Explicit dynamical systems on the Sierpiński carpet

Abstract

We apply Boroński and Oprocha’s inverse limit construction of dynamical systems on the Sierpiński carpet by using the initial systems of $n$-Chamanara surfaces and their $n$-baker transformations, $n\ge 2$. 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 Boroński and Oprocha’s proof showing that dynamical systems on the carpet do not have the Bowen specification property.