Transitive mappings on the Cantor fan

TL;DR

This paper constructs transitive homeomorphisms on the Cantor fan, a simpler topological continuum, using four novel techniques, expanding the class of known transitive continua.

Contribution

It introduces four new methods to create transitive homeomorphisms on the Cantor fan, a simpler continuum, broadening understanding of dynamical systems on such spaces.

Findings

Constructed a transitive homeomorphism on the Cantor fan.

Developed four distinct techniques for such constructions.

Showed how Mahavier products can produce a Lelek fan from the Cantor fan.

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. On the other hand, 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 Sierpi\' nski 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 four different techniques, each of them giving a unique construction of a transitive homeomorphism on the Cantor fan:} two techniques using quotient spaces of products of compact metric spaces and Cantor sets, and two 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.