Anosov Diffeomorphisms and $${\gamma}$$ γ -Tilings

Abstract

We consider a toral Anosov automorphism \({G_\gamma:{\mathbb{T}}_\gamma\to{\mathbb{T}}_\gamma}\) given by \({G_\gamma(x,y)=(ax+y,x)}\) in the \({ }\) base, where \({a\in\mathbb{N} \backslash\{1\}}\), \({\gamma=1/(a+1/(a+1/\ldots))}\), \({v=(\gamma,1)}\) and \({w=(-1,\gamma)}\) in the canonical base of \({{\mathbb{R}}^2}\) and \({{\mathbb{T}}_\gamma={\mathbb{R}}^2/(v{\mathbb{Z}} \times w{\mathbb{Z}})}\). We introduce the notion of \({\gamma}\)-tilings to prove the existence of a one-to-one correspondence between (i) marked smooth conjugacy classes of Anosov diffeomorphisms, with invariant measures absolutely continuous with respect to the Lebesgue measure, that are in the isotopy class of \({G_\gamma}\); (ii) affine classes of \({\gamma}\)-tilings; and (iii) \({\gamma}\)-solenoid functions. Solenoid functions provide a parametrization of the infinite dimensional space of the mathematical objects described in these equivalences.