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Robustness of ergodic properties of non-autonomous piecewise expanding maps

Published online by Cambridge University Press:  25 September 2017

MATTEO TANZI ,
TIAGO PEREIRA  and
SEBASTIAN VAN STRIEN
MATTEO TANZI
Affiliation:
Department of Mathematics, South Kensington Campus, London SW7 2AZ, UK email m.tanzi13@imperial.ac.uk, tiago.pereira@imperial.ac.uk, s.van-strien@imperial.ac.uk
TIAGO PEREIRA
Affiliation:
Department of Mathematics, South Kensington Campus, London SW7 2AZ, UK email m.tanzi13@imperial.ac.uk, tiago.pereira@imperial.ac.uk, s.van-strien@imperial.ac.uk Institute of Mathematical and Computer Sciences, Universidade de São Paulo, São Carlos 13566-590, São Paulo, Brazil
SEBASTIAN VAN STRIEN
Affiliation:
Department of Mathematics, South Kensington Campus, London SW7 2AZ, UK email m.tanzi13@imperial.ac.uk, tiago.pereira@imperial.ac.uk, s.van-strien@imperial.ac.uk
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Abstract

Recently, there has been an increasing interest in non-autonomous composition of perturbed hyperbolic systems: composing perturbations of a given hyperbolic map $F$ results in statistical behaviour close to that of $F$. We show this fact in the case of piecewise regular expanding maps. In particular, we impose conditions on perturbations of this class of maps that include situations slightly more general than what has been considered so far, and prove that these are stochastically stable in the usual sense. We then prove that the evolution of a given distribution of mass under composition of time-dependent perturbations (arbitrarily—rather than randomly—chosen at each step) close to a given map $F$ remains close to the invariant mass distribution of $F$. Moreover, for almost every point, Birkhoff averages along trajectories do not fluctuate wildly. This result complements recent results on memory loss for non-autonomous dynamical systems.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 4 , April 2019 , pp. 1121 - 1152
Copyright
© Cambridge University Press, 2017 

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