Conditioned stochastic stability of equilibrium states on uniformly expanding repellers

Abstract

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 contribute to the rigorous foundation for the existence of ‘natural measures’, which were proposed by Kantz and Grassberger [Repellers, semi-attractors, and long-lived chaotic transients. Phys. D , vol. 17(1), 1985, pp. 75–86] to aid the understanding of chaotic transients.