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Linear response for random dynamical systems

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journal contribution
posted on 27.01.2020, 10:17 by Wael BahsounWael Bahsoun, Marks Ruziboev, Benoit Saussol
We study for the first time linear response for random compositions of maps, chosen independently according to a distribution P. We are interested in the following question: how does an absolutely continuous stationary measure (acsm) of a random system change when P changes smoothly to Pε? For a wide class of one dimensional random maps, we prove differentiability of acsm with respect to ε; moreover, we obtain a linear response formula. Our results cover random maps whose transfer operator does not necessarily admit a spectral gap. We apply our results to iid compositions, with respect to various distributions Pε, of uniformly expanding circle maps, Gauss-R´enyi maps (random continued fractions) and Pomeau-Manneville maps. Our results yield an exact formula for the invariant density of random continued fractions; while for Pomeau-Manneville maps our results provide a precise relation between their linear response under certain random perturbations and their linear response under deterministic perturbations.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Advances in Mathematics

Volume

364

Publisher

Elsevier

Version

AM (Accepted Manuscript)

Rights holder

© Elsevier

Publisher statement

This paper was accepted for publication in the journal Advances in Mathematics and the definitive published version is available at https://doi.org/10.1016/j.aim.2020.107011

Acceptance date

24/01/2020

Publication date

2020-02-05

Copyright date

2020

ISSN

0001-8708

Language

en

Depositor

Dr Wael Bahsoun Deposit date: 24 January 2020

Article number

107011

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