A dynamical system with a \(Z^ 2\) centralizer
The paper is concerned with Chacon \(\mathbb{Z}^2\) actions. The purpose of the authors is to study the particular action of the nontrivial subgroup \(\mathbb{Z}\) of \(\mathbb{Z}^2\). They describe the prototypical example of a Chacon \(\mathbb{Z}^2\) action and develop a symbolic version of its construction. Denoting Chacon \(\mathbb{Z}^2\) action by \(\{T^iS^j\mid(i,j)\in\mathbb{Z}^2\}\), the authors give attention to its study in terms of \(T\). Two main results are obtained: -- If \(P\) is the time zero partition of \(X\subset\{0,1\}^{\mathbb{Z}^2}\), then \(P\) is a generating partition under \(T\). -- The centralizer of \(T\) is the \(\mathbb{Z}^2\) group \(\{T^iS^j\mid(i,j)\in\mathbb{Z}^2\}\).
scientific article; zbMATH DE number 1031652
scientific article