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

Abstract

We consider a toral Anosov automorphism G γ : T γ → T γ given by G γ (x, y) = (ax + y, x) in the < v, w > base, where a ∈ N\{1}, γ = 1/(a + 1/(a + 1/ . . .)), v = (γ , 1) and w = (−1, γ ) in the canonical base of R 2 and T γ = R 2 /(vZ × wZ). We introduce the notion of γ -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 γ ; (ii) affine classes of γ -tilings; and (iii) γ -solenoid functions. Solenoid functions provide a parametrization of the infinite dimensional space of the mathematical objects described in these equivalences.