Cover times in dynamical systems

Natalia Jurga; Mike Todd

doi:10.1007/s11856-026-2912-9

Cover times in dynamical systems

Natalia Jurga^*
, Mike Todd

^*Corresponding author for this work

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

Abstract

We introduce the notion of cover time to dynamical systems. This quantifies the rate at which orbits become dense in the state space and can be viewed as a global, rather than the more standard local, notion of recurrence for a system. Using transfer operator tools for systems with holes and inducing techniques, we obtain an asymptotic formula for the expected cover time in terms of the decay rate of the measure of the ball of minimum measure. We apply this to a wide class of uniformly hyperbolic and non-uniformly hyperbolic interval maps, including the Gauss map and Manneville–Pomeau maps.

Original language	English
Number of pages	60
Journal	Israel Journal of Mathematics
Volume	Online first
Early online date	6 Apr 2026
DOIs	https://doi.org/10.1007/s11856-026-2912-9
Publication status	E-pub ahead of print - 6 Apr 2026

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© 2026 The Author(s). This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium, provided the appropriate credit is given to the original authors and the source, and a link is provided to the Creative Commons license, indicating if changes were made.
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Fractals, dimension theory: Fractals, dimension theory and random matrix products
Jurga, N. (PI)
Leverhulme Trust
1/05/22 → 30/04/25
Project: Standard
Fourier analytic techniques: Fourier analytic techniques in geometry and analysis
Fraser, J. (PI) & Falconer, K. (CoI)
EPSRC
1/02/18 → 11/06/21
Project: Standard

Cite this

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title = "Cover times in dynamical systems",

abstract = "We introduce the notion of cover time to dynamical systems. This quantifies the rate at which orbits become dense in the state space and can be viewed as a global, rather than the more standard local, notion of recurrence for a system. Using transfer operator tools for systems with holes and inducing techniques, we obtain an asymptotic formula for the expected cover time in terms of the decay rate of the measure of the ball of minimum measure. We apply this to a wide class of uniformly hyperbolic and non-uniformly hyperbolic interval maps, including the Gauss map and Manneville–Pomeau maps.",

author = "Natalia Jurga and Mike Todd",

note = "Funding: NJ was supported by an EPSRC Standard Grant (EP/R015104/1) and Leverhulme Early Career Fellowship (ECF-2021-385). MT was partially supported by the FCT (Funda{\c c}{\~a}o para a Ci{\^e}ncia e a Tecnologia) project 2022.07167.PTDC.",

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N2 - We introduce the notion of cover time to dynamical systems. This quantifies the rate at which orbits become dense in the state space and can be viewed as a global, rather than the more standard local, notion of recurrence for a system. Using transfer operator tools for systems with holes and inducing techniques, we obtain an asymptotic formula for the expected cover time in terms of the decay rate of the measure of the ball of minimum measure. We apply this to a wide class of uniformly hyperbolic and non-uniformly hyperbolic interval maps, including the Gauss map and Manneville–Pomeau maps.

AB - We introduce the notion of cover time to dynamical systems. This quantifies the rate at which orbits become dense in the state space and can be viewed as a global, rather than the more standard local, notion of recurrence for a system. Using transfer operator tools for systems with holes and inducing techniques, we obtain an asymptotic formula for the expected cover time in terms of the decay rate of the measure of the ball of minimum measure. We apply this to a wide class of uniformly hyperbolic and non-uniformly hyperbolic interval maps, including the Gauss map and Manneville–Pomeau maps.

U2 - 10.1007/s11856-026-2912-9

DO - 10.1007/s11856-026-2912-9

M3 - Article

SN - 0021-2172

VL - Online first

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

ER -

Cover times in dynamical systems

Abstract

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Projects

Fractals, dimension theory: Fractals, dimension theory and random matrix products

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

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