Minimal and proximal examples of d̄-stable and d̄-approachable shift spaces

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985556%3A_____%2F25%3A00636737" target="_blank" >RIV/67985556:_____/25:00636737 - isvavai.cz</a>

Výsledek na webu

<a href="https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/minimal-and-proximal-examples-of-bar-dstable-and-bar-dapproachable-shift-spaces/A32C9FF11308F7847A3F9F6BB051C011" target="_blank" >https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/minimal-and-proximal-examples-of-bar-dstable-and-bar-dapproachable-shift-spaces/A32C9FF11308F7847A3F9F6BB051C011</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1017/etds.2024.43" target="_blank" >10.1017/etds.2024.43</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Minimal and proximal examples of d̄-stable and d̄-approachable shift spaces

Popis výsledku v původním jazyce

We study shift spaces over a finite alphabet that can be approximated by mixing shifts of finite type in the sense of (pseudo)metrics connected to Ornstein's (d) over bar metric ((d) over bar -approachable shift spaces). The class of (d) over bar -approachable shifts can beconsidered as a topological analog of measure-theoretical Bernoulli systems. The notionof (d) over bar -approachability, together with a closely connected notion of Itshows-shadowing, was introduced by Konieczny, Kupsa, and Kwietniak [Ergod. Th. & Dynam. Sys.43(3) (2023),943-970]. These notions were developed with the aim of significantly generalizing specification properties. Indeed, many popular variants of the specification property, includingthe classic one and the almost/weak specification property, ensure (d) over bar -approachability and (d) over bar -shadowing. Here, we study further properties and connections between (d) over bar -shadowing and (d) over bar -approachability. We prove that (d) over bar -shadowing implies (d) over bar -stability (a notion recently introduced by Tim Austin). We show that for surjective shift spaces with the (d) over bar -shadowingproperty the Hausdorff pseudodistance (d) over bar (H) between shift spaces induced by (d) over bar is the sameas the Hausdorff distance between their simplices of invariant measures with respect tothe Hausdorff distance induced by Ornstein's metric (d) over bar between measures. We prove thatwithout Itshows-shadowing this need not to be true (it is known that the former distance alwaysbounds the latter). We provide examples illustrating these results, including minimal examples and proximal examples of shift spaces with the (d) over bar -shadowing property. The existence of such shift spaces was announced in the earlier paper mentioned above. It shows that (d) over bar -shadowing indeed generalizes the specification property.

Název v anglickém jazyce

Minimal and proximal examples of d̄-stable and d̄-approachable shift spaces

Popis výsledku anglicky

We study shift spaces over a finite alphabet that can be approximated by mixing shifts of finite type in the sense of (pseudo)metrics connected to Ornstein's (d) over bar metric ((d) over bar -approachable shift spaces). The class of (d) over bar -approachable shifts can beconsidered as a topological analog of measure-theoretical Bernoulli systems. The notionof (d) over bar -approachability, together with a closely connected notion of Itshows-shadowing, was introduced by Konieczny, Kupsa, and Kwietniak [Ergod. Th. & Dynam. Sys.43(3) (2023),943-970]. These notions were developed with the aim of significantly generalizing specification properties. Indeed, many popular variants of the specification property, includingthe classic one and the almost/weak specification property, ensure (d) over bar -approachability and (d) over bar -shadowing. Here, we study further properties and connections between (d) over bar -shadowing and (d) over bar -approachability. We prove that (d) over bar -shadowing implies (d) over bar -stability (a notion recently introduced by Tim Austin). We show that for surjective shift spaces with the (d) over bar -shadowingproperty the Hausdorff pseudodistance (d) over bar (H) between shift spaces induced by (d) over bar is the sameas the Hausdorff distance between their simplices of invariant measures with respect tothe Hausdorff distance induced by Ornstein's metric (d) over bar between measures. We prove thatwithout Itshows-shadowing this need not to be true (it is known that the former distance alwaysbounds the latter). We provide examples illustrating these results, including minimal examples and proximal examples of shift spaces with the (d) over bar -shadowing property. The existence of such shift spaces was announced in the earlier paper mentioned above. It shows that (d) over bar -shadowing indeed generalizes the specification property.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2025

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Ergodic Theory and Dynamical Systems

ISSN

0143-3857

e-ISSN

1469-4417

Svazek periodika

45

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

31

Strana od-do

396-426

Kód UT WoS článku

001309809400001

EID výsledku v databázi Scopus

2-s2.0-85204494007