Explicit Dynamical Systems on the Sierpi\'nski Curve

TL;DR

This paper constructs dynamical systems on the Sierpiński carpet using inverse limits of specific surface transformations, demonstrating all positive entropy values and showing the absence of the Bowen specification property.

Contribution

It introduces a new method for creating dynamical systems on the Sierpiński carpet with arbitrary positive entropy and simplifies existing proofs regarding their dynamical properties.

Findings

All positive real numbers can be realized as entropy values.

Dynamical systems on the carpet lack the Bowen specification property.

The paper provides a new construction method using inverse limits of surface transformations.

Abstract

We apply Boro\'nski and Oprocha's inverse limit construction of dynamical systems on the Sierpi\'nski carpet by using the initial systems of $n-$Chamanara surfaces and their $n-$baker transformations, $n \geq 2$. We show that all positive real numbers are realized as metric entropy values of dynamical systems on the carpet. We also produce a simplification of Boro\'nski and Oprocha's proof showing that dynamical systems on the carpet do not have the Bowen specification property.