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Sufficient conditions for non-zero entropy of closed relations

Part of: Special properties Maps and general types of spaces defined by maps Topological dynamics

Published online by Cambridge University Press:  15 February 2024

IZTOK BANIČ Open the ORCID record for IZTOK BANIČ [Opens in a new window] ,
RENE GRIL ROGINA ,
JUDY KENNEDY  and
VAN NALL
IZTOK BANIČ*
Affiliation:
Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor 2000, Slovenia (e-mail: rene.gril@student.um.si) Institute of Mathematics, Physics and Mechanics, Ljubljana 1000, Slovenia Andrej Marušič Institute, University of Primorska, Koper 6000, Slovenia
RENE GRIL ROGINA
Affiliation:
Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor 2000, Slovenia (e-mail: rene.gril@student.um.si)
JUDY KENNEDY
Affiliation:
Department of Mathematics, Lamar University, Beaumont, TX 77710,USA (e-mail: kennedy9905@gmail.com)
VAN NALL
Affiliation:
Department of Mathematics, University of Richmond, Richmond, VA 23173, USA (e-mail: vnall@richmond.edu)
*
e-mail: iztok.banic@um.si
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Abstract

We introduce the notions of returns and well-aligned sets for closed relations on compact metric spaces and then use them to obtain non-trivial sufficient conditions for such a relation to have non-zero entropy. In addition, we give a characterization of finite relations with non-zero entropy in terms of Li–Yorke and DC2 chaos.

Keywords

closed relationsMahavier productsdynamical systemsentropyLi-Yorke chaosdistributional chaos

MSC classification

Primary: 37B45: Continua theory in dynamics
Secondary: 54C60: Set-valued maps 54F15: Continua and generalizations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 29
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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