Zhihong Xia, Hua Zhang
We prove a $C^r$ closing lemma for a class of partially hyperbolic symplectic diffeomorphisms. We show that for a generic $C^r$ symplectic diffeomorphism, $r =1, 2, ...,$, with two dimensional center and close to a product map, the set of all periodic points is dense.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We prove the ergodic Closing Lemma for Nonsingular Endomorphisms.
For every $r\in\mathbb{N}_{\geq2}\cup\{\infty\}$, we prove the $C^r$-closing lemma for general and conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle. In particular, it implies periodic points are dense for $C^r$-generic conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle.
Asaoka & Irie recently proved a $C^{\infty}$ closing lemma of Hamiltonian diffeomorphisms of closed surfaces. We reformulated their techniques into a more general perturbation lemma for area-preserving diffeomorphism and proved a $C^{\infty}$ closing lemma for area-preserving diffeomorphisms on a torus that is isotopic to identity. i.e., we show that the set of periodic orbits is dense for a generic diffeomorphism isotopic to identity area-preserving diffeomorphism on torus. The main tool is the flux vector of area-preserving diffeomorphisms which is, different from Hamiltonian cases, non-zero…
We prove that for a polynomial diffeomorphism of C^2 , the support of any invariant measure, apart from a few obvious cases, is contained in the closure of the set of saddle periodic points.
We investigate the open Closing Lemma problem for vector fields on the 2-dimensional torus. Under the assumption of bounded type rotation number, the $C^r$ Closing Lemma is verified for smooth vector fields that are area-preserving at all saddle points. Namely, given such a $C^r$ vector field $X$, $r\geq 4$, with a non-trivially recurrent point $p$, there exists a vector field $Y$ arbitrarily near to $X$ in the $C^r$ topology and obtained from $X$ by a twist perturbation, such that $p$ is a periodic point of $Y$. The proof relies on a new result in 1-dimensional dynamics on the non-existence o…
$C^r$-Chain closing lemma for certain partially hyperbolic diffeomorphisms
本文证明了部分双曲微分同胚的 $C^r$-链闭包引理，并由此得出通有情形下周期点稠密及链传递蕴含传递性。
An effective closing lemma for unipotent flows
本文针对完美实群商空间上的幂幺流，建立了一个有效闭合引理，为有效幂幺动力学研究提供新工具。
Here we show that for a C^2 surface diffeomorphism that satisfy the hypothesis of Hayashi connecting lemma either can be approximated, in the C^1 topology, by a diffeomorphism exhibiting a homoclinic tangency or the diffeomorphism already presented transversal homoclinic orbits.
We develop methods for studying the smooth closing lemma for Reeb flows in any dimension using contact homology. As an application, we prove a conjecture of Irie, stating that the strong closing lemma holds for Reeb flows on ellipsoids. Our methods also apply to other Reeb flows, and we illustrate this for a class of examples introduced by Albers-Geiges-Zehmisch.
The main result in this paper is the $C^{\infty}$ closing lemma for a large family of Hamiltonian flows on $4$-dimensional symplectic manifolds, which includes classical Hamiltonian systems. First we prove the $C^{\infty}$ closing lemma and the $C^r$ general density theorem for geodesic flows on closed Finsler surfaces by combining a result of Asaoka-Irie with the dual lens map technique. Then we extend our results to Hamitonian flows with certain restriction. We also list some applications of our results in differential geometry and contact topology.
We prove a $C^\infty$ closing lemma for Hamiltonian diffeomorphisms of closed surfaces. This is a consequence of a $C^\infty$ closing lemma for Reeb flows on closed contact three-manifolds, which was recently proved as an application of spectral invariants in embedded contact homology. A key new ingredient of this paper is an analysis of an area-preserving map near its fixed point, which is based on some classical results in Hamiltonian dynamics: existence of KAM invariant circles for elliptic fixed points, and convergence of the Birkhoff normal form for hyperbolic fixed points.
We prove the strong $C^\infty$ closing property, as formulated by Irie, for a class of Hamiltonian diffeomorphisms which includes all pseudo-rotations of projective spaces as well as all Anosov-Katok pseudo-rotations.
In the paper by D.~Burago, S.~Ivanov and A.~Novikov, "A survival guide for feeble fish", it has been shown that a fish with limited velocity can reach any point in the (possibly unbounded) ocean provided that the fluid velocity field is incompressible, bounded and has vanishing mean drift. This result extends some known global controllability theorems though being substantially nonconstructive. We give a fish a different recipe of how to survive in a turbulent ocean, and show its relationship to structural stability of dynamical systems by providing a constructive way to change slightly the ve…
Strong closing lemmas in Hamiltonian dynamics
本文探讨了利用谱不变量证明的辛动力学中强闭合引理，涵盖低维和高维情况，以及与极小超曲面的类比结果。
A quantitative closing Lemma and Partner orbits on Riemannian manifolds with negative curvature
该研究证明负截面曲率流形的定量闭引理，并将其应用于自交闭测地线的伙伴及伪伙伴轨道研究，为相关动力系统与几何问题提供定量分析工具。
We provide an affirmative answer to the Cr Closing Lemma, r>1, for a large class of flows defined on every closed surface.
Ergodic closing lemmas and invariant Lagrangians
受Mañé引理启发，研究高维哈密顿系统中$C^\infty$闭包引理，证明在Floer条件下不变拉格朗日子流形可产生统计可控的周期轨道，并给出零截面情形的收敛结果。
Elementary spectral invariants and quantitative closing lemmas for contact three-manifolds
本研究利用受点约束全纯曲线最大最小能量的构造变体，定义接触三维流形ECH谱的初等替代方案，将其应用于三维Reeb动力学，得到三维Reeb向量场的定量闭合引理，并在无理椭球例子中得到精确结果。
PFH spectral invariants and $C^\infty$ closing lemmas
本文建立保面积曲面微分同胚的PFH谱不变量理论，并借此证明定量$C^\infty$闭合引理，进而推出环面通有保面积微分同胚的周期点稠密性及谱不变量渐近Weyl定律。
We prove that, on connected compact manifolds, both C1-generic conservative diffeomorphisms and C1-generic transitive diffeomorphisms are topologically mixing. This is obtained through a description of the periods of a homoclinic class and by a control of the period of the periodic points given by the closing lemma.
In this paper, we develop an approach to the problem of closing lemma based on KAM normal form. The new approach differs from existing $C^1$ perturbation approach and spectral approach, and can handle the high regularity, high dimensional cases and even Riemannian metric perturbations. Moreover, the proof is constructive and effective. We apply the method to the original nearly integrable setting of Poincaré and confirm several old and new conjectures with weak formulations. First, for Poincaré's original setting of nearly integrable systems, we prove that for typical perturbations, periodic o…
We prove a very general Weyl-type law for Periodic Floer Homology, estimating the action of twisted Periodic Floer Homology classes over essentially any coefficient ring in terms of the grading and the degree, and recovering the Calabi invariant of Hamiltonians in the limit. We also prove a strong non-vanishing result, showing that under a monotonicity assumption which holds for a dense set of maps, the Periodic Floer Homology has infinite rank. An application of these results yields that a $C^{\infty}$-generic area-preserving diffeomorphism of a closed surface has a dense set of periodic poin…
A proof of the $C^r$ closing lemma and stability conjecture
本文针对动力系统长期存在的有限周期轨道猜想，提出了一种适用于时变与时不变向量场的 $C^r$ 闭合周期轨道微扰方法，证明了 $C^r$ 闭合引理与稳定性猜想。
We prove a perturbation (pasting) lemma for conservative (and symplectic) systems. This allows us to prove that $C^{\infty}$ volume preserving vector fields are $C^1$-dense in $C^{1}$ volume preserving vector fields (After the conclusion of this work, Ali Tahzibi pointed out to us that this result was proved in 1979 by Carlos Zuppa, although his proof is different from ours.). Moreover, we obtain that $C^1$ robustly transitive conservative flows in three-dimensional manifolds are Anosov and we conclude that there are no geometrical Lorenz-like sets for conservative flows. Also, by-product of t…
J. Mather characterized uniform hyperbolicity of a discrete dynamical system as equivalent to invertibility of an operator on the set of all sequences bounded in norm in the tangent bundle of an orbit. We develop a similar characterization of nonuniform hyperbolicity and show that it is equivalent to invertibility of the same operator on a larger, Fréchet space. We apply it to obtain a condition for a diffeomorphism on the boundary of the set of Anosov diffeomorphisms to be nonuniformly hyperbolic. Finally we generalise the Shadowing lemma in the same context.
Several perturbation tools are established in the volume preserving setting allowing for the pasting, extension, localized smoothing and local linearization of vector fields. The pasting and local linearization hold in all classes of regularity ranging from $C^{1}$ to $C^{\infty}$ (Hölder included). For diffeomorphisms, a conservative linearized version of Franks lemma is proved in the $C^{r,α}$ ($r\in\mathbb{Z}^+$, $0<α<1$) and $C^{\infty}$ settings, the resulting diffeomorphism having the same regularity as the original one.
We introduce a class of volume-contracting surface diffeomorphisms whose dynamics is intermediate between one-dimensional dynamics and general surface dynamics. For that type of systems one can associate to the dynamics a reduced one-dimensional model and it is proved a type of $C^\infty-$closing lemma on the support of every ergodic measure. We also show that this class contains Hénon maps with Jacobian in $(-1/4,1/4)$.
A Recurrence-type Strong Borel--Cantelli Lemma for Axiom A Diffeomorphisms
本文在3重相关性指数衰减的度量保测动力系统中，针对缓慢趋于零的序列$(M_k)$证明了递归型强Borel–Cantelli引理，并将其推广至公理A微分同胚及平衡态情形。
The aim of this short note is to explain how the arguments of the "closing lemma with time control" of F. Abdenur and S. Crovisier (arXiv:1111.4206) can be used to answer Question 1 of the article "Instability for the rotation set of homeomorphisms of the torus homotopic to the identity" of S. Addas-Zanata.
A partially hyperbolic diffeomorphism $f$ has quasi-shadowing property if for any pseudo orbit ${x_k}_{k\in \mathbb{Z}}$, there is a sequence of points ${y_k}_{k\in \mathbb{Z}}$ tracing it in which $y_{k+1}$ is obtained from $f(y_k)$ by a motion $τ$ along the center direction. We show that any partially hyperbolic diffeomorphism has quasi-shadowing property, and if $f$ has $C^1$ center foliation then we can require $τ$ to move the points along the center foliation. As applications, we show that any partially hyperbolic diffeomorphism is topologically quasi-stable under $C^0$-perturbation. When…
Let $N$ be a smooth manifold and $f:N\to N$ be a $C^l$, $l\geq 2$ diffeomorphism. Let $M$ be a normally hyperbolic invariant manifold, not necessarily compact. We prove an analogue of the $λ$-lemma in this case.
Stable manifolds, Horseshoes and Lyapunov exponents for $C^1$ diffeomorphisms without domination
本文提出了一种针对$C^1$微分同胚的非均匀超双曲理论，通过引入新的共振块，研究稳定流形、蛇形结构及李亚普诺夫指数的性质。
For smooth hyperbolic dynamical systems and smooth weights, we relate Ruelle transfer operators with dynamical Fredholm determinants and dynamical zeta functions: First, we establish bounds for the essential spectral radii of the transfer operator on new spaces of anisotropic distributions, improving our previous results. Then we give a new proof of Kitaev's lower bound for the radius of convergence of the dynamical Fredholm determinant. In addition we show that the zeroes of the determinant in the corresponding disc are in bijection with the eigenvalues of the transfer operator on our spaces …
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 Rüssmann 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 Hölder) norm must be close to Identity. From our result and a structure theorem by Fr…
We prove that in a compact manifold of dimension $n\geq 2$, a $C^{1+α}$ volume-preserving diffeomorphisms that are robustly transitive in the $C^1$-topology have a dominated splitting. Also we prove that for 3-dimensional compact manifolds, an isolated robustly transitive invariant set for a divergence-free vector field can not have a singularity. In particular, we prove that robustly transitive divergence-free vector fields in 3-dimensional manifolds are Anosov. For this, we prove some ``pasting'' lemma, which allows to make perturbations in conservative systems.
We prove that the Oseledets splittings of an ergodic hyperbolic measure of a $C^{1+ r}$ diffeomorphism can be approximated by that of atomic measures on hyperbolic periodic orbits. This removes the assumption on simple spectrum in \cite{Liang} and strengthens Katok's closing lemma.
We study how physical measures vary with the underlying dynamics in the open class of $C^r$, $r>1$, strong partially hyperbolic diffeomorphisms for which the central Lyapunov exponents of every Gibbs $u$-state is positive. If transitive, such a diffeomorphism has a unique physical measure that persists and varies continuously with the dynamics. A main ingredient in the proof is a new Pliss-like Lemma which, under the right circumstances, yields frequency of hyperbolic times close to one. Another novelty is the introduction of a new characterization of Gibbs $cu$-states. Both of these may be of…
Quasi-shadowing property for nonuniformly partially hyperbolic systems
本文研究非一致部分双曲系统，建立自适应伪轨道运动的拟跟踪性质，并考察拟 specification 与拟闭合性质，进而将 Katok 周期轨道增长结果推广到任意遍历测度。
We prove a criteria for uniform hyperbolicity based on the periodic points of the transformation. More precisely, if a mild (non uniform) hyperbolicity condition holds for the periodic points of any diffeomorphism in a residual subset of a $C^1$-open set $\SU$ then there exists an open and dense subset $\SA\subset \SU$ of Axiom A diffeomorphisms. Moreover, we also prove a noninvertible version of Ergodic Closing Lemma which we use to prove a counterpart of this result for local diffeomorphisms. As a simple corollary of our techniques, we have that an arbitrary $\mathrm{C}^1$-class local diffeo…
Let ($M$, $Ω$) be a smooth symplectic manifold and $f:M\rightarrow M$ be a symplectic diffeomorphism of class $C^l$ ($l\geq 3$). Let $N$ be a compact submanifold of $M$ which is boundaryless and normally hyperbolic for $f$. We suppose that $N$ is controllable and that its stable and unstable bundles are trivial. We consider a $C^1$-submanifold $\d$ of $M$ whose dimension is equal to the dimension of a fiber of the unstable bundle of $T_NM$. We suppose that $\d$ transversely intersects the stable manifold of $N$. Then, we prove that for all $\varepsilon>0$, and for $n$ $\in$ $\N$ large enough, …
We prove a Closing Lemma for nonuniformly hyperbolic measures of meromorphic maps. We prove also a theorem of approximation of the dynamics of such measures by Bernoulli coding maps.
We prove that any perturbation of the symplectic part of the derivative of a Poisson diffeomorphism can be realized as the derivative of a $C^1$-close Poisson diffeomorphism. We also show that a similar property holds for the Poincaré map of a Hamiltonian on a Poisson manifold. These results are the conservative counterparts of the Franks lemma, a perturbation tool used in several contexts most notably in the theory of smooth dynamical systems.
Using probabilistic methods, we prove new rigidity results for groups and pseudo-groups of diffeomorphisms of one dimensional manifolds with intermediate regularity class ({\em i.e.} between $C^1$ and $C^2$). In particular, we demonstrate some generalizations of Denjoy's Theorem and the classical Kopell's Lemma for Abelian groups. After that, these techniques are applied to the study of codimension 1 foliations. We obtain for instance several generalized versions of Sacksteder's Theorem in class $C^1$. We conclude with some remarks about the stationary measure.
Statistical Limit Theorems for Axiom A Diffeomorphisms: Exponential Mixing, Central Limit Theorem, and Large Deviations
本文研究Axiom A微分同胚的平衡态的统计极限定理，通过Part I中建立的Ruelle转移算子的谱间隙和Part III中的马尔可夫分区编码，证明了体积引理、指数相关衰减、中心极限定理、几乎必然不变原理以及大偏差原理，并展示了这些结果如何从单一的谱机制中推导出来，明确依赖于双曲性数据。
KAM splittings and equidistributed periodic orbits for stable hypersurfaces
本文证明辛4流形上满足同调条件的稳定超曲面可被闭特征等分布的超曲面C∞逼近，结合KAM理论与定量闭合引理，并推广到三维流形上的体积保持向量场。
Uniform Hyperbolicity and Symbolic Dynamics: Markov Partitions, Shadowing, and the Coding of Axiom A Diffeomorphisms
本文研究均匀双曲集的几何理论，通过定量界证明五个主要定理，包括稳定流形定理、谱分解定理、影子引理、马尔可夫划分的存在性以及符号映射的构造，揭示了双曲动力系统的结构与性质。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。