The Specification Property on the Lelek Fan

TL;DR

This paper demonstrates that the Lelek fan, a smooth one-dimensional continuum, can exhibit diverse dynamical behaviors similar to symbolic spaces, using Mahavier products of closed relations.

Contribution

It extends the study of dynamical systems with separation properties from symbolic spaces to the Lelek fan, providing a unified framework via Mahavier products.

Findings

Constructed Mahavier products homeomorphic to the Lelek fan

Obtained shift maps with diverse dynamical behaviors on the Lelek fan

Demonstrated sharper separation of dynamical properties on a smooth continuum

Abstract

Recent work of Piotr Oprocha and his collaborators has provided a number of delicate examples of dynamical systems separating specification, shadowing, and periodic-point density, primarily in symbolic or totally disconnected spaces. The goal of the present paper is to demonstrate that similar - and in some cases sharper - separations occur on the Lelek fan, a smooth one-dimensional continuum. Our constructions rely on Mahavier products of closed relations. By carefully choosing relations on the unit interval, we obtain Mahavier products that are homeomorphic to the Lelek fan whose associated shift maps display diverse dynamical behavior. This approach yields a unified framework for producing and analyzing examples on a familiar continuum.