On dynamics of the Sierpi\'nski carpet

TL;DR

This paper demonstrates that the Sierpiński carpet can be equipped with a homeomorphism exhibiting strong mixing behavior, but lacking Bowen's specification property, advancing understanding of its dynamical complexity.

Contribution

It introduces a specific homeomorphism on the Sierpiński carpet with strong mixing properties and shows it does not satisfy Bowen's specification, highlighting new dynamical features.

Findings

Sierpiński carpet admits a strongly mixing homeomorphism

Constructed example lacks Bowen's specification property

Advances understanding of fractal dynamics

Abstract

We prove that the Sierpi\'nski curve admits a homeomorphism with strong mixing properties. We also prove that the constructed example does not have Bowen's specification property.