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Statistical properties of compact group extensions of non-uniformly expanding dynamical systems
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Matharu, Barinder Singh (2019) Statistical properties of compact group extensions of non-uniformly expanding dynamical systems. PhD thesis, University of Warwick.
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WRAP_Theses_Matharu_2019.pdf - Submitted Version - Requires a PDF viewer. Download (1261Kb) | Preview |
Official URL: http://webcat.warwick.ac.uk/record=b3492775~S15
Abstract
We prove statistical limit theorems for Birkhoff sums of the form Pn−1 k=0 φn ◦ Th(n) , where Th(n) are a sequence of compact group extensions with non-uniformly expanding base and φn are a sequence of equivariant H¨older observables. This is done by extending the methods of Korepanov, Kosloff, and Melbourne to construct two new martingale-coboundary decompositions.
Even in the case of a fixed observable and compact group extension, these decompositions enable us not only to reprove existing results in the literature, but also to obtain far reaching consequences. Using our primary martingale-coboundary decomposition, we give a new proof of a central limit theorem and weak invariance principle under very general conditions, and obtain moment estimates which are optimal given our setup. Still in the case of a fixed observable and compact group extension, we use our secondary martingale-coboundary decomposition to prove an almost sure invariance principle with excellent error rates.
As an application, we prove a homogenisation result for discrete fast-slow dynamical systems with additive noise, where the fast dynamics are generated by a family of compact group extensions with non-uniformly expanding base.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Limit theorems (Probability theory), Group extensions (Mathematics), Dynamics, Decomposition (Mathematics) | ||||
Official Date: | October 2019 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Melbourne, Ian | ||||
Sponsors: | Engineering and Physical Sciences Research Council | ||||
Format of File: | |||||
Extent: | vii, 134 leaves : illustrations (black and white) | ||||
Language: | eng |
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