Almeida, João P.; Fisher, Albert M.; Pinto, Alberto A.; Rand, David A. Anosov diffeomorphisms. (English) Zbl 1338.37039 Discrete Contin. Dyn. Syst. 2013, Suppl., 837-845 (2013). The paper explicitly constructs a one-to-one correspondence between \(C^{1^+}\) conjugacy classes of \(C^{1^+}\) Anosov diffeomorphisms on the \(2\)-torus and pairs of \(C^{1^+}\) stable and unstable self-renormalizable sequences. All of the smooth information of the foliations of \(C^{1^+}\) Anosov diffeomorphisms is encoded in the one-dimensional smooth self-renormalizable sequences.The one-to-one correspondence is an extension of earlier work for hyperbolic diffeomorphisms on surfaces done by two of the authors. This extension makes use of the Adler-Tresser-Worfolk decomposition of linear Anosov diffeomorphisms of the \(2\)-torus [R. Adler et al., Trans. Am. Math. Soc. 349, No. 4, 1633–1652 (1997; Zbl 0947.37027)].Another one-to-one correspondence is explicitly constructed between \(C^{1^+}\) conjugacy classes of Anosov diffeomorphisms on the \(2\)-torus and pairs of \(C^{1^+}\) circle diffeomorphisms that are \(C^{1^+}\) periodic points of renormalization with respect to certain \(C^{1^+}\) structures. Reviewer: Lennard Bakker (Provo) MSC: 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37E10 Dynamical systems involving maps of the circle Keywords:Anosov diffeomorphisms; self-renormalizable structures; train-tracks; Markov maps; circle diffeomorphisms Citations:Zbl 0947.37027 PDFBibTeX XMLCite \textit{J. P. Almeida} et al., Discrete Contin. Dyn. Syst. 2013, 837--845 (2013; Zbl 1338.37039) Full Text: Link