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

Result code in 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>

Result on the web

<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>

Result language

angličtina

Original language name

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

Original language description

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.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

10101 - Pure mathematics

Project

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Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2025

Confidentiality

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

Name of the periodical

Ergodic Theory and Dynamical Systems

ISSN

0143-3857

e-ISSN

1469-4417

Volume of the periodical

45

Issue of the periodical within the volume

2

Country of publishing house

US - UNITED STATES

Number of pages

31

Pages from-to

396-426

UT code for WoS article

001309809400001

EID of the result in the Scopus database

2-s2.0-85204494007