A trichotomy for hitting times and escape rates for a class of unimodal maps

Mark Demers; Mike Todd

doi:10.1017/etds.2025.10201

A trichotomy for hitting times and escape rates for a class of unimodal maps

Mark Demers, Mike Todd^*

^*Corresponding author for this work

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

Abstract

We consider local escape rates and hitting time statistics for unimodal interval maps of Misiurewicz–Thurston type. We prove that for any point z in the interval, there is a local escape rate and hitting time statistics that is one of three types. While it is key that we cover all points z, the particular interest here is when z is periodic and in the postcritical orbit that yields the third part of the trichotomy. We also prove generalized asymptotic escape rates of the form first shown by Bruin, Demers and Todd.

Original language	English
Number of pages	35
Journal	Ergodic Theory and Dynamical Systems
Volume	FirstView
Early online date	11 Aug 2025
DOIs	https://doi.org/10.1017/etds.2025.10201
Publication status	E-pub ahead of print - 11 Aug 2025

Keywords

Hitting times
Escape rate
Unimodal maps
Misiurewicz-Thurston maps
Inducing techniques

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10.1017/etds.2025.10201Licence: CC BY

Demers_2025_ETaDS_Trichotomy-hitting-times-escape-rates-unimodal-maps_CC
© The Author(s), 2025. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
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Cite this

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abstract = "We consider local escape rates and hitting time statistics for unimodal interval maps of Misiurewicz–Thurston type. We prove that for any point z in the interval, there is a local escape rate and hitting time statistics that is one of three types. While it is key that we cover all points z, the particular interest here is when z is periodic and in the postcritical orbit that yields the third part of the trichotomy. We also prove generalized asymptotic escape rates of the form first shown by Bruin, Demers and Todd.",

keywords = "Hitting times, Escape rate, Unimodal maps, Misiurewicz-Thurston maps, Inducing techniques",

author = "Mark Demers and Mike Todd",

note = "Funding: M.D. was partially supported by NSF grants DMS 2055070 and DMS 2350079. M.T. 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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AU - Demers, Mark

AU - Todd, Mike

N1 - Funding: M.D. was partially supported by NSF grants DMS 2055070 and DMS 2350079. M.T. was partially supported by the FCT (Fundação para a Ciência e a Tecnologia) project 2022.07167.PTDC.

PY - 2025/8/11

Y1 - 2025/8/11

N2 - We consider local escape rates and hitting time statistics for unimodal interval maps of Misiurewicz–Thurston type. We prove that for any point z in the interval, there is a local escape rate and hitting time statistics that is one of three types. While it is key that we cover all points z, the particular interest here is when z is periodic and in the postcritical orbit that yields the third part of the trichotomy. We also prove generalized asymptotic escape rates of the form first shown by Bruin, Demers and Todd.

AB - We consider local escape rates and hitting time statistics for unimodal interval maps of Misiurewicz–Thurston type. We prove that for any point z in the interval, there is a local escape rate and hitting time statistics that is one of three types. While it is key that we cover all points z, the particular interest here is when z is periodic and in the postcritical orbit that yields the third part of the trichotomy. We also prove generalized asymptotic escape rates of the form first shown by Bruin, Demers and Todd.

KW - Hitting times

KW - Escape rate

KW - Unimodal maps

KW - Misiurewicz-Thurston maps

KW - Inducing techniques

U2 - 10.1017/etds.2025.10201

DO - 10.1017/etds.2025.10201

M3 - Article

SN - 0143-3857

VL - FirstView

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

ER -

A trichotomy for hitting times and escape rates for a class of unimodal maps

Abstract

Keywords

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