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

Abstract

For a ${\mathbb{Z}}^d$-topological dynamical system $(X,T,{\mathbb{Z}}^d)$, an isomomorphism is a self-homeomorphism $\ddot{I}•:X\to X$ such that for some matrix $M\in \mathrm{GL}(d,\mathbb{Z})$ and any $n\in {\mathbb{Z}}^d$, $\ddot{I}•\circ T^n=T^{Mn}\circ \ddot{I}•$, 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.

30 pages, 4 figures