Anosov Diffeomorphisms and -Tilings

Abstract

We consider a toral Anosov automorphism G(gamma) : T-gamma --> T-gamma given by G(gamma) (x, y) = (ax + y, x) in the < v, w > base, where , a is an element of N\{1}, gamma = 1/(a + 1/(a + 1/...)), v = (gamma, 1) and w = (-1, gamma) in the canonical base of R-2 and T-gamma = R-2 / (vZ x wZ). 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.