A transitive homeomorphism on the Lelek fan

TL;DR

This paper constructs a transitive homeomorphism on the Lelek fan, a path-connected but not locally connected continuum, expanding the class of known dynamical systems with such properties.

Contribution

It introduces the first example of a transitive homeomorphism on the Lelek fan, a complex topological continuum, and also constructs a non-invertible transitive map on it.

Findings

Existence of a transitive homeomorphism on the Lelek fan.

Construction of a non-invertible transitive map on the Lelek fan.

Extension of dynamical systems theory to new topological structures.

Abstract

Let $X$ be a continuum and let $\varphi:X\rightarrow X$ be a homeomorphism. To construct a dynamical system $(X,\varphi)$ 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 $(X,\varphi)$ 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), Wazewski's universal dendrite and the Sierpinski carpet are such examples. In this paper, we present an example of a dynamical system $(X,\varphi)$, where $\varphi$ 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.