Transitive mappings on 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 (X,f) is a Lelek fan.