Orbits closeness for slowly mixing dynamical systems

Jerome Rousseau; Mike Todd

doi:10.1017/etds.2023.50

Orbits closeness for slowly mixing dynamical systems

Jerome Rousseau, Mike Todd^*

^*Corresponding author for this work

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

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Abstract

Given a dynamical system, we prove that the shortest distance between two n-orbits scales like n to a power even when the system has slow mixing properties, thus building and improving on results of Barros, Liao and the first author [On the shortest distance between orbits and the longest common substring problem. Adv. Math. 344 (2019), 311–339]. We also extend these results to flows. Finally, we give an example for which the shortest distance between two orbits has no scaling limit.

Original language	English
Pages (from-to)	1192 - 1208
Number of pages	17
Journal	Ergodic Theory and Dynamical Systems
Volume	44
Issue number	4
Early online date	24 Jul 2023
DOIs	https://doi.org/10.1017/etds.2023.50
Publication status	Published - Apr 2024

Keywords

Shortest distance
Longest common substring
Correlation dimension
Inducing schemes

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10.1017/etds.2023.50Licence: CC BY

Rousseau_2023_ETDS_Orbits-closeness_CC
Copyright © The Author(s), 2023. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Final published version, 266 KBLicence: CC BY

Cite this

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title = "Orbits closeness for slowly mixing dynamical systems",

abstract = "Given a dynamical system, we prove that the shortest distance between two n-orbits scales like n to a power even when the system has slow mixing properties, thus building and improving on results of Barros, Liao and the first author [On the shortest distance between orbits and the longest common substring problem. Adv. Math. 344 (2019), 311–339]. We also extend these results to flows. Finally, we give an example for which the shortest distance between two orbits has no scaling limit.",

keywords = "Shortest distance, Longest common substring, Correlation dimension, Inducing schemes",

author = "Jerome Rousseau and Mike Todd",

note = "Funding: Both authors were partially supported by FCT projects PTDC/MAT-PUR/28177/2017 and by CMUP (UIDB/00144/2020), which is funded by FCT with national (MCTES) and European structural funds through the programs FEDER, under the partnership agreement PT2020. JR was also partially supported by CNPq and PTDC/MATPUR/4048/2021, and with national funds.",

year = "2024",

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doi = "10.1017/etds.2023.50",

language = "English",

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pages = "1192 -- 1208",

journal = "Ergodic Theory and Dynamical Systems",

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publisher = "Cambridge University Press",

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T1 - Orbits closeness for slowly mixing dynamical systems

AU - Rousseau, Jerome

AU - Todd, Mike

N1 - Funding: Both authors were partially supported by FCT projects PTDC/MAT-PUR/28177/2017 and by CMUP (UIDB/00144/2020), which is funded by FCT with national (MCTES) and European structural funds through the programs FEDER, under the partnership agreement PT2020. JR was also partially supported by CNPq and PTDC/MATPUR/4048/2021, and with national funds.

PY - 2024/4

Y1 - 2024/4

N2 - Given a dynamical system, we prove that the shortest distance between two n-orbits scales like n to a power even when the system has slow mixing properties, thus building and improving on results of Barros, Liao and the first author [On the shortest distance between orbits and the longest common substring problem. Adv. Math. 344 (2019), 311–339]. We also extend these results to flows. Finally, we give an example for which the shortest distance between two orbits has no scaling limit.

AB - Given a dynamical system, we prove that the shortest distance between two n-orbits scales like n to a power even when the system has slow mixing properties, thus building and improving on results of Barros, Liao and the first author [On the shortest distance between orbits and the longest common substring problem. Adv. Math. 344 (2019), 311–339]. We also extend these results to flows. Finally, we give an example for which the shortest distance between two orbits has no scaling limit.

KW - Shortest distance

KW - Longest common substring

KW - Correlation dimension

KW - Inducing schemes

U2 - 10.1017/etds.2023.50

DO - 10.1017/etds.2023.50

M3 - Article

SN - 0143-3857

VL - 44

SP - 1192

EP - 1208

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

IS - 4

ER -

Orbits closeness for slowly mixing dynamical systems

Abstract

Keywords

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