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Conditioned stochastic stability of equilibrium states on uniformly hyperbolic sets

Bernat Bassols Cornudella, Matheus M. Castro

TL;DR

The paper develops a rigorous framework for conditioned stochastic stability of equilibrium states on uniformly hyperbolic sets by studying quasi-ergodic measures of small-noise perturbations. It combines perturbative transfer-operator theory on anisotropic Banach spaces with Conley-style dynamical filtrations to produce local and global results, proving convergence to equilibrium states associated with the modified potential $\psi=\phi-\log|\det DT|_{E^u}|$. The authors provide constructive methods to obtain quasi-ergodic measures from dominant eigenfunctions and illustrate the theory on the Hénon repeller and Arnold’s cat map, thus linking transient chaos statistics to thermodynamic formalism. This yields a robust, spectrally grounded approach to characterizing natural measures on repellers and offers practical avenues for data-driven approximation of these measures. The global extension to Axiom A systems with multiple basic sets enhances the utility of the framework for complex dynamical scenarios with competing invariant sets.

Abstract

We establish the conditioned stochastic stability of equilibrium states for Hölder 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 $e^φ$-weighted process generated by $\e$-small random perturbations of the deterministic dynamics converges to the unique equilibrium state associated with the potential $φ- \log \left|\det \left. D T\right|_{E^u}\right|$. 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.

Conditioned stochastic stability of equilibrium states on uniformly hyperbolic sets

TL;DR

The paper develops a rigorous framework for conditioned stochastic stability of equilibrium states on uniformly hyperbolic sets by studying quasi-ergodic measures of small-noise perturbations. It combines perturbative transfer-operator theory on anisotropic Banach spaces with Conley-style dynamical filtrations to produce local and global results, proving convergence to equilibrium states associated with the modified potential . The authors provide constructive methods to obtain quasi-ergodic measures from dominant eigenfunctions and illustrate the theory on the Hénon repeller and Arnold’s cat map, thus linking transient chaos statistics to thermodynamic formalism. This yields a robust, spectrally grounded approach to characterizing natural measures on repellers and offers practical avenues for data-driven approximation of these measures. The global extension to Axiom A systems with multiple basic sets enhances the utility of the framework for complex dynamical scenarios with competing invariant sets.

Abstract

We establish the conditioned stochastic stability of equilibrium states for Hölder 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 -weighted process generated by -small random perturbations of the deterministic dynamics converges to the unique equilibrium state associated with the potential . 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.

Paper Structure

This paper contains 23 sections, 25 theorems, 101 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Conditioned Markov processes
  3. Conditioned stochastic stability
  4. Equilibrium states and natural measures
  5. Setup, results and examples
  6. The deterministic dynamics
  7. The random perturbations
  8. Main results
  9. Examples
  10. The Hénon repeller
  11. Arnold's cat map
  12. The anisotropic Banach space Bᵗˢ
  13. Charts and cone systems adapted to (T,V)
  14. The space Bᵗˢ in Rᵈ
  15. The space Bᵗˢ on M
  16. ...and 8 more sections

Key Result

Theorem 2.6

Assume that $(T,g,\Lambda)$ satisfies Hypothesis hyp:local and let $V$ be an isolating neighbourhood of $\Lambda$. Let $e^\phi = g$ on $\Lambda$. Then for $\varepsilon>0$ small enough, $X_\varepsilon^{\phi}$ admits a unique quasi-ergodic measure $\nu_\varepsilon^\phi$ on $\overline{V}$ such that $\L

Figures (1)

  • Figure 1: (a) Initial configuration, graph $G$. (b) $G_1 = G \setminus \mathscr G_{i_0} = \mathscr{G}_{i_1} \sqcup \mathscr{G}_{i_2}$.

Theorems & Definitions (74)

  • Definition 1.1: Quasi-ergodic measure on $\mathcal{U}\subset M$
  • Remark 1.2
  • Definition 2.1: Locally maximal hyperbolic set, isolating neighbourhood
  • Definition 2.2: Hyperbolic basic set HasselblattKatok2003
  • Remark 2.3
  • Definition 2.4: Set of non-wandering points, $NW(T)$
  • Definition 2.5: Axiom A
  • Theorem 2.6
  • Theorem 2.7: Local conditioned stochastic stability, $\partial = M \setminus \overline{V}$
  • Remark 2.8
  • ...and 64 more