibers. This generalizes deterministic spectral theory for Anosov systems [BaladiTsujii2007, GouezelLiverani2006, Baladi2018] to the random fibre context without requiring uniform mixing times.

Main Theorems

Two main theorems are established:

• Existence and Uniqueness: For any uniform Hölder random potential, there is a unique $(\theta, \Omega, \mathbb{P})$9-relative equilibrium state among all $T_\omega$0-invariant measures projecting to $T_\omega$1. The constructed measure is characterized by the eigenfunction obtained from the cone spectral theory.
• Quenched Exponential Decay of Correlations: For almost every $T_\omega$2, there are random constants $T_\omega$3 and uniform rate $T_\omega$4 such that for all $T_\omega$5,

$T_\omega$6

and the same holds for the pullback correlation along inverse iterates. Under the exponential mixing time tail (stronger hypothesis), $T_\omega$7 can be taken in $T_\omega$8 for all $T_\omega$9, giving strong control over the correlation constants.

Variational Principle and Weak Gibbs Property

The authors prove a variational principle equating pressure and the sum of relative entropy and $\mathcal{C}^2$0-integral for the equilibrium state. The construction is supported by a weak Gibbs property: for almost every $\mathcal{C}^2$1, the equilibrium measure satisfies

$\mathcal{C}^2$2

up to subexponential distortion factors, where $\mathcal{C}^2$3 are random Bowen balls.

Structural Examples and Applicability

The paper gives concrete families satisfying all hypotheses, including random compositions of hyperbolic toral automorphisms, random Anosov diffeomorphisms with locally constant base dynamics, and skew products with trivial or strongly mixing base transformations.

Implications and Future Directions

This work generalizes deterministic thermodynamic formalism for uniformly hyperbolic systems to a class of invertible random dynamical systems where fibrewise mixing may be only typical and the mixing time non-uniform. The primary advance is the cone technique enabling a fiberwise spectral analysis without requiring uniform mixing, and proving both uniqueness and strong statistical properties for equilibrium states.

The results unify and extend prior work on non-invertible expanding random maps, deterministic Anosov systems, and random interval maps. Invariant measures constructed via these methods admit strong statistical laws (including large deviations and CLTs under further hypotheses), and the random variability in correlation decay constants is tightly quantified via exponential moment bounds when the mixing time has exponential tails.

Potential future avenues include:

• Extension to non-uniformly hyperbolic random diffeomorphisms without pinning the stable direction's dimension;
• Fluctuation results for time averages and other limit theorems;
• Analysis of stochastic stability for equilibrium states and transfer operator spectra under random perturbations (building on [GouezelLiverani2006, DragicevicFroylandGonzalezTokmanVaienti2020TAMS]);
• Applicability to random compositions in high-dimensional settings and to random partially hyperbolic systems;
• Development of anisotropic Banach space frameworks in the random context paralleling deterministic spectral approaches.

Conclusion

The paper establishes a robust thermodynamic formalism for skew products with uniformly hyperbolic, fibrewise-diffeomorphic fibres, under minimal assumptions on mixing time uniformity. By extending projective cone contraction methods to the random invertible setting, the work delivers existence, uniqueness, and strong statistical properties for equilibrium states under broad conditions. The cone contraction principle and spectral decomposition for the random transfer operator cocycle provide foundational tools for future stochastic analysis of dynamical systems with random or non-uniform characteristics.

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References (inline, not exhaustive):

• "Decay of correlations for invertible maps with hyperbolic structure" [liverani1995decay].
• "Decay of correlations for open billiards" [LSV1998].
• "Anisotropic spaces and the stability of the essential spectrum for transfer operators" [GouezelLiverani2006].
• "Transfer operators and decay of correlations" [BaladiTsujii2007].
• "Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps" [Baladi2018].
• "Stochastic stability of Lyapunov exponents for random dynamical systems" [viana1997stochastic].
• "Exponential decay of correlations for invertible random dynamical systems" [liu2024exponential].