A Regularity Lemma for Functions of Several Variables

Journé, Jean-Lin

doi:10.4171/rmi/69

Cited by 101 publications

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“…Therefore, the conjugacy between φ and ψ is C r along the stable and unstable leaves of the transverse stable and unstable foliations with uniformly smooth leaves. Hence, by Journé (see [7]) the conjugacy is C r−ε for all small ε > 0, which ends the proof of part (ii) of this theorem. ✷…”

Section: Teichmüller Spaces For Hyperbolic Surface Dynamics 1925supporting

confidence: 59%

Teichmüller spaces and HR structures for hyperbolic surface dynamics

Pinto

Rand

2002

Ergod. Th. Dynam. Sys.

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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.

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Section: Teichmüller Spaces For Hyperbolic Surface Dynamics 1925supporting

confidence: 59%

Teichmüller spaces and HR structures for hyperbolic surface dynamics

Pinto

Rand

2002

Ergod. Th. Dynam. Sys.

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“…Since the leaves are 1-dimensional, the fact that h preserves these densities implies that h is C ∞ along strong stable leaves and along strong unstable leaves. Since strong stable, strong unstable, and center leaves span complementary directions, it follows (Journé [28]) that h is C ∞ in the ambient space, as stated.…”

Section: Rigidity and Center Lyapunov Exponentsmentioning

confidence: 88%

Extremal Lyapunov exponents: an invariance principle and applications

2010

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Abstract. We propose a new approach to analyzing dynamical systems that combine hyperbolic and non-hyperbolic ("center") behavior, e.g. partially hyperbolic diffeomorphisms. A number of applications illustrate its power.We find that any ergodic automorphism of the 4-torus with two eigenvalues in the unit circle is stably Bernoulli among symplectic maps. Indeed, any nearby symplectic map has no zero Lyapunov exponents, unless it is volume preserving conjugate to the automorphism itself. Another main application is to accessible skew-product maps preserving area on the fibers. We prove, in particular, that if the genus of the fiber is at least 2 then the Lyapunov exponents must be different from zero and vary continuously with the map.These, and other dynamical conclusions, originate from a general Invariance Principle we prove in here. It is formulated in terms of smooth cocycles, that is, fiber bundle morphisms acting by diffeomorphisms on the fibers. The extremal Lyapunov exponents measure the smallest and largest exponential rates of growth of the derivative along the fibers. The Invariance Principle states that if these two numbers coincide then the fibers carry some amount of structure which is transversely invariant, that is, invariant under certain canonical families of homeomorphisms between fibers.

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“…For each x, y ∈ W we will define an isometric identification I x,y : T x W → T y W of the respective tangent spaces. 23 In addition, for each x ∈ W we consider a chart Υ x : B ρ * ⊆ T x W → W where B ρ * = {ξ ∈ R ds : ξ < ρ * } with a radius ρ * > 0 that can be chosen the same for all W and all x ∈ W . We will use Υ x to define "balls centred at x" in W .…”

Section: Proofsmentioning

confidence: 99%

Ruelle Perron Frobenius spectrum for Anosov maps

2002

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Abstract. We extend a number of results from one dimensional dynamics based on spectral properties of the Ruelle-Perron-Frobenius transfer operator to Anosov diffeomorphisms on compact manifolds. This allows to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the transfer operator is quasicompact. (Information on the existence of an SRB measure, its smoothness properties and statistical properties readily follow from such a result.) In dimension d = 2 we show that the transfer operator associated to smooth random perturbations of the map is close, in a proper sense, to the unperturbed transfer operator. This allows to obtain easily very strong spectral stability results, which in turn imply spectral stability results for smooth deterministic perturbations as well. Finally, we are able to implement an Ulam type finite rank approximation scheme thus reducing the study of the spectral properties of the transfer operator to a finite dimensional problem.

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A Regularity Lemma for Functions of Several Variables

Cited by 101 publications

References 0 publications

Teichmüller spaces and HR structures for hyperbolic surface dynamics

Teichmüller spaces and HR structures for hyperbolic surface dynamics

Extremal Lyapunov exponents: an invariance principle and applications

Ruelle Perron Frobenius spectrum for Anosov maps

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