Teichmüller spaces and HR structures for hyperbolic surface dynamics

Abstract

Abstract. We construct a Teichmüller space for the C 1+ -conjugacy classes of hyperbolic dynamical systems on surfaces. After introducing the notion of an HR structure which associates an affine structure with each of the stable and unstable laminations, we show that there is a one-to-one correspondence between these HR structures and the C 1+ -conjugacy classes. As part of the proof we construct a canonical representative dynamical system for each HR structure. This has the smoothest holonomies of any representative of the corresponding C 1+ -conjugacy class. Finally, we introduce solenoid functions and show that they provide a good Teichmüller space. IntroductionIn this paper we study the flexibility of smooth hyperbolic dynamics on surfaces. By the flexibility of a given topological model of hyperbolic dynamics we mean the extent of different smooth realizations of this model. Thus a typical result provides a Teichmüller space or a moduli space to parametrize these realizations. In this paper we will construct Teichmüller spaces for hyperbolic sets of surface diffeomorphisms with one-dimensional stable and unstable manifolds including Anosov diffeomorphisms, attractors and Smale horseshoes. In a later paper we extend these results to pseudo-Anosov systems.To be effective it is important that these Teichmüller spaces should be easily characterized. For example, for Anosov diffeomorphisms of the torus that are either C ∞ or C 2 and preserve a smooth invariant measure, the eigenvalue spectrum is known to be a complete invariant of smooth conjugacy [4,5]. However, it is not clear which eigenvalue spectra are realized by such systems. Thus these do not make up a good Teichmüller space. The Teichmüller spaces that we construct do not suffer from this and they usually consist of easily characterized Hölder functions. Moreover, for hyperbolic systems on surfaces other than Anosov systems not only are the eigenvalue spectra difficult to characterize, they are also only a complete invariant of Lipschitz conjugacy.

Anosov Diffeomorphisms and $${\gamma}$$ γ -Tilings

“…The proofs of these results follow closely those in [21]. One of the main ingredients in the proof of Theorem 3 is the solenoid function of an Anosov map (see [26,28]), which is similar to the scaling function of a hyperbolic Cantor set (see [7,33]). The ratios between the lengths of consecutive intervals in an admissible tiling are precisely the values of the solenoid function in a certain sequence of points.…”

“…In fact, if two of these Anosov diffeomorphisms determine a same γ -tiling, then they determine the same sets of γ -tilings. Furthermore, we observe that a solenoid function determines a ratio function (see [26,28]) that measures the ratio of the asymptotic lengths of any pair of leaves with a common endpoint. Hence, any two γ -tilings are obtained from the same Anosov diffeomorphism, but using unstable lines passing through different fixed points, if, and only if, they determine the same ratio function.…”

Smoothness of Holonomies for Codimension 1 Hyperbolic Dynamics

“…In general terms, it allows one to reduce many questions about 2-dimensional dynamics to simpler questions about 1-dimensional dynamics. In related work, we use it: (i) to construct a Teichmüller space for all C 1+ conjugacy classes of hyperbolic sets of surface diffeomorphisms with 1-dimensional stable and unstable manifolds [8], (ii) to construct all such systems with an invariant measure with a given geometric measure class (such as all Anosov diffeomorphisms with an invariant measure that is absolutely continuous with respect to 2-dimensional Lebesgue measure) [10], and (iii) to prove that if the basic holonomies of such dynamical systems are sufficiently smooth, then the system is rigid in the sense that it is C 1+ conjugate to an affine model [9]. Although the results that we prove have been in the folklore for some time, there appears to be no published statement or proof of them.…”

“…However, in the case of hyperbolic sets on surfaces, we are able to show that our laminations are C 1+α in the strongest sense. In [8] we show how to use the stable and unstable ratio functions to construct uniformly bounded C 1+α orthogonal charts in which the images of the stable and unstable manifolds are respectively horizontal and vertical lines.…”

Renormalization of Circle Diffeomorphism Sequences and Markov Sequences

“…formation of the expanding and contracting laminations of the Anosov diffeomorphisms [10][11][12][13][14][15][16][17][18]). …”