Conditioned stochastic stability of equilibrium states on uniformly hyperbolic sets

TL;DR

This paper studies the stability of equilibrium states under small random perturbations on hyperbolic sets, revealing how transient dynamics behave and converge to specific measures as noise diminishes.

Contribution

It introduces a novel conditioned stochastic stability framework for hyperbolic sets, extending analysis to non-attracting regions and multiple basic sets in Axiom A systems.

Findings

Quasi-ergodic measures converge to equilibrium states as noise vanishes.

The framework applies to Axiom A diffeomorphisms with multiple basic sets.

Provides a rigorous characterization of measures on hyperbolic repellers.

Abstract

We establish the conditioned stochastic stability of equilibrium states for H\"older potentials on uniformly hyperbolic sets. While standard stochastic stability characterises measures on attractors, we analyse the statistics of transient dynamics on non-attracting sets by conditioning small random perturbations of the dynamics to not escape from our regions of interest. We prove that as the noise intensity vanishes, the quasi-ergodic measure of the $e^\phi$-weighted process generated by $\e$-small random perturbations of the deterministic dynamics converges to the unique equilibrium state associated with the potential $\phi - \log \left|\det \left. D T\right|_{E^u}\right|$. The results are obtained via perturbative spectral analysis of transfer operators acting on anisotropic Banach spaces and topological hyperbolic dynamics arguments. Furthermore, we extend this framework globally to Axiom A diffeomorphisms with multiple basic sets using dynamical filtrations. This work provides a rigorous characterisation of natural measures on uniformly hyperbolic repellers, which are fundamental in the context of transient chaos.