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Anosov Diffeomorphism

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Karel Dekimpe – 1st expert on this subject based on the ideXlab platform

  • admitting an Anosov Diffeomorphism
    , 2016
    Co-Authors: Karel Dekimpe, Wim Malfait

    Abstract:

    A nilmanifold admits an Anosov Diffeomorphism if and only if its fundamental group (which is finitely generated, torsion-free and nilpotent) supports an automorphism having no eigenvalues of absolute value one. Here we concentrate on nilpotency class 2 and fundamental groups whose commuta, tor subgroup is of maximal torsion-free rank. We prove that the corresponding nilmanifold admits an Anosov Diffeomorphism if and only if the torsion-free rank of the abelianization of its fundamental group is greater than or equal to 3.

  • Existence of Anosov Diffeomorphisms on infra-nilmanifolds modeled on free nilpotent Lie groups
    Topological Methods in Nonlinear Analysis, 2015
    Co-Authors: Karel Dekimpe, Jonas Dere

    Abstract:

    An infra-nilmanifold is a manifold which is constructed as a~quotient space $\Gamma\setminus G$ of a simply connected nilpotent Lie group $G$, where $\Gamma$ is a discrete group acting properly discontinuously and cocompactly on~$G$ via so called affine maps. The manifold $\Gamma\setminus G$ is said to be modeled on the Lie group~$G$. This class of manifolds is conjectured to be the only class of closed manifolds allowing an Anosov Diffeomorphism. However, it is far from obvious which of these infra-nilmanifolds actually do admit an Anosov Diffeomorphism. In this paper we completely solve this question for infra-nilmanifolds modeled on a free $c$-step nilpotent Lie group.

  • Existence of Anosov Diffeomorphisms on infra-nilmanifolds modeled on free nilpotent Lie groups
    arXiv: Dynamical Systems, 2013
    Co-Authors: Karel Dekimpe, Jonas Dere

    Abstract:

    An infra-nilmanifold is a manifold which is constructed as a quotient space $\Gamma\backslash G$ of a simply connected nilpotent Lie group $G$, where $\Gamma$ is a discrete group acting properly discontinuously and cocompactly on $G$ via so called affine maps. The manifold $\Gamma\backslash G$ is said to be modeled on the Lie group $G$. This class of manifolds is conjectured to be the only class of closed manifolds allowing an Anosov Diffeomorphism. However, it is far from obvious which of these infra–nilmanifolds actually do admit an Anosov Diffeomorphism. In this paper we completely solve this question for infra-nilmanifolds modeled on a free $c$–step nilpotent Lie group.

Masato Tsujii – 2nd expert on this subject based on the ideXlab platform

  • prequantum transfer operator for symplectic Anosov Diffeomorphism
    Astérisque, 2015
    Co-Authors: Frederic Faure, Masato Tsujii

    Abstract:

    We define the prequantization of a symplectic Anosov Diffeomorphism f:M-> M, which is a U(1) extension of the Diffeomorphism f preserving an associated specific connection, and study the spectral properties of the associated transfer operator, called prequantum transfer operator. This is a model for the transfer operators associated to geodesic flows on negatively curved manifolds (or contact Anosov flows). We restrict the prequantum transfer operator to the N-th Fourier mode with respect to the U(1) action and investigate the spectral property in the limit N->infinity, regarding the transfer operator as a Fourier integral operator and using semi-classical analysis. In the main result, we show a ” band structure ” of the spectrum, that is, the spectrum is contained in a few separated annuli and a disk concentric at the origin. We show that, with the special (Holder continuous) potential V0=1/2 log |det Df_x|_{E_u}|, the outermost annulus is the unit circle and separated from the other parts. For this, we use an extension of the transfer operator to the Grassmanian bundle. Using Atiyah-Bott trace formula, we establish the Gutzwiller trace formula with exponentially small reminder for large time. We show also that, for a potential V such that the outermost annulus is separated from the other parts, most of the eigenvalues in the outermost annulus concentrate on a circle of radius exp where denotes the spatial average on M. The number of these eigenvalues is given by the “Weyl law”, that is, N^d.Vol(M) with d=1/2. dim(M) in the leading order. We develop a semiclassical calculus associated to the prequantum operator by defining quantization of observables Op(psi) in an intrinsic way. We obtain that the semiclassical Egorov formula of quantum transport is exact. We interpret all these results from a physical point of view as the emergence of quantum dynamics in the classical correlation functions for large time. We compare these results with standard quantization (geometric quantization) in quantum chaos.

  • prequantum transfer operator for Anosov Diffeomorphism preliminary version
    arXiv: Mathematical Physics, 2012
    Co-Authors: Frederic Faure, Masato Tsujii

    Abstract:

    This is a preliminary version and some other results will appear in the next version. We define the prequantization of a symplectic Anosov Diffeomorphism, which is a U(1) extension of the Diffeomorphism preserving an associated specific connection. We study the spectrum of the associated transfer operator, called prequantum transfer operator, restricted to the N-th Fourier mode with respect to the U(1) action on P. We investigate the spectral property in the limit N to infinity, regarding the transfer operator as a Fourier integral operator and using semi-classical analysis. In the main result, we show a ” band structure ” of the spectrum, that is, the spectrum is contained in a few separated annuli and a disk concentric at the origin.

Jonas Dere – 3rd expert on this subject based on the ideXlab platform

  • Existence of Anosov Diffeomorphisms on infra-nilmanifolds modeled on free nilpotent Lie groups
    Topological Methods in Nonlinear Analysis, 2015
    Co-Authors: Karel Dekimpe, Jonas Dere

    Abstract:

    An infra-nilmanifold is a manifold which is constructed as a~quotient space $\Gamma\setminus G$ of a simply connected nilpotent Lie group $G$, where $\Gamma$ is a discrete group acting properly discontinuously and cocompactly on~$G$ via so called affine maps. The manifold $\Gamma\setminus G$ is said to be modeled on the Lie group~$G$. This class of manifolds is conjectured to be the only class of closed manifolds allowing an Anosov Diffeomorphism. However, it is far from obvious which of these infra-nilmanifolds actually do admit an Anosov Diffeomorphism. In this paper we completely solve this question for infra-nilmanifolds modeled on a free $c$-step nilpotent Lie group.

  • which infra nilmanifolds admit an expanding map or an Anosov Diffeomorphism
    , 2015
    Co-Authors: Jonas Dere

    Abstract:

    Expanding maps and Anosov Diffeomorphisms are important types of dynamical systems since they were among the first examples with structural stability and chaotic behavior. Every closed manifold admitting an expanding map is homeomorphic to an infra-nilmanifold and it is conjectured that the same is true for manifolds admitting an Anosov Diffeomorphism. This motivates the research of expanding maps and Anosov Diffeomorphisms on infra-nilmanifolds. Although, up to homeomorphism, infra-nilmanifolds are the only closed manifolds supporting an expanding map, not every infra-nilmanifold admits an expanding map. Similarly the existence of an Anosov Diffeomorphism on an infra-nilmanifold puts strong conditions on its fundamental group. This dissertation studies which infra-nilmanifolds admit an expanding map or an Anosov Diffeomorphism. Because of the algebraic nature of infra-nilmanifolds, these questions are translated into studying the group morphisms of their fundamental groups, which are exactly the almost-Bieberbach groups. The main results of this essay give algebraic methods for deciding whether a given infra-nilmanifold admits an expanding map or an Anosov Diffeomorphism. The proofs in this dissertation combine methods from different branches in mathematics, including (geometric) group theory, number theory, Lie algebras, linear algebraic groups and representation theory of finite groups. The first part of this thesis gives the necessary background about the definitions and results in these areas which are needed throughout the following chapters. The emphasis of this first part is on self-maps of infra-nilmanifolds and the relation to expanding maps and Anosov Diffeomorphisms. The second part focuses on the situation of expanding maps. The main result gives an algebraic criterion to decide whether an infra-nilmanifold admits an expanding map or not. This criterion only depends on the covering Lie group and more specific on the existence of a positive grading on the corresponding Lie algebra. The proof of this result consists of two steps which use different

  • A new method for constructing Anosov Lie algebras
    Transactions of the American Mathematical Society, 2015
    Co-Authors: Jonas Dere

    Abstract:

    It is conjectured that every closed manifold admitting an Anosov Diffeomorphism is, up to homeomorphism, finitely covered by a nilmanifold. Motivated by this conjecture, an important problem is to determine which nilmanifolds admit an Anosov Diffeomorphism. The main theorem of this article gives a general method for constructing Anosov Diffeomorphisms on nilmanifolds. As a consequence, we give new examples which were overlooked in a corollary of the classification of low-dimensional nilmanifolds with Anosov Diffeomorphisms and a correction to this statement is proven. This method also answers some open questions about the existence of Anosov Diffeomorphisms which are minimal in some sense.status: publishe