Dynamics on fences

TL;DR

This paper introduces a general method to transfer dynamical properties from Cantor set homeomorphisms to fence-like spaces, unifying their study and revealing new links between topology and dynamics.

Contribution

It provides a systematic construction linking Cantor set dynamics to fence spaces, enabling analysis of their structural and dynamical features.

Findings

Transferred properties include minimality, recurrence, and orbit structure.

Unified framework for dynamics on various fence-like spaces.

New connections between topological structure and dynamical behavior.

Abstract

Homeomorphisms of the Cantor set play a central role in topology, dynamical systems and descriptive set theory. In parallel, several classes of fence-like spaces - such as the hairy Cantor set, hairy arcs, Cantor bouquets in complex dynamics, the Lelek fan in topology and Fra\"iss\'e fence in descriptive set theory - have recently been studied for their rich structural and dynamical properties. In this paper, we introduce a general construction that associates to each homeomorphism of the Cantor set a canonically defined homeomorphism of a corresponding fence space. This construction lifts dynamical properties from the Cantor set to these fence-like spaces, allowing one to systematically transfer features such as minimality, recurrence, and orbit structure. As a consequence, we obtain a unified framework for studying dynamics on a broad class of fence-like spaces and establish new connections between their topological structure and induced dynamical behavior.