Lelek-like Fans: Endpoint-dense Continua Supporting Topologically Mixing Maps

TL;DR

This paper constructs a diverse family of non-smooth Lelek-like fans with dense endpoints and demonstrates that each admits topologically mixing dynamics, including both non-invertible maps and homeomorphisms.

Contribution

It introduces uncountably many non-homeomorphic non-smooth fans with dense endpoints and proves their compatibility with topologically mixing maps and homeomorphisms.

Findings

Constructed uncountably many non-homeomorphic non-smooth fans with dense endpoints.

Proved each fan admits a topologically mixing non-invertible map.

Proved each fan admits a topologically mixing homeomorphism.

Abstract

The Lelek fan is the only smooth fan that has a dense set of end-points. In this paper, we study non-smooth fans with this property; i.e., we construct an uncountable family of pairwise non-homeomorphic such fans. Furthermore, we prove that each of them admits a topologically mixing non-invertible mapping as well as a topologically mixing homeomorphism.