On distributional limit laws for recurrence

Mark Holland; Mike Todd

doi:10.1088/1361-6544/ade044

On distributional limit laws for recurrence

Mark Holland^*, Mike Todd

^*Corresponding author for this work

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

Abstract

For a probability measure preserving dynamical system (𝒳,f,µ), the Poincaré Recurrence theorem asserts that µ-almost every orbit is recurrent with respect to its initial condition. This motivates study of the statistics of the process X_n(x) = d(fⁿ(x), x)), and real-valued functions thereof. For a wide class of non-uniformly expanding dynamical systems, we show that the time-n counting process R_n(x) associated to the number recurrences below a certain radii sequence r_n(τ) follows an averaged Poisson distribution G(τ). Furthermore, we obtain quantitative results on almost sure rates for the recurrence statistics of the process X_n.

Original language	English
Article number	075028
Number of pages	26
Journal	Nonlinearity
Volume	38
Issue number	7
DOIs	https://doi.org/10.1088/1361-6544/ade044
Publication status	Published - 3 Jul 2025

Keywords

Recurrence
Return time statistics
Poisson limits
Almost sure limits

Access to Document

10.1088/1361-6544/ade044Licence: CC BY

Holland_2025_Nonlinearity_On-distributional-limit-laws-recurrence_CC
© 2025 The Author(s). Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 license (https://creativecommons.org/licenses/by/4.0/). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Final published version, 761 KBLicence: CC BY

Cite this

@article{10d4c0c833eb4b588ac9b7acf00ef67a,

title = "On distributional limit laws for recurrence",

abstract = "For a probability measure preserving dynamical system (𝒳,f,µ), the Poincar{\'e} Recurrence theorem asserts that µ-almost every orbit is recurrent with respect to its initial condition. This motivates study of the statistics of the process Xn(x) = d(fn(x), x)), and real-valued functions thereof. For a wide class of non-uniformly expanding dynamical systems, we show that the time-n counting process Rn(x) associated to the number recurrences below a certain radii sequence rn(τ) follows an averaged Poisson distribution G(τ). Furthermore, we obtain quantitative results on almost sure rates for the recurrence statistics of the process Xn.",

keywords = "Recurrence, Return time statistics, Poisson limits, Almost sure limits",

author = "Mark Holland and Mike Todd",

note = "Funding: MT was partially supported by the FCT (Funda{\c c}ao para a Ciencia e a Tecnologia) Project 2022.07167.PTDC. MH acknowledges University of St Andrews for hospitality. Both authors gratefully acknowledge LMS Scheme 4 funding",

year = "2025",

month = jul,

day = "3",

doi = "10.1088/1361-6544/ade044",

language = "English",

volume = "38",

journal = "Nonlinearity",

issn = "0951-7715",

publisher = "IOP Publishing Ltd.",

number = "7",

}

TY - JOUR

T1 - On distributional limit laws for recurrence

AU - Holland, Mark

AU - Todd, Mike

N1 - Funding: MT was partially supported by the FCT (Fundaçao para a Ciencia e a Tecnologia) Project 2022.07167.PTDC. MH acknowledges University of St Andrews for hospitality. Both authors gratefully acknowledge LMS Scheme 4 funding

PY - 2025/7/3

Y1 - 2025/7/3

N2 - For a probability measure preserving dynamical system (𝒳,f,µ), the Poincaré Recurrence theorem asserts that µ-almost every orbit is recurrent with respect to its initial condition. This motivates study of the statistics of the process Xn(x) = d(fn(x), x)), and real-valued functions thereof. For a wide class of non-uniformly expanding dynamical systems, we show that the time-n counting process Rn(x) associated to the number recurrences below a certain radii sequence rn(τ) follows an averaged Poisson distribution G(τ). Furthermore, we obtain quantitative results on almost sure rates for the recurrence statistics of the process Xn.

AB - For a probability measure preserving dynamical system (𝒳,f,µ), the Poincaré Recurrence theorem asserts that µ-almost every orbit is recurrent with respect to its initial condition. This motivates study of the statistics of the process Xn(x) = d(fn(x), x)), and real-valued functions thereof. For a wide class of non-uniformly expanding dynamical systems, we show that the time-n counting process Rn(x) associated to the number recurrences below a certain radii sequence rn(τ) follows an averaged Poisson distribution G(τ). Furthermore, we obtain quantitative results on almost sure rates for the recurrence statistics of the process Xn.

KW - Recurrence

KW - Return time statistics

KW - Poisson limits

KW - Almost sure limits

U2 - 10.1088/1361-6544/ade044

DO - 10.1088/1361-6544/ade044

M3 - Article

SN - 0951-7715

VL - 38

JO - Nonlinearity

JF - Nonlinearity

IS - 7

M1 - 075028

ER -

On distributional limit laws for recurrence

Abstract

Keywords

Access to Document

Fingerprint

Cite this