The Experts below are selected from a list of 294 Experts worldwide ranked by ideXlab platform
Kenichiro Yamamoto - One of the best experts on this subject based on the ideXlab platform.
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Ergodic measure-expansive Diffeomorphisms
Dynamical Systems-an International Journal, 2014Co-Authors: Kazuhiro Sakai, Naoya Sumi, Kenichiro YamamotoAbstract:In this paper, we consider the set of Diffeomorphisms which are measure-expansive for any ergodic measure, and study the set from the viewpoint of geometric theory of dynamical systems. It is proved that (1) there exists a non-empty C1-open set of robustly non-hyperbolic and transitive Diffeomorphisms such that each element of the set is measure-expansive for any ergodic measure, and that (2) C1-generically, a diffeomorphism is measure-expansive for any generic ergodic measure.
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measure expansive Diffeomorphisms
Journal of Mathematical Analysis and Applications, 2014Co-Authors: Kazuhiro Sakai, Naoya Sumi, Kenichiro YamamotoAbstract:Abstract In this paper, measure-expansive Diffeomorphisms are considered and the characterizations of the C 1 -interiors of the set of measure-expansive Diffeomorphisms are obtained.
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Diffeomorphisms satisfying the specification property
Proceedings of the American Mathematical Society, 2010Co-Authors: Kazuhiro Sakai, Naoya Sumi, Kenichiro YamamotoAbstract:Let f be a diffeomorphism of a closed C ∞ manifold M. In this paper, we introduce the notion of the C 1 -stable specification property for a closed f-invariant set Λof M, and we prove that f/ Λ satisfies a C 1 -stable specification property if and only if Λ is a hyperbolic elementary set. As a corollary, the C 1 -interior of the set of Diffeomorphisms of M satisfying the specification property is characterized as the set of transitive Anosov Diffeomorphisms.
Kazuhiro Sakai - One of the best experts on this subject based on the ideXlab platform.
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Ergodic measure-expansive Diffeomorphisms
Dynamical Systems-an International Journal, 2014Co-Authors: Kazuhiro Sakai, Naoya Sumi, Kenichiro YamamotoAbstract:In this paper, we consider the set of Diffeomorphisms which are measure-expansive for any ergodic measure, and study the set from the viewpoint of geometric theory of dynamical systems. It is proved that (1) there exists a non-empty C1-open set of robustly non-hyperbolic and transitive Diffeomorphisms such that each element of the set is measure-expansive for any ergodic measure, and that (2) C1-generically, a diffeomorphism is measure-expansive for any generic ergodic measure.
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measure expansive Diffeomorphisms
Journal of Mathematical Analysis and Applications, 2014Co-Authors: Kazuhiro Sakai, Naoya Sumi, Kenichiro YamamotoAbstract:Abstract In this paper, measure-expansive Diffeomorphisms are considered and the characterizations of the C 1 -interiors of the set of measure-expansive Diffeomorphisms are obtained.
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Diffeomorphisms satisfying the specification property
Proceedings of the American Mathematical Society, 2010Co-Authors: Kazuhiro Sakai, Naoya Sumi, Kenichiro YamamotoAbstract:Let f be a diffeomorphism of a closed C ∞ manifold M. In this paper, we introduce the notion of the C 1 -stable specification property for a closed f-invariant set Λof M, and we prove that f/ Λ satisfies a C 1 -stable specification property if and only if Λ is a hyperbolic elementary set. As a corollary, the C 1 -interior of the set of Diffeomorphisms of M satisfying the specification property is characterized as the set of transitive Anosov Diffeomorphisms.
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The oe-property of Diffeomorphisms
Discrete and Continuous Dynamical Systems, 1998Co-Authors: Kazuhiro SakaiAbstract:In this paper, the $C^1$ interior of the set of all Diffeomorphisms satisfying the OE-property is characterized as the set of all Diffeomorphisms satisfying Axiom A and the strong transversality condition. Thus the $C^1$ interior of the set of all Diffeomorphisms satisfying the OE-property is equal to the $C^1$ interior of the set of all Diffeomorphisms satisfying the shadowing property.
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Continuum-wise expansive Diffeomorphisms
Publicacions Matematiques, 1997Co-Authors: Kazuhiro SakaiAbstract:In this paper, we show that the $C^1$ interior of the set of all continuum-wise expansive Diffeomorphisms of a closed manifold coincides with the $C^1$ interior of the set of all expansive Diffeomorphisms. And the $C^1$ interior of the set of all continuum-wise fully expansive Diffeomorphisms on a surface is investigated. Furthermore, we have necessary and sufficient conditions for a diffeomorphism belonging to these open sets to be Anosov.
Bassam Fayad - One of the best experts on this subject based on the ideXlab platform.
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On mixing Diffeomorphisms of the disc
Inventiones Mathematicae, 2019Co-Authors: Artur Avila, Bassam Fayad, Patrice Le Calvez, Disheng Xu, Zhiyuan ZhangAbstract:We prove that a Ck, k≥2 pseudo-rotation f of the disc with non-Brjuno rotation number is Ck−1-rigid. The proof is based on two ingredients: (1) we derive from Franks’ Lemma on free discs that a pseudo-rotation with small rotation number compared to its C1 norm must be close to the identity map; (2) using Pesin theory, we obtain an effective finite information version of the Katok closing lemma for an area preserving surface diffeomorphism f, that provides a controlled gap in the possible growth of the derivatives of f between exponential and sub-exponential. Our result on rigidity, together with a KAM theorem by Russmann, allow to conclude that analytic pseudo-rotations of the disc or the sphere are never topologically mixing. Due to a structure theorem by Franks and Handel of zero entropy surface Diffeomorphisms, it follows that an analytic conservative diffeomorphism of the disc or the sphere that is topologically mixing must have positive topological entropy.
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On mixing Diffeomorphisms of the disc
Inventiones mathematicae, 2019Co-Authors: Artur Avila, Bassam Fayad, Patrice Le Calvez, Disheng Xu, Zhiyuan ZhangAbstract:We prove that a $$C^k$$ C k , $$k\ge 2$$ k ≥ 2 pseudo-rotation f of the disc with non-Brjuno rotation number is $$C^{k-1}$$ C k - 1 -rigid. The proof is based on two ingredients: (1) we derive from Franks’ Lemma on free discs that a pseudo-rotation with small rotation number compared to its $$C^1$$ C 1 norm must be close to the identity map; (2) using Pesin theory, we obtain an effective finite information version of the Katok closing lemma for an area preserving surface diffeomorphism f , that provides a controlled gap in the possible growth of the derivatives of f between exponential and sub-exponential. Our result on rigidity, together with a KAM theorem by Rüssmann, allow to conclude that analytic pseudo-rotations of the disc or the sphere are never topologically mixing. Due to a structure theorem by Franks and Handel of zero entropy surface Diffeomorphisms, it follows that an analytic conservative diffeomorphism of the disc or the sphere that is topologically mixing must have positive topological entropy.
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on mixing Diffeomorphisms of the disk
arXiv: Dynamical Systems, 2015Co-Authors: Artur Avila, Bassam Fayad, Disheng Xu, Le P Calvez, Zhiyuan ZhangAbstract:We prove that a real analytic pseudo-rotation $f$ of the disc or the sphere is never topologically mixing. When the rotation number of $f$ is of Brjuno type, the latter follows from a KAM theorem of R\"ussmann on the stability of real analytic elliptic fixed points. In the non-Brjuno case, we prove that a pseudo-rotation of class $C^k$, $k\geq 2$, is $C^{k-1}$-rigid using the simple observation, derived from Franks' Lemma on free discs, that a pseudo-rotation with small rotation number compared to its $C^1$ (or H\"older) norm must be close to Identity. From our result and a structure theorem by Franks and Handel (on zero entropy surface Diffeomorphisms) it follows that an analytic conservative diffeomorphism of the disc or the sphere that is topologically mixing must have positive topological entropy. In our proof we need an a priori limit on the growth of the derivatives of the iterates of a pseudo-rotation that we obtain via an effective finite information version of the Katok closing lemma for an area preserving surface diffeomorphism $f$, that provides a controlled gap in the possible growth of the derivatives of $f$ between exponential and sub-exponential.
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weak mixing disc and annulus Diffeomorphisms with arbitrary liouville rotation number on the boundary
Annales Scientifiques De L Ecole Normale Superieure, 2005Co-Authors: Bassam Fayad, Maria SaprykinaAbstract:Abstract Let M be an m-dimensional differentiable manifold with a nontrivial circle action S = { S t } t ∈ R , S t + 1 = S t , preserving a smooth volume μ. For any Liouville number α we construct a sequence of area-preserving Diffeomorphisms H n such that the sequence H n ○ S α ○ H n −1 converges to a smooth weak mixing diffeomorphism of M. The method is a quantitative version of the approximation by conjugations construction introduced in [Trans. Moscow Math. Soc. 23 (1970) 1]. For m = 2 and M equal to the unit disc D 2 = { x 2 + y 2 ⩽ 1 } or the closed annulus A = T × [ 0 , 1 ] this result proves the following dichotomy: α ∈ R ∖ Q is Diophantine if and only if there is no ergodic diffeomorphism of M whose rotation number on the boundary equals α (on at least one of the boundaries in the case of A ). One part of the dichotomy follows from our constructions, the other is an unpublished result of Michael Herman asserting that if α is Diophantine, then any area preserving diffeomorphism with rotation number α on the boundary (on at least one of the boundaries in the case of A ) displays smooth invariant curves arbitrarily close to the boundary which clearly precludes ergodicity or even topological transitivity.
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weak mixing disc and annulus Diffeomorphisms with arbitrary liouville rotation number on the boundary
arXiv: Dynamical Systems, 2005Co-Authors: Bassam Fayad, Maria SaprykinaAbstract:Let $M$ be an $m$-dimensional differentiable manifold with a nontrivial circle action ${\mathcal S}= {\lbrace S_t \rbrace}_{t \in\RR}, S_{t+1}=S_t$, preserving a smooth volume $\mu$. For any Liouville number $\a$ we construct a sequence of area-preserving Diffeomorphisms $H_n$ such that the sequence $H_n\circ S_\a\circ H_n^{-1}$ converges to a smooth weak mixing diffeomorphism of $M$. The method is a quantitative version of the approximation by conjugations construction introduced in \cite{AK}. For $m=2$ and $M$ equal to the unit disc $\DD^2=\{x^2+y^2\leq 1\}$ or the closed annulus $\AAA=\TT\times [0,1]$ this result proves the following dichotomy: $\a \in \RR \setminus\QQ$ is Diophantine if and only if there is no ergodic diffeomorphism of $M$ whose rotation number on the boundary equals $\alpha$ (on at least one of the boundaries in the case of $\AAA$). One part of the dichotomy follows from our constructions, the other is an unpublished result of Michael Herman asserting that if $\a$ is Diophantine, then any area preserving diffeomorphism with rotation number $\a$ on the boundary (on at least one of the boundaries in the case of $\AAA$) displays smooth invariant curves arbitrarily close to the boundary which clearly precludes ergodicity or even topological transitivity.
Laurent Freidel - One of the best experts on this subject based on the ideXlab platform.
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Diffeomorphisms and spin foam models
Nuclear Physics, 2003Co-Authors: Laurent Freidel, David LouapreAbstract:Abstract We study the action of Diffeomorphisms on spin foam models. We prove that in 3 dimensions, there is a residual action of the Diffeomorphisms that explains the naive divergences of state sum models. We present the gauge fixing of this symmetry and show that it explains the original renormalization of Ponzano–Regge model. We discuss the implication this action of Diffeomorphisms has on higher dimensional spin foam models and especially the finite ones.
Nicholas Ayache - One of the best experts on this subject based on the ideXlab platform.
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a log euclidean framework for statistics on Diffeomorphisms
Medical Image Computing and Computer-Assisted Intervention, 2006Co-Authors: Vincent Arsigny, Olivier Commowick, Xavier Pennec, Nicholas AyacheAbstract:In this article, we focus on the computation of statistics of invertible geometrical deformations (i.e., Diffeomorphisms), based on the generalization to this type of data of the notion of principal logarithm. Remarkably, this logarithm is a simple 3D vector field, and is well-defined for Diffeomorphisms close enough to the identity. This allows to perform vectorial statistics on Diffeomorphisms, while preserving the invertibility constraint, contrary to Euclidean statistics on displacement fields. We also present here two efficient algorithms to compute logarithms of Diffeomorphisms and exponentials of vector fields, whose accuracy is studied on synthetic data. Finally, we apply these tools to compute the mean of a set of Diffeomorphisms, in the context of a registration experiment between an atlas an a database of 9 T1 MR images of the human brain.
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statistics on Diffeomorphisms in a log euclidean framework
1st MICCAI Workshop on Mathematical Foundations of Computational Anatomy: Geometrical Statistical and Registration Methods for Modeling Biological Sha, 2006Co-Authors: Vincent Arsigny, Olivier Commowick, Xavier Pennec, Nicholas AyacheAbstract:In this article, we focus on the computation of statistics of invertible geometrical deformations (i.e., Diffeomorphisms), based on the generalization to this type of data of the notion of principal logarithm. Remarkably, this logarithm is a simple 3D vector field, and can be used for Diffeomorphisms close enough to the identity. This allows to perform vectorial statistics on Diffeomorphisms, while preserving the invertibility constraint, contrary to Euclidean statistics on displacement fields.