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Equidistribution for nonuniformly expanding dynamical systems, and application to the almost sure invariance principle

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Korepanov, Alexey (2018) Equidistribution for nonuniformly expanding dynamical systems, and application to the almost sure invariance principle. Communications in Mathematical Physics, 359 (3). pp. 1123-1138. doi:10.1007/s00220-017-3062-z ISSN 0010-3616.

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Official URL: http://dx.doi.org/10.1007/s00220-017-3062-z

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Abstract

Let T:M→MT:M→M be a nonuniformly expanding dynamical system, such as logistic or intermittent map. Let v:M→Rdv:M→Rd be an observable and vn=∑n−1k=0v∘Tkvn=∑k=0n−1v∘Tk denote the Birkhoff sums. Given a probability measure μμ on M, we consider vn as a discrete time random process on the probability space (M,μ)(M,μ) . In smooth ergodic theory there are various natural choices of μμ , such as the Lebesgue measure, or the absolutely continuous T-invariant measure. They give rise to different random processes. We investigate relation between such processes. We show that in a large class of measures, it is possible to couple (redefine on a new probability space) every two processes so that they are almost surely close to each other, with explicit estimates of “closeness”. The purpose of this work is to close a gap in the proof of the almost sure invariance principle for nonuniformly hyperbolic transformations by Melbourne and Nicol.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH): Ergodic theory, Measure theory
Journal or Publication Title: Communications in Mathematical Physics
Publisher: Springer
ISSN: 0010-3616
Official Date: May 2018
Dates:
DateEvent
May 2018Published
12 December 2017Available
7 November 2017Accepted
Volume: 359
Number: 3
Page Range: pp. 1123-1138
DOI: 10.1007/s00220-017-3062-z
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
Date of first compliant deposit: 3 January 2018
Date of first compliant Open Access: 12 December 2018
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
ERC AdG 320977H2020 European Research Councilhttp://dx.doi.org/10.13039/100010663

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