Golden tilings.

The authors consider the golden Anosov automorphism GA on the torus T 2 , defined in such a way that the ratio of its unstable and stable eigenvalues is the golden number (1 + √ 5)/2. They study the space G of all C 1+α diffeomorphisms that are topologically conjugate to GA and possess an invariant measure absolutely continuous with respect to Lebesgue. More precisely, they study the infinite dimensional space of smooth conjugacy classes in G. The purpose of the paper is to prove that this space of conjugacy classes can be parametrized in two ways: (i) by the affine classes of the so-called ̀̀golden tilingś́, and (ii) by the so- called ̀̀solenoid functionś́. In the first main theorem of the paper (Theorem 1.1) the authors prove that there exists a well-defined one-to-one map that associates to each C 1+α diffeomorphism G ∈ G a sequence, which is called golden, and describes the infinitesimal geometric structure of the dynamics along an invariant unstable leaf of G. Afterwards, the concept of rigidity of a golden sequence ai is defined, when the values of ai are rigidly determined, related to the golden number and according to the Fibonacci decomposition of the index i. The second main theorem (Theorem 1.3) states that if G ∈ G has a C 1+Zigmund complete system of unstable holonomies, then the golden sequence of G must be rigid. Afterwards, the authors apply the notion of tiling. This is a sequence Ii of closed intervals in R, with pairwise disjoint interiors and such that their union is R. In particular, a tiling is golden if |Ii+1|/|Ii | is a golden sequence. With this tool, they prove that the space of smooth conjugacy classes in G is in one-to-one correspondence with the affine classes of golden tilings. Finally, the authors consider the notion of “solenoid functions”. These functions are constructed from the microscopic (bounded) distortion by iteration of G ∈ G to pairs of unstable leaf segments. They prove that the solenoid functions are also in a one-to-one correspondence with the smooth conjugacy classes of G.