Escape of entropy for countable Markov shifts

Godofredo Iommi; Mike Todd; Aníbal Velozo

doi:10.1016/j.aim.2022.108507

Escape of entropy for countable Markov shifts

Godofredo Iommi, Mike Todd, Aníbal Velozo^*

^*Corresponding author for this work

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

Abstract

In this paper we study ergodic theory of countable Markov shifts. These are dynamical systems defined over non-compact spaces. Our main result relates the escape of mass, the measure theoretic entropy, and the entropy at infinity of the system. This relation has several consequences. For example we obtain that the entropy map is upper semi-continuous and that the ergodic measures form an entropy dense subset. Our results also provide new proofs of results describing the existence and stability of the measure of maximal entropy. We relate the entropy at infinity with the Hausdorff dimension of the set of recurrent points that escape on average. Of independent interest, we prove a version of Katok’s entropy formula in this non-compact setting.

Original language	English
Article number	108507
Number of pages	54
Journal	Advances in Mathematics
Volume	405
Early online date	10 Jun 2022
DOIs	https://doi.org/10.1016/j.aim.2022.108507
Publication status	Published - 27 Aug 2022

Keywords

Entropy
Countable
Markov shifts
Escape of mass

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10.1016/j.aim.2022.108507

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Copyright © 2022 Elsevier Inc. All rights reserved. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1016/j.aim.2022.108507
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Cite this

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title = "Escape of entropy for countable Markov shifts",

abstract = "In this paper we study ergodic theory of countable Markov shifts. These are dynamical systems defined over non-compact spaces. Our main result relates the escape of mass, the measure theoretic entropy, and the entropy at infinity of the system. This relation has several consequences. For example we obtain that the entropy map is upper semi-continuous and that the ergodic measures form an entropy dense subset. Our results also provide new proofs of results describing the existence and stability of the measure of maximal entropy. We relate the entropy at infinity with the Hausdorff dimension of the set of recurrent points that escape on average. Of independent interest, we prove a version of Katok{\textquoteright}s entropy formula in this non-compact setting.",

keywords = "Entropy, Countable, Markov shifts, Escape of mass",

author = "Godofredo Iommi and Mike Todd and An{\'i}bal Velozo",

note = "Funding: GI was partially supported by CONICYT PIA ACT172001 and by Proyecto Fondecyt 1190194. AV was supported by Proyecto Fondecyt Iniciaci{\'o}n 11220409.",

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T1 - Escape of entropy for countable Markov shifts

AU - Iommi, Godofredo

AU - Todd, Mike

AU - Velozo, Aníbal

N1 - Funding: GI was partially supported by CONICYT PIA ACT172001 and by Proyecto Fondecyt 1190194. AV was supported by Proyecto Fondecyt Iniciación 11220409.

PY - 2022/8/27

Y1 - 2022/8/27

N2 - In this paper we study ergodic theory of countable Markov shifts. These are dynamical systems defined over non-compact spaces. Our main result relates the escape of mass, the measure theoretic entropy, and the entropy at infinity of the system. This relation has several consequences. For example we obtain that the entropy map is upper semi-continuous and that the ergodic measures form an entropy dense subset. Our results also provide new proofs of results describing the existence and stability of the measure of maximal entropy. We relate the entropy at infinity with the Hausdorff dimension of the set of recurrent points that escape on average. Of independent interest, we prove a version of Katok’s entropy formula in this non-compact setting.

AB - In this paper we study ergodic theory of countable Markov shifts. These are dynamical systems defined over non-compact spaces. Our main result relates the escape of mass, the measure theoretic entropy, and the entropy at infinity of the system. This relation has several consequences. For example we obtain that the entropy map is upper semi-continuous and that the ergodic measures form an entropy dense subset. Our results also provide new proofs of results describing the existence and stability of the measure of maximal entropy. We relate the entropy at infinity with the Hausdorff dimension of the set of recurrent points that escape on average. Of independent interest, we prove a version of Katok’s entropy formula in this non-compact setting.

KW - Entropy

KW - Countable

KW - Markov shifts

KW - Escape of mass

U2 - 10.1016/j.aim.2022.108507

DO - 10.1016/j.aim.2022.108507

M3 - Article

SN - 0001-8708

VL - 405

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 108507

ER -

Escape of entropy for countable Markov shifts

Abstract

Keywords

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