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

Iztok Banic, Goran Erceg, Ivan Jelic, Judy Kennedy

TL;DR

The paper addresses the existence and dynamical richness of Lelek-like fans, i.e., dendroids with dense end-points, extending chaotic dynamics from the classical Lelek fan to a broad non-smooth class. It develops a quotient- and Mahavier-dynamics framework to produce an uncountable family of pairwise non-homeomorphic Lelek-like fans, each supporting both a topologically mixing non-invertible map and a topologically mixing homeomorphism. Key contributions include a constructive uncountable family via quotients $L/_{\sim_{\mathbf i}}$ and a demonstration that these quotients retain Lelek-like structure while preserving mixing behavior. The results highlight the versatility of Mahavier dynamics and quotient constructions in generating diverse spaces with strong mixing properties on dendroidal continua.

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.

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

TL;DR

The paper addresses the existence and dynamical richness of Lelek-like fans, i.e., dendroids with dense end-points, extending chaotic dynamics from the classical Lelek fan to a broad non-smooth class. It develops a quotient- and Mahavier-dynamics framework to produce an uncountable family of pairwise non-homeomorphic Lelek-like fans, each supporting both a topologically mixing non-invertible map and a topologically mixing homeomorphism. Key contributions include a constructive uncountable family via quotients and a demonstration that these quotients retain Lelek-like structure while preserving mixing behavior. The results highlight the versatility of Mahavier dynamics and quotient constructions in generating diverse spaces with strong mixing properties on dendroidal continua.

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.
Paper Structure (6 sections, 27 theorems, 100 equations, 9 figures)

This paper contains 6 sections, 27 theorems, 100 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Non-smooth fans
  3. An uncountable family of Lelek-like fans
  4. Mahavier dynamical systems
  5. Topological mixing on Lelek-like fans
  6. Acknowledgement

Key Result

Theorem 2.6

Let $X$ be a fan with the top $v$. Then there is a family of arcs $\mathcal{L}$ in $X$ such that

Figures (9)

  • Figure 2: A fan with a non-connected non-smooth set.
  • Figure 3: An aligned triple in $A$
  • Figure 4: An interruption
  • Figure 5: The relation $\sim_{(B,C,D)}$ on $B\cup C\cup D$
  • Figure 6: The interruption $q_{\sim}(B)$ in $L/_{\sim}$
  • ...and 4 more figures

Theorems & Definitions (103)

  • Definition 2.1
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • Theorem 2.12
  • proof
  • ...and 93 more