Large normalizers of ${\mathbb Z}^{d}$-odometers systems and realization on substitutive subshifts

TL;DR

This paper characterizes the normalizer groups of ${f Z}^d$-odometers and minimal ${f Z}^d$-substitutive subshifts, revealing complex structures and providing the first example of a minimal zero-entropy subshift with maximal normalizer group.

Contribution

It offers a detailed description of isomorphisms for ${f Z}^d$-odometers and minimal ${f Z}^d$-substitutive subshifts, including the first example of a minimal zero-entropy subshift with the largest normalizer group.

Findings

Complete description of isomorphisms for ${f Z}^2$-odometers.

Identification of the normalizer group structure for certain subshifts.

First known example of a minimal zero-entropy subshift with maximal normalizer group.

Abstract

For a ${\mathbb Z}^{d}$-topological dynamical system $(X, T, {\mathbb Z}^{d})$, an isomomorphism is a self-homeomorphism $\phi : X\to X$ such that for some matrix $M\in {\rm GL}(d,{\mathbb Z})$ and any ${n}\in {\mathbb Z}^{d}$, $\phi\circ T^{{n}}=T^{M{n}}\circ \phi$, where $T^{n}$ denote the self-homeomorphism of $X$ given by the action of ${n}\in {\mathbb Z}^d$. The collection of all the isomorphisms forms a group that is the normalizer of the set of transformations $T^{n}$. In the one-dimensional case, isomorphisms correspond to the notion of flip conjugacy of dynamical systems and by this fact are also called reversing symmetries. These isomorphisms are not well understood even for classical systems. We present a description of them for odometers and more precisely for constant-base ${\mathbb Z}^{2}$-odometers, which is surprisingly not simple. We deduce a complete description of the isomorphisms of some minimal ${\mathbb Z}^{d}$-substitutive subshifts. This enables us to provide the first example known of a minimal zero-entropy subshift with the largest possible normalizer group.