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| Inspired in the works of Y. Jiang and A. Pinto and D. Sullivan, A. Pinto et al. introduced the notion of golden tiling and proved the existence of a natural correspondence between golden tilings, smooth conjugacy classes of Anosov diffeomorphisms with invariant measure absolutely continuous with respect to Lebesgue measure and solenoid functions. Here we extend their result and introduce the notion of $\gamma$-tiling. Like the golden tilings, the $\gamma$-tilings record the infinitesimal geometric structure determined by the dynamics of an Anosov diffeomorphism G along the unstable leaf that is invariant under the action of G. The properties of $\gamma$-tilings are defined using a decomposition of natural numbers that we call $\gamma$-Fibonacci decomposition. The main contribution of this work consists in understanding the way how this $\gamma$-Fibonacci decomposition encodes the combinatorics determined by the Markov partition of G along the unstable leaf. We exhibit a natural correspondence between $\gamma$-tilings, smooth conjugacy classes of Anosov diffeomorphisms with invariant measure absolutely continuous with respect to Lebesgue measure and solenoid functions |
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