The Experts below are selected from a list of 243 Experts worldwide ranked by ideXlab platform
Karel Dekimpe - One of the best experts on this subject based on the ideXlab platform.
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admitting an Anosov Diffeomorphism
2016Co-Authors: Karel Dekimpe, Wim MalfaitAbstract: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.
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Existence of Anosov Diffeomorphisms on infra-nilmanifolds modeled on free nilpotent Lie groups
Topological Methods in Nonlinear Analysis, 2015Co-Authors: Karel Dekimpe, Jonas DereAbstract: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.
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Existence of Anosov Diffeomorphisms on infra-nilmanifolds modeled on free nilpotent Lie groups
arXiv: Dynamical Systems, 2013Co-Authors: Karel Dekimpe, Jonas DereAbstract: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.
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constructing infra nilmanifolds admitting an Anosov Diffeomorphism
Advances in Mathematics, 2011Co-Authors: Karel Dekimpe, Kelly VerheyenAbstract:Abstract In this paper we establish an algebraic characterization of those infra-nilmanifolds modeled on a free c -step nilpotent Lie group and with an abelian holonomy group admitting an Anosov Diffeomorphism. We also develop a new method for constructing examples of infra-nilmanifolds having an Anosov Diffeomorphism.
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Anosov Diffeomorphisms on nilmanifolds modelled on a free nilpotent Lie group
Dynamical Systems-an International Journal, 2010Co-Authors: Karel Dekimpe, Kelly VerheyenAbstract:In this article we give an elementary proof, using standard arguments from algebraic number theory, of the fact that a nilmanifold modelled on a free c-step nilpotent Lie group on n generators admits an Anosov Diffeomorphism if and only if n > c. In fact, we need to show that for any integer n > 1, there exists a matrix A ∈ GL(n, ℤ), such that any product of less than n eigenvalues of A is of modulus ≠ 1.
Masato Tsujii - One of the best experts on this subject based on the ideXlab platform.
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prequantum transfer operator for symplectic Anosov Diffeomorphism
Astérisque, 2015Co-Authors: Frederic Faure, Masato TsujiiAbstract: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.
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prequantum transfer operator for Anosov Diffeomorphism preliminary version
arXiv: Mathematical Physics, 2012Co-Authors: Frederic Faure, Masato TsujiiAbstract: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 - One of the best experts on this subject based on the ideXlab platform.
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Existence of Anosov Diffeomorphisms on infra-nilmanifolds modeled on free nilpotent Lie groups
Topological Methods in Nonlinear Analysis, 2015Co-Authors: Karel Dekimpe, Jonas DereAbstract: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.
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which infra nilmanifolds admit an expanding map or an Anosov Diffeomorphism
2015Co-Authors: Jonas DereAbstract: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
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A new method for constructing Anosov Lie algebras
Transactions of the American Mathematical Society, 2015Co-Authors: Jonas DereAbstract: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
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A new method for constructing Anosov Lie algebras
arXiv: Dynamical Systems, 2013Co-Authors: Jonas DereAbstract:It is conjectured that every 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 counterexamples to 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.
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Existence of Anosov Diffeomorphisms on infra-nilmanifolds modeled on free nilpotent Lie groups
arXiv: Dynamical Systems, 2013Co-Authors: Karel Dekimpe, Jonas DereAbstract: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.
Frederic Faure - One of the best experts on this subject based on the ideXlab platform.
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prequantum transfer operator for symplectic Anosov Diffeomorphism
Astérisque, 2015Co-Authors: Frederic Faure, Masato TsujiiAbstract: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.
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prequantum transfer operator for Anosov Diffeomorphism preliminary version
arXiv: Mathematical Physics, 2012Co-Authors: Frederic Faure, Masato TsujiiAbstract: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.
Alistair Windsor - One of the best experts on this subject based on the ideXlab platform.
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smooth dependence on parameters of solutions to cohomologyequations over Anosov systems with applications to cohomologyequations on Diffeomorphism groups
Discrete and Continuous Dynamical Systems, 2010Co-Authors: Rafael De La Llave, Alistair WindsorAbstract:We consider the dependence on parameters of the solutions of cohomology equations over Anosov Diffeomorphisms. We show that the solutions depend on parameters as smoothly as the data. As a consequence we prove optimal regularity results for the solutions of cohomology equations taking value in Diffeomorphism groups. These results are motivated by applications to rigidity theory, dynamical systems, and geometry. In particular, in the context of Diffeomorphism groups we show: Let $f$ be a transitive Anosov Diffeomorphism of a compact manifold $M$. Suppose that $\eta \in C$k+α$(M,$Diff$^r(N))$ for a compact manifold $N$, $k,r \in \N$, $r \geq 1$, and $0 < \alpha \leq \Lip$. We show that if there exists a $\varphi\in C$k+α$(M,$Diff$^1(N))$ solving $ \varphi_{f(x)} = \eta_x \circ \varphi_x$ then in fact $\varphi \in C$k+α$(M,$Diff$^r(N))$. The existence of this solutions for some range of regularities is studied in the literature.
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livsic theorems for non commutative groups including Diffeomorphism groups and results on the existence of conformal structures for Anosov systems
Ergodic Theory and Dynamical Systems, 2010Co-Authors: Rafael De La Llave, Alistair WindsorAbstract:The celebrated Livsic theorem [A. N. Livsic, Certain properties of the homology of Y -systems, Mat. Zametki 10 (1971), 555–564; A. N. Livsic, Cohomology of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1296–1320] states that given a manifold M , a Lie group G , a transitive Anosov Diffeomorphism f on M and a Holder function η : M ↦ G whose range is sufficiently close to the identity, it is sufficient for the existence of ϕ: M ↦ G satisfying η ( x )=ϕ( f ( x ))ϕ( x ) −1 that a condition—obviously necessary—on the cocycle generated by η restricted to periodic orbits is satisfied. In this paper we present a new proof of the main result. These methods allow us to treat cocycles taking values in the group of Diffeomorphisms of a compact manifold. This has applications to rigidity theory. The localization procedure we develop can be applied to obtain some new results on the existence of conformal structures for Anosov systems.
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smooth dependence on parameters of solution of cohomology equations over Anosov systems and applications to cohomology equations on Diffeomorphism groups
arXiv: Dynamical Systems, 2008Co-Authors: Rafael De La Llave, Alistair WindsorAbstract:We consider the dependence on parameters of the solutions of cohomology equations over Anosov Diffeomorphisms. We show that the solutions depend on parameters as smoothly as the data. As a consequence we prove optimal regularity results for the solutions of equations taking value in Diffeomorphism groups. These results are motivated by applications to rigidity theory, dynamical systems, and geometry. In particular, in the context of Diffeomorphism groups we show: Let $f$ be a transitive Anosov Diffeomorphism of a compact manifold $M$. Suppose that $\eta \in C^{\reg}(M,\Diff^r(N))$ for a compact manifold $N$, $k,r \in \N$, $r \geq 1$, and $0 < \alpha \leq \Lip$. We show that if there exists a $\varphi\in C^{\reg}(M,\Diff^1(N))$ solving \begin{equation*} \varphi_{f(x)} = \eta_x \circ \varphi_x \end{equation*} then in fact $\varphi \in C^{\reg}(M,\Diff^r(N))$.
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liv v s ic theorems for non commutative groups including Diffeomorphism groups and results on the existence of conformal structures for Anosov systems
arXiv: Dynamical Systems, 2007Co-Authors: Rafael De La Llave, Alistair WindsorAbstract:The celebrated Livsic theorem states that given M a manifold, a Lie group G, a transitive Anosov Diffeomorphism f on M and a Holder function \eta: M \mapsto G whose range is sufficiently close to the identity, it is sufficient for the existence of \phi :M \mapsto G satisfying \eta(x) = \phi(f(x)) \phi(x)^{-1} that a condition -- obviously necessary -- on the cocycle generated by \eta restricted to periodic orbits is satisfied. In this paper we present a new proof of the main result. These methods allow us to treat cocycles taking values in the group of Diffeomorphisms of a compact manifold. This has applications to rigidity theory. The localization procedure we develop can be applied to obtain some new results on the existence of conformal structures for Anosov systems.
