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

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