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Conditioned stochastic stability of equilibrium states on uniformly expanding repellers

Bernat Bassols Cornudella, Matheus Manzatto de Castro, Jeroen S. W. Lamb

TL;DR

<3-5 sentence high-level summary> This work develops a rigorous framework for conditioned stochastic stability of invariant measures near repellers. It introduces quasi-ergodic measures of $e^\phi$-weighted Markov processes, conditioned on survival near a repeller, and shows that, in uniformly expanding repellers, these conditioned statistics converge to the deterministic equilibrium state arising from thermodynamic formalism. Furthermore, the paper proves that any equilibrium state on repellers can be approximated by quasi-ergodic measures, providing local and global results with open holes and concrete dynamical examples. Overall, it furnishes a robust zero-noise perspective on transient dynamics, connects conditioned stochastic stability to natural measures, and extends stochastic stability ideas from attractors to repellers via a weight/kill framework and spectral analysis.

Abstract

We propose a notion of conditioned stochastic stability of invariant measures on repellers: we consider whether quasi-ergodic measures of absorbing Markov processes, generated by random perturbations of the deterministic dynamics and conditioned upon survival in a neighbourhood of a repeller, converge to an invariant measure in the zero-noise limit. Under suitable choices of the random perturbation, we find that equilibrium states on uniformly expanding repellers are conditioned stochastically stable. In the process, we establish a rigorous foundation for the existence of ``natural measures'', which were proposed by Kantz and Grassberger in 1984 to aid the understanding of chaotic transients.

Conditioned stochastic stability of equilibrium states on uniformly expanding repellers

TL;DR

<3-5 sentence high-level summary> This work develops a rigorous framework for conditioned stochastic stability of invariant measures near repellers. It introduces quasi-ergodic measures of -weighted Markov processes, conditioned on survival near a repeller, and shows that, in uniformly expanding repellers, these conditioned statistics converge to the deterministic equilibrium state arising from thermodynamic formalism. Furthermore, the paper proves that any equilibrium state on repellers can be approximated by quasi-ergodic measures, providing local and global results with open holes and concrete dynamical examples. Overall, it furnishes a robust zero-noise perspective on transient dynamics, connects conditioned stochastic stability to natural measures, and extends stochastic stability ideas from attractors to repellers via a weight/kill framework and spectral analysis.

Abstract

We propose a notion of conditioned stochastic stability of invariant measures on repellers: we consider whether quasi-ergodic measures of absorbing Markov processes, generated by random perturbations of the deterministic dynamics and conditioned upon survival in a neighbourhood of a repeller, converge to an invariant measure in the zero-noise limit. Under suitable choices of the random perturbation, we find that equilibrium states on uniformly expanding repellers are conditioned stochastically stable. In the process, we establish a rigorous foundation for the existence of ``natural measures'', which were proposed by Kantz and Grassberger in 1984 to aid the understanding of chaotic transients.
Paper Structure (25 sections, 44 theorems, 164 equations)

This paper contains 25 sections, 44 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Conditioned stochastic stability
  3. Thermodynamic formalism and weighted Markov processes
  4. Context of the results
  5. Outline
  6. Setup, main results and examples
  7. The deterministic dynamics
  8. The random perturbations
  9. Main results
  10. Examples
  11. The logistic map
  12. The complex quadratic map
  13. The Boole map
  14. Some direct consequences of Hypothesis H1
  15. The local problem
  16. ...and 10 more sections

Key Result

Theorem A

Given a $\mathcal{C}^2$ map $T$ on $M$ and a suitable open set $\partial \subset M$, with $\Lambda$ as in equation eq:lambda, assume that Then $\nu_0$ is conditioned stochastically stable on $M \setminus \partial$.

Theorems & Definitions (122)

  • Theorem A
  • Definition 1.1
  • Definition 1.2
  • Theorem B
  • Remark 1.3
  • Corollary B1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.5
  • ...and 112 more