Fleming-Vázquez, Nicholas (2022) Functional correlation bounds and optimal iterated moment bounds for slowly-mixing nonuniformly hyperbolic maps. Communications in Mathematical Physics, 391 . pp. 173-198. doi:10.1007/s00220-022-04325-w ISSN 0010-3616.

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Abstract

Consider a nonuniformly hyperbolic map T:M→M modelled by a Young tower with tails of the form O(n−β), β>2. We prove optimal moment bounds for Birkhoff sums ∑n−1i=0v∘Ti and iterated sums ∑0≤i<j<nv∘Tiw∘Tj, where v,w:M→R are (dynamically) Hölder observables. Previously iterated moment bounds were only known for β>5. Our method of proof is as follows; (i) prove that T satisfies an abstract functional correlation bound, (ii) use a weak dependence argument to show that the functional correlation bound implies moment estimates. Such iterated moment bounds arise when using rough path theory to prove deterministic homogenisation results. Indeed, by a recent result of Chevyrev, Friz, Korepanov, Melbourne & Zhang we have convergence to an Itô diffusion for fast-slow systems of the form

x(n)k+1=x(n)k+n−1a(x(n)k,yk)+n−1/2b(x(n)k,yk),yk+1=Tyk

in the optimal range β>2.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH):	Differentiable dynamical systems, Functional analysis, Mappings (Mathematics), Ergodic theory
Journal or Publication Title:	Communications in Mathematical Physics
Publisher:	Springer
ISSN:	0010-3616
Official Date:	April 2022
Dates:	Date Event April 2022 Published 2 February 2022 Available 13 December 2021 Accepted 21 July 2021 Submitted
Volume:	391
Number of Pages:	26
Page Range:	pp. 173-198
DOI:	10.1007/s00220-022-04325-w
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access (Creative Commons)
Date of first compliant deposit:	3 March 2022
Date of first compliant Open Access:	7 March 2022
RIOXX Funder/Project Grant:	Project/Grant ID RIOXX Funder Name Funder ID UNSPECIFIED University of Warwick, Mathematics Institute http://dx.doi.org/10.13039/501100000741

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