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Communications in Mathematical Physics

, Volume 345, Issue 2, pp 435–456 | Cite as

Anosov Diffeomorphisms and \({\gamma}\)-Tilings

  • João P. Almeida
  • Alberto A. Pinto
Article
  • 79 Downloads

Abstract

We consider a toral Anosov automorphism \({G_\gamma:{\mathbb{T}}_\gamma\to{\mathbb{T}}_\gamma}\) given by \({G_\gamma(x,y)=(ax+y,x)}\) in the \({ < v,w > }\) base, where \({a\in\mathbb{N} \backslash\{1\}}\), \({\gamma=1/(a+1/(a+1/\ldots))}\), \({v=(\gamma,1)}\) and \({w=(-1,\gamma)}\) in the canonical base of \({{\mathbb{R}}^2}\) and \({{\mathbb{T}}_\gamma={\mathbb{R}}^2/(v{\mathbb{Z}} \times w{\mathbb{Z}})}\). We introduce the notion of \({\gamma}\)-tilings to prove the existence of a one-to-one correspondence between (i) marked smooth conjugacy classes of Anosov diffeomorphisms, with invariant measures absolutely continuous with respect to the Lebesgue measure, that are in the isotopy class of \({G_\gamma}\); (ii) affine classes of \({\gamma}\)-tilings; and (iii) \({\gamma}\)-solenoid functions. Solenoid functions provide a parametrization of the infinite dimensional space of the mathematical objects described in these equivalences.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.LIAAD–INESC TEC and Department of Mathematics, School of Technology and ManagementPolytechnic Institute of BragançaBragançaPortugal
  2. 2.LIAAD–INESC TEC and Department of Mathematics, Faculty of SciencesUniversity of PortoPortoPortugal

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