Conditioned stochastic stability of equilibrium states on uniformly expanding repellers

TL;DR

This paper introduces a new concept of conditioned stochastic stability for invariant measures on repellers, demonstrating that equilibrium states on uniformly expanding repellers are stable under certain random perturbations, and establishing a foundation for natural measures.

Contribution

It defines and analyzes conditioned stochastic stability for invariant measures, providing rigorous support for the existence of natural measures in chaotic dynamics.

Findings

Equilibrium states on uniformly expanding repellers are stochastically stable under specific perturbations.

The paper establishes a rigorous foundation for natural measures in chaotic transient analysis.

Demonstrates convergence of quasi-ergodic measures to invariant measures in the zero-noise limit.

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.