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Transitive mappings on the Cantor fan

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

Published online by Cambridge University Press:  24 February 2025

IZTOK BANIČ Open the ORCID record for IZTOK BANIČ [Opens in a new window] ,
GORAN ERCEG Open the ORCID record for GORAN ERCEG [Opens in a new window] ,
JUDY KENNEDY ,
CHRIS MOURON  and
VAN NALL
IZTOK BANIČ*
Affiliation:
Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor 2000, Slovenia Institute of Mathematics, Physics and Mechanics, Ljubljana 1000, Slovenia Andrej Marušič Institute, University of Primorska, Koper 6000, Slovenia
GORAN ERCEG
Affiliation:
Faculty of Science, University of Split, Split 21000, Croatia (e-mail: goran.erceg@pmfst.hr)
JUDY KENNEDY
Affiliation:
Department of Mathematics, Lamar University, Beaumont 77710, Texas, USA (e-mail: kennedy9905@gmail.com)
CHRIS MOURON
Affiliation:
Rhodes College, Memphis 38112, Tennessee, USA (e-mail: mouronc@rhodes.edu)
VAN NALL
Affiliation:
Department of Mathematics, University of Richmond, Richmond 23173, Virginia, USA (e-mail: vnall@richmond.edu)
*
e-mail: iztok.banic@um.si
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Abstract

Many continua that admit a transitive homeomorphism may be found in the literature. The circle is probably the simplest non-degenerate continuum that admits such a homeomorphism. However, most of the known examples of such continua have a complicated topological structure. For example, they are indecomposable (such as the pseudo-arc or the Knaster bucket-handle continuum), or they are not indecomposable but have some other complicated topological structure, such as a dense set of ramification points (such as the Sierpiński carpet) or a dense set of end-points (such as the Lelek fan). In this paper, we continue our mission of finding continua with simpler topological structures that admit a transitive homeomorphism. We construct a transitive homeomorphism on the Cantor fan. In our approach, we use two different techniques, each of them giving two constructions of a transitive homeomorphism on the Cantor fan: one technique using quotient spaces of products of compact metric spaces and Cantor sets, and one using Mahavier products of closed relations on compact metric spaces. We also demonstrate how our technique using Mahavier products of closed relations may be used to construct a transitive function f on a Cantor fan X such that $\varprojlim (X,f)$ is a Lelek fan.

Keywords

closed relationsMahavier productstransitive dynamical systemsCantor fansLelek fans

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 - 35
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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