A transitive homeomorphism on the Lelek fan

Let X be a continuum and let be a homeomorphism. To construct a dynamical system with interesting dynamical properties, the continuum X often needs to have some complicated topological structure. In this paper, we are interested in one such dynamical property: transitivity. By now, various examples...

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Bibliographic Details
Published in:Journal of difference equations and applications 2023-04, Vol.29 (4), p.393-418
Main Authors: Banič, Iztok, Erceg, Goran, Kennedy, Judy
Format: Article
Language:English
Subjects:
Closed relations
Dynamical systems
fans
Lelek fans
Mahavier products
Topology
transitive dynamical systems
transitive homeomorphisms
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Summary:Let X be a continuum and let be a homeomorphism. To construct a dynamical system with interesting dynamical properties, the continuum X often needs to have some complicated topological structure. In this paper, we are interested in one such dynamical property: transitivity. By now, various examples of continua X have been constructed in such a way that the dynamical system is transitive. Mostly, they are examples of continua that are not path-connected, such as the pseudo-arc or the pseudo-circle, or they are examples of locally connected continua (and every locally connected continuum is path-connected), Sierpiński carpet is  such an example. In this paper, we present an example of a dynamical system , where φ is a homeomorphism on the continuum X and X is a path-connected but not locally connected continuum. We construct a transitive homeomorphism on the Lelek fan. As a by-product, a non-invertible transitive map on the Lelek fan is also constructed.
ISSN:1023-6198
1563-5120
DOI:10.1080/10236198.2023.2208242