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Quenched stochastic stability for eventually expanding-on-average random interval map cocycles

Published online by Cambridge University Press:  25 January 2018

GARY FROYLAND ,
CECILIA GONZÁLEZ-TOKMAN  and
RUA MURRAY
GARY FROYLAND
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia email g.froyland@unsw.edu.au
CECILIA GONZÁLEZ-TOKMAN
Affiliation:
School of Mathematics and Physics, University of Queensland, St Lucia QLD 4072, Australia email cecilia.gt@uq.edu.au
RUA MURRAY
Affiliation:
School of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand email rua.murray@canterbury.ac.nz
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Abstract

The paper by Froyland, González-Tokman and Quas [Stability and approximation of random invariant densities for Lasota–Yorke map cocycles. Nonlinearity27(4) (2014), 647] established fibrewise stability of random absolutely continuous invariant measures (acims) for cocycles of random Lasota–Yorke maps under a variety of perturbations, including ‘Ulam’s method’, a popular numerical method for approximating acims. The expansivity requirements of Froyland et al were that the cocycle (or powers of the cocycle) should be ‘expanding on average’ before applying a perturbation, such as Ulam’s method. In the present work, we make a significant theoretical and computational weakening of the expansivity hypotheses of Froyland et al, requiring only that the cocycle be eventually expanding on average, and importantly, allowing the perturbation to be applied after each single step of the cocycle. The family of random maps that generate our cocycle need not be close to a fixed map and our results can handle very general driving mechanisms. We provide a detailed numerical example of a random Lasota–Yorke map cocycle with expanding and contracting behaviour and illustrate the extra information carried by our fibred random acims, when compared to annealed acims or ‘physical’ random acims.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 10 , October 2019 , pp. 2769 - 2792
Copyright
© Cambridge University Press, 2018 

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