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Anosov diffeomorphisms. (English) Zbl 1338.37039
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.
MSC:
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37E10 Dynamical systems involving maps of the circle
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