Anosov Diffeomorphisms and ${\gamma}$-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 ${ < v,w > }$ 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.