Tilings and Anosov diffeomorphisms

Abstract

A. Pinto and D. Sullivan [4] proved a one-to-one correspondence between: (i) C1+ conjugacy classes of expanding circle maps; (ii) solenoid functions and (iii) Pinto-Sullivan’s dyadic tilings on the real line. A. Pinto [1,3] introduced the notion of golden tilings and proved a one-to-one correspondence between (i) smooth conjugacy classes of Anosov diffeomorphisms, with an invariant measure absolutely continuous with respect to the Lebesgue measure, that are topologically conjugate to the linear automorphism G(x; y) = (x + y; x), (ii) affine classes of golden tilings and (iii) solenoid functions. Here we extend this last result and we exhibit a natural one-to-one correspondence between (i) smooth conjugacy classes of Anosov diffeomorphisms, with an invariant measure absolutely continuous with respect to the Lebesgue measure, that are topologically conjugate to the linear automorphism G(x; y) = (ax+y; ax), where a 2 N, (ii) affine classes of tilings in the real line and (iii) solenoid functions. The solenoid functions give a parametrization of the infinite dimensional space consisting of the mathematical objects described in the above equivalences