scholarly journals Foliations and conjugacy, II: the Mendes conjecture for time-one maps of flows

2020 ◽  
pp. 1-18
Author(s):  
JORGE GROISMAN ◽  
ZBIGNIEW NITECKI
Keyword(s):  
Conjugacy Classes ◽  
Topological Conjugacy ◽  
Anosov Diffeomorphism ◽  
Anosov Diffeomorphisms

Abstract A diffeomorphism of theplane is Anosov if it has a hyperbolic splitting at every point of the plane. In addition to linear hyperbolic automorphisms, translations of the plane also carry an Anosov structure (the existence of Anosov structures for plane translations was originally shown by White). Mendes conjectured that these are the only topological conjugacy classes for Anosov diffeomorphisms in the plane. Very recently, Matsumoto gave an example of an Anosov diffeomorphism of the plane, which is a Brouwer translation but not topologically conjugate to a translation, disproving Mendes’ conjecture. In this paper we prove that Mendes’ claim holds when the Anosov diffeomorphism is the time-one map of a flow, via a theorem about foliations invariant under a time-one map. In particular, this shows that the kind of counterexample constructed by Matsumoto cannot be obtained from a flow on the plane.

scholarly journals Foliations and conjugacy: Anosov structures in the plane

2014 ◽  
Vol 35 (4) ◽  
pp. 1229-1242 ◽  
Author(s):  
JORGE GROISMAN ◽  
ZBIGNIEW NITECKI
Keyword(s):  
Equivalence Classes ◽  
Conjugacy Classes ◽  
Topological Conjugacy ◽  
Stable And Unstable Manifolds ◽  
Anosov Diffeomorphisms ◽  
Unstable Manifolds ◽  
Unbounded Orbits

AbstractIn a non-compact setting, the notion of hyperbolicity, together with the associated structure of stable and unstable manifolds (for unbounded orbits), is highly dependent on the choice of metric used to define it. We consider the simplest version of this, the analogue for the plane of Anosov diffeomorphisms, studied earlier by White and Mendes. The two known topological conjugacy classes of such diffeomorphisms are linear hyperbolic automorphisms and translations. We show that if the structure of stable and unstable manifolds is required to be preserved by these conjugacies, the number of distinct equivalence classes of Anosov diffeomorphisms in the plane becomes infinite.

On the integrability of intermediate distributions for Anosov diffeomorphisms

1995 ◽  
Vol 15 (2) ◽  
pp. 317-331 ◽  
Author(s):  
M. Jiang ◽  
Ya B. Pesin ◽  
R. de la Llave
Keyword(s):  
Finite Number ◽  
Lyapunov Exponents ◽  
Periodic Orbits ◽  
Three Dimensional ◽  
Dimensional Torus ◽  
Anosov Diffeomorphism ◽  
Anosov Diffeomorphisms ◽  
Stable Foliation

AbstractWe study the integrability of intermediate distributions for Anosov diffeomorphisms and provide an example of a C∞-Anosov diffeomorphism on a three-dimensional torus whose intermediate stable foliation has leaves that admit only a finite number of derivatives. We also show that this phenomenon is quite abundant. In dimension four or higher this can happen even if the Lyapunov exponents at periodic orbits are constant.

Rigidity of symplectic Anosov diffeomorphisms on low dimensional tori

1991 ◽  
Vol 11 (3) ◽  
pp. 427-441 ◽  
Author(s):  
L. Flaminio ◽  
A. Katok
Keyword(s):  
Anosov Diffeomorphism ◽  
Anosov Diffeomorphisms ◽  
Hyperbolic Automorphism ◽  
Low Dimensional

AbstractWe show that any symplectic Anosov diffeomorphism of a four torus T4 with sufficiently smooth stable and unstable foliations is smoothly conjugate to a linear hyperbolic automorphism of T4.

scholarly journals Diffeomorphisms Hölder conjugate to Anosov diffeomorphisms

2009 ◽  
Vol 30 (2) ◽  
pp. 441-456 ◽  
Author(s):  
ANDREY GOGOLEV
Keyword(s):  
Anosov Diffeomorphism ◽  
Anosov Diffeomorphisms ◽  
Hölder Exponents ◽  
Phd Thesis

AbstractWe show by means of a counterexample that a C1+Lip diffeomorphism Hölder conjugate to an Anosov diffeomorphism is not necessarily Anosov. Also we include a result from the 2006 PhD thesis of Fisher: a C1+Lip diffeomorphism Hölder conjugate to an Anosov diffeomorphism is Anosov itself provided that Hölder exponents of the conjugacy and its inverse are sufficiently large.

scholarly journals Manifolds which do not Admit Anosov Diffeomorphisms

1973 ◽  
Vol 49 ◽  
pp. 111-115 ◽  
Author(s):  
Kenichi Shiraiwa
Keyword(s):  
Necessary Conditions ◽  
Differentiable Manifold ◽  
Anosov Diffeomorphism ◽  
Anosov Diffeomorphisms

In [3], M. W. Hirsch obtained some necessary conditions for the existence of an Anosov diffeomorphism on a differentiable manifold. As an application, he constructed many manifolds which do not admit Anosov diffeomorphisms.

scholarly journals On one-sided topological conjugacy of topological Markov shifts and gauge actions on Cuntz–Krieger algebras

2021 ◽  
pp. 1-8
Author(s):  
KENGO MATSUMOTO
Keyword(s):  
Conjugacy Classes ◽  
Topological Conjugacy ◽  
Markov Shifts

Abstract We characterize topological conjugacy classes of one-sided topological Markov shifts in terms of the associated Cuntz–Krieger algebras and their gauge actions with potentials.

scholarly journals On the equidistribution of unstable curves for pseudo-Anosov diffeomorphisms of compact surfaces

2021 ◽  
pp. 1-26
Author(s):  
GIOVANNI FORNI
Keyword(s):  
Banach Spaces ◽  
Open Interval ◽  
Orientable Surface ◽  
Anosov Diffeomorphism ◽  
Anosov Diffeomorphisms ◽  
Invariant Foliation ◽  
Higher Genus ◽  
Compact Orientable Surface ◽  
Compact Surfaces

Abstract We prove that the asymptotics of ergodic integrals along an invariant foliation of a toral Anosov diffeomorphism, or of a pseudo-Anosov diffeomorphism on a compact orientable surface of higher genus, is determined (up to a logarithmic error) by the action of the diffeomorphism on the cohomology of the surface. As a consequence of our argument and of the results of Giulietti and Liverani [Parabolic dynamics and anisotropic Banach spaces. J. Eur. Math. Soc. (JEMS)21(9) (2019), 2793–2858] on horospherical averages, toral Anosov diffeomorphisms have no Ruelle resonances in the open interval $(1, e^{h_{\mathrm {top}}})$ .

Sinai–Ruelle–Bowen measures for N-dimensional derived from Anosov diffeomorphisms

1993 ◽  
Vol 13 (1) ◽  
pp. 21-44 ◽  
Author(s):  
Maria Carvalho
Keyword(s):  
Anosov Diffeomorphism ◽  
Anosov Diffeomorphisms ◽  
Axiom A ◽  
Srb Measures ◽  
Saddle Node ◽  
Basic Set ◽  
Hyperbolic Attractors ◽  
Generic Bifurcation

AbstractThis paper is about the existence of transitive non-hyperbolic attractors with corresponding SRB measures for arcs of diffeomorphisms crossing the boundary of the Axiom A systems, obtained through an elementary generic bifurcation (Hopf, saddle-node or flip) on a transitive Anosov diffeomorphism or an attracting basic set.

scholarly journals Real analytic actions of SL (2, ℝ) on a surface

1983 ◽  
Vol 3 (3) ◽  
pp. 447-499 ◽  
Author(s):  
Dennis C. Stowe
Keyword(s):  
Vector Space ◽  
Lie Groups ◽  
Conjugacy Classes ◽  
Topological Conjugacy ◽  
The Real ◽  
Finite Dimensional ◽  
Real Analytic ◽  
Compact Surfaces

AbstractReal analytic actions of connected Lie groups locally isomorphic to SL (2, ℝ) on compact surfaces, possibly with boundary, are classified up to topological conjugacy and up to real analytic conjugacy. Finite dimensional universal unfoldings of the real analytic conjugacy relation are also constructed. These are local transversals to the conjugacy classes in the space of actions. Sometimes the unfolding is a variety but not a manifold, and thus the space of actions is not naturally modelled on a vector space. We find many rigid actions and some unexpected bifurcation.

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