Precise asymptotics on the Birkhoff sums for dynamical systems

Lulu Fang; Hiroki Takahasi; Yiwei Zhang

doi:10.1088/1361-6544/ac20a2

Precise asymptotics on the Birkhoff sums for dynamical systems

Lulu Fang, Hiroki Takahasi, Yiwei Zhang

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We establish two precise asymptotic results on the Birkhoff sums for dynamical systems. These results are parallel to that on the partial sums of independent and identically distributed random variables previously obtained by Hsu and Robbins, Erdös, Heyde. We apply our results to the Gauss map and obtain new precise asymptotics in the theorem of Lévy on the regular continued fraction expansion of irrational numbers in (0, 1).

Original language	English
Pages (from-to)	7095-7108
Number of pages	14
Journal	Nonlinearity
Volume	34
Issue number	10
DOIs	https://doi.org/10.1088/1361-6544/ac20a2
Publication status	Published - 2021 Oct

Keywords

Continued fraction
Expanding Markov interval maps
Lévy’s theorem
Precise asymptotics

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics
General Physics and Astronomy
Applied Mathematics

Access to Document

10.1088/1361-6544/ac20a2

Cite this

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title = "Precise asymptotics on the Birkhoff sums for dynamical systems",

abstract = "We establish two precise asymptotic results on the Birkhoff sums for dynamical systems. These results are parallel to that on the partial sums of independent and identically distributed random variables previously obtained by Hsu and Robbins, Erd{\"o}s, Heyde. We apply our results to the Gauss map and obtain new precise asymptotics in the theorem of L{\'e}vy on the regular continued fraction expansion of irrational numbers in (0, 1).",

keywords = "Continued fraction, Expanding Markov interval maps, L{\'e}vy{\textquoteright}s theorem, Precise asymptotics",

author = "Lulu Fang and Hiroki Takahasi and Yiwei Zhang",

note = "Funding Information: We would like to thank two anonymous referees for their invaluable comments and suggestions that have helped to improve the original manuscript. We would also like to thank Shanghai Center for Mathematics Science, and the 2019 Fall Program of Low Dimensional Dynamics, where part of this work was written. L Fang is supported by NSFC No. 11801591. H Takahasi is supported by the JSPS KAKENHI 19K21835 and 20H01811. Y Zhang is supported by NSFC Nos. 11701200, 11871262, and Hubei Key Laboratory of Engineering Modeling and Scientific Computing in HUST. Publisher Copyright: {\textcopyright} 2021 IOP Publishing Ltd & London Mathematical Society.",

year = "2021",

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doi = "10.1088/1361-6544/ac20a2",

language = "English",

volume = "34",

pages = "7095--7108",

journal = "Nonlinearity",

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T1 - Precise asymptotics on the Birkhoff sums for dynamical systems

AU - Fang, Lulu

AU - Takahasi, Hiroki

AU - Zhang, Yiwei

N1 - Funding Information: We would like to thank two anonymous referees for their invaluable comments and suggestions that have helped to improve the original manuscript. We would also like to thank Shanghai Center for Mathematics Science, and the 2019 Fall Program of Low Dimensional Dynamics, where part of this work was written. L Fang is supported by NSFC No. 11801591. H Takahasi is supported by the JSPS KAKENHI 19K21835 and 20H01811. Y Zhang is supported by NSFC Nos. 11701200, 11871262, and Hubei Key Laboratory of Engineering Modeling and Scientific Computing in HUST. Publisher Copyright: © 2021 IOP Publishing Ltd & London Mathematical Society.

PY - 2021/10

Y1 - 2021/10

N2 - We establish two precise asymptotic results on the Birkhoff sums for dynamical systems. These results are parallel to that on the partial sums of independent and identically distributed random variables previously obtained by Hsu and Robbins, Erdös, Heyde. We apply our results to the Gauss map and obtain new precise asymptotics in the theorem of Lévy on the regular continued fraction expansion of irrational numbers in (0, 1).

AB - We establish two precise asymptotic results on the Birkhoff sums for dynamical systems. These results are parallel to that on the partial sums of independent and identically distributed random variables previously obtained by Hsu and Robbins, Erdös, Heyde. We apply our results to the Gauss map and obtain new precise asymptotics in the theorem of Lévy on the regular continued fraction expansion of irrational numbers in (0, 1).

KW - Continued fraction

KW - Expanding Markov interval maps

KW - Lévy’s theorem

KW - Precise asymptotics

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JO - Nonlinearity

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Precise asymptotics on the Birkhoff sums for dynamical systems

Abstract

Keywords

ASJC Scopus subject areas

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