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CLASSIFICATION OF ONE DIMENSIONAL DYNAMICAL SYSTEMS BY COUNTABLE STRUCTURES

Part of: Set theory Connections with other structures, applications

Published online by Cambridge University Press:  03 October 2022

HENK BRUIN Open the ORCID record for HENK BRUIN [Opens in a new window]  and
BENJAMIN VEJNAR Open the ORCID record for BENJAMIN VEJNAR [Opens in a new window]
HENK BRUIN
Affiliation:
FACULTY OF MATHEMATICS UNIVERSITY OF VIENNA OSKAR MORGENSTERNPLATZ 1 1090 VIENNA, AUSTRIA E-mail: henk.bruin@univie.ac.at
BENJAMIN VEJNAR*
Affiliation:
FACULTY OF MATHEMATICS AND PHYSICS CHARLES UNIVERSITY PRAGUE, CZECHIA
*
E-mail: vejnar@karlin.mff.cuni.cz
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Abstract

We study the complexity of the classification problem of conjugacy on dynamical systems on some compact metrizable spaces. Especially we prove that the conjugacy equivalence relation of interval dynamical systems is Borel bireducible to isomorphism equivalence relation of countable graphs. This solves a special case of Hjorth’s conjecture which states that every orbit equivalence relation induced by a continuous action of the group of all homeomorphisms of the closed unit interval is classifiable by countable structures. We also prove that conjugacy equivalence relation of Hilbert cube homeomorphisms is Borel bireducible to the universal orbit equivalence relation.

Keywords

conjugacyclassificationBorel reductioncountable structuresuniversal orbit equivalence relation

MSC classification

Primary: 54H20: Topological dynamics
Secondary: 03E15: Descriptive set theory
Type
Article
Information
The Journal of Symbolic Logic , Volume 88 , Issue 2 , June 2023 , pp. 562 - 578
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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