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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
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Official URL: http://dx.doi.org/10.1007/s00220-017-3062-z
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 | ||||||||
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| 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 | ||||||||
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| 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 | ||||||||
| RIOXX Funder/Project Grant: |
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