Smoothness of Holonomies for Codimension 1 Hyperbolic Dynamics

Pinto, Alberto A.; Rand, D.A.J.

doi:10.1112/s0024609301008670

Cited by 23 publications

(22 citation statements)

References 14 publications

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“…Similarly, we can define the unstable lamination atlas L u = L u (G, ρ). By Theorem 2.1 in [27], the basic unstable and stable holonomies are C 1+ with respect to the lamination atlas L s .…”

Section: Lamination Atlasmentioning

confidence: 95%

Anosov and Circle Diffeomorphisms

Almeida

Fisher²,

Pinto

et al. 2011

Dynamics, Games and Science I

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We present an infinite dimensional space of C 1+ smooth conjugacy classes of circle diffeomorphisms that are C 1+ fixed points of renormalization. We exhibit a one-to-one correspondence between these C 1+ fixed points of renormalization and C 1+ conjugacy classes of Anosov diffeomorphisms. IntroductionThe link between Anosov diffeomorphisms and diffeomorphisms of the circle is due to D. Sullivan and E. Ghys through the observation that the holonomies of Anosov diffeomorphisms give rise to C 1+ circle diffeomorphisms that are C 1+ fixed points of renormalization (see also [1]). A. Pinto and D. Rand [21] proved that this observation gives an one-to-one correspondence between the corresponding smooth

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Section: Lamination Atlasmentioning

confidence: 95%

Anosov and Circle Diffeomorphisms

Almeida

Fisher²,

Pinto

et al. 2011

Dynamics, Games and Science I

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“…By Theorem 2.2 in [14], the basic ι-holonomies satisfy (2.2) and so (2.3), with respect to the charts in the lamination atlas A ι (ρ). Hence, for some 0 < β 2 ≤ 1, we have…”

Section: ) Where the Constant Of Proportionality In The O Term Only mentioning

confidence: 99%

“…By Corollary 2.4 in [14], the mapsθ y,s andθ x,u have C 1+α 1 extensionsθ y,s andθ x,u which vary Hölder continuously with y and x respectively, for some 0 < α 1 < 1. Thus, we define By the Whitney Extension Lemma (see [1]), the map u has a C 1+ extensionũ with…”

Section: Definition 52mentioning

confidence: 99%

“…The C r embeddings λ : I 1 ⊂ R → M of the full ι-leaf segments I can be reparametrized such that they are isometries between the Euclidean metric on I 1 and the induced ρ metric on I . By construction, these reparametrizations λ : [14], the lamination atlas A ι (ρ) has bounded geometry. By Theorem 2.2 in [14], the lamination atlas A ι (ρ) is C 1+α -foliated, for some 0 < α ≤ 1, and the ι-basic holonomies satisfy (2.2), with respect to the charts in this atlas.…”

Section: Definition 24mentioning

confidence: 99%

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

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confidence: 99%