报告题目：微分动力系统2021年度学术会议

微分动力系统2021年度学术会议苏州大学数学科学学院2021年7月26—2021年7月31日微分动力系统2021年度学术会议会议通知由苏州大学数学科学学院主办的动力系统前沿理论研讨会，计划于2021年7月26日至7月31日在苏州大学举行。本次旨在交流微分动力系统中的最新成果、展望拟研究的重要问题。现将会议的有关事项通知如下：一、会议主题会议将围绕微分动力系统的前沿问题和最新成果展开交流和讨论，并进一步探讨未来拟研究的一些重要问题。本会议受到自然科学基金NSFC 11822109资助。二、会议日程7月26日下午 注册报到；7月27日至7月31日 学术报告三、会议地点会议地点：苏州南林饭店园中楼二楼浮翠厅住宿地点：苏州南林饭店住宿地址：苏州市姑苏区十全街滚绣坊20号四、注册费所有参会人员均免收注册费，食宿由会议统一安排。五、联系方式联系人：杨大伟     [email protected]         臧运涛     [email protected]                苏州大学数学科学学院                                                     2021年7月1日会议日程2021年7月27日 （周二）12：00-22：00报到晚餐2021年7月28日 （周三）Session 1. 主持人 ：曹永罗9：00-9：50孙文祥(北京大学)Liapunov exponents in Liao perturbations9：50-9：55提问9：55-10：10茶歇10：10-11：00章梅荣(清华大学)旋转p-Laplacian周期特征值的结构问题11：00-11：05提问11：05-11：55甘少波(北京大学)Centralizers of DA on T^3: rigidity vs triviality11：55-12：00提问12：00午餐Session 2. 主持人 ：朱圣芝14：30-15：20文晓(北京航空航天大学)No-shadowing for singular hyperbolic sets with a singularity15：20-15：25提问15：25-15：40茶歇15：40-16：30史逸(北京大学)Spectrum rigidity and integrability for Anosov diffeomorphisms16：30-16：35提问16：35-17：25王晓东(上海交通大学)Quasi-attractors of differentiable dynamical systems17：25-17：30提问17：30晚餐2021年7月29日 （周四）Session 1.  主持人 ：谢践生9：00-9：50王林(山西财经大学)Subsystems with shadowing property for Zk-actions9：50-9：55提问9：55-10：10茶歇10：10-11：00朱玉峻(厦门大学)Entropy along unstable and stable foliations11：00-11：05提问11：05-11：55周云华(重庆大学)Shadowing for nonuniformly hyperbolic maps in Hilbert spaces11：55-12：00提问12：00午餐Session 2.  主持人 ：汤信佳14：30-15：20李治平(盐城师范学院)Quasi-shadowing property for flows15：20-15：25提问15：25-15：40茶歇15：40-16：30王昕晟(厦门大学)Local entropy via preimage structure16：30-16：35提问16：35-17：25瞿聪聪(浙江万里学院)Exceptional sets for average conformal dynamical systems17：25-17：30提问17：30晚餐2021年7月30日 （周五）Session 1.  主持人 ：廖刚9：00-9：50田学廷(复旦大学)Some research on ergodic average9：50-9：55提问9：55-10：10茶歇10：10-11：00梁超(中央财经大学)density and openness properties of non-uniformly hyperbolic diffeomorphism11：00-11：05提问11：05-11：55张金华(北京航空航天大学)The structure of the set of non-hyperbolic ergodic measures for certain class of partially hyperbolic diffeomorphisms11：55-12：00茶歇12：00午餐Session 2.  主持人：王广瓦14：30-15：20吴万楼(江苏师范大学)Shrinking target for some dynamical systems15：20-15：25提问15：25-15：40茶歇15：40-16：30臧运涛(华东师范大学)Entropy and volume growth16：30-16：35提问16：35-17：25陈晔星(苏州大学)Topological entropy of irregular sets for typical cocycles17：25-17：30提问17：30晚餐2021年7月31日 （周六）Session 1.  主持人 ：陈剑宇9：00-9：50王方(首都师范大学)流形的拓扑结构与正定拉格朗日系统的复杂性9：50-9：55提问9：55-10：10茶歇10：10-11：00吴伟胜(厦门大学)Measure of maximal entropy for geodesic flows and closed geodesics11：00-11：05提问11：05-11：55刘飞(山东科技大学)一类(非紧致)无共轭点黎曼流形测地流的动力学性质11：55-12：00提问12：00午餐Session 2.  主持人 ：李明14：30-15：20任宪坤(重庆大学)Saturated properties for amenable group actions15：20-15：25提问15：25-15：40茶歇15：40-16：30王娟(上海工程技术大学)Dimension approximation for diffeomorphisms preserving hyperbolic measures16：30-16：35提问16：35-17：25潘娟(重庆大学)Shadowing properties of weakly hyperbolic $\mathbb{Z}^d$-actions17：25-17：30提问17：30晚餐报告题目及摘要报告人：王娟题目：Dimension approximation for diffeomorphisms preserving hyperbolic measures摘要：For a $C^{1+\alpha}$ diffeomorphism $f$ preserving a hyperbolic ergodic SRB measure $\mu$, Katok's remarkable results assert that $\mu$ can be approximated by a sequence of hyperbolic sets $\{\Lambda_n\}_{n\geq1}$. In this work, we investigate the Hausdorff dimension for $\Lambda_n$. We also extend Katok's remarkable results to $C^1$ diffeomorphisms with dominated splitting.报告人：任宪坤题目：Saturated properties for amenable group actions摘要：In this talk, we will discuss upper capacity entropy, packing entropy on saturated subsets for amenable group actions with weak specification property. We give different variational principles of these two entropies. This is joint work with Wenda Zhang.  报告人：王林题目：Subsystems with shadowing property for Zk-actions摘要：In this talk, subsystems with shadowing property for Zk-actions are investigated. Let α be a continuous Zk-action on a compact metric space X. We introduce the notions of pseudo orbit and shadowing property for α along subsets, particularly subspaces, of Rk. Combining with another important property ``expansiveness" for subsystems of α which was introduced and systematically investigated by Boyle and Lind, we show that if α has the shadowing property and is expansive along a subspace V of Rk, then so does for α along any subspace W of Rk containing V. Let α be a smooth Zk-action on a closed Riemannian manifold M, µ an ergodic probability measure and Γ the Oseledec set. We show that, under a basic assumption on the Lyapunov spectrum, α has the shadowing property and is expansive on Γ along any subspace V of Rk containing a regular vector; furthermore, α has the quasi-shadowing property on Γ along any 1-dimensional subspace V of Rk containing a first-type singular vector. As an application, we also consider the 1-dimensional subsystems (i.e., flows) with shadowing property for the Rk-action on the suspension manifold induced by α.报告人：王晓东题目：Quasi-attractors of differentiable dynamical systems摘要：The notion of quasi-attractor generalizes that of attractor. Quasi-attractors always exist by Conley's theory while attractors may not. For C1-generic differentiable dynamical systems, quasi-attractors attract most (generic) orbits. In this talk, we will review the research on quasi-attractors of dynamical systems beyond uniform hyperbolicity. Some open problems on quasi-attractors will also be mentioned.报告人：文晓题目：No-shadowing for singular hyperbolic sets with a singularity摘要:  We prove that every singular hyperbolic chain transitive set with a singularity does not admit the shadowing property.  Using this result we show that if a star flow has the shadowing property on its chain recurrent set then it satisfies Axiom A and the no-cycle conditions; and that if a multisingular hyperbolic set has the shadowing property then it is hyperbolic. This is a joint work with Lan Wen.报告人：张金华题目：The structure of the set of non-hyperbolic ergodic measures for certain class of partially hyperbolic diffeomorphisms摘要：For a large class of robustly transitive partially hyperbolic diffeomorphisms, we show that the set of non-hyperbolic ergodic measures is convex and each non-hyperbolic ergodic measure is approached by horseshoes in weak*-topology and in entropy; in particular, the entropy varies continuously and Katok’s intermediate entropy conjecture holds for such systems. The talk is based on joint works with C. Bonatti and D. Yang.报告人：刘飞题目： 一类(非紧致)无共轭点黎曼流形测地流的动力学性质摘要：测地流是一类非常重要的微分动力系统。众所周知，紧致负曲率黎曼流形上的测地流是一致双曲流，具有包括传递性、混合性、可扩性、遍历性、碎轨连接性质（specification）、封闭引理、局部乘积结构、最大熵测度存在、唯一等诸多复杂的动力学性态。当我们把视线投向更为一般的流形（比如非正曲率流形）时，这些性质常常不再成立。因此一个自然的问题是：在非正曲率流形（甚至更为一般的无焦点流形或无共轭点流形）上，测地流的哪些性质被保留下来了呢？这个问题导致了秩（rank）这一概念的提出。1998 年，德国数学家 G. Knieper 发表了一篇现在已成为经典的论文，他证明了紧致的秩1非正曲率黎曼流形的测地流具有唯一的最大熵测度。在Knieper工作的基础上，我们证明了紧致的秩1无焦点黎曼流形的测地流是拓扑传递的（与朱雄峰合作），具有唯一的最大熵测度（与王方、吴伟胜合作），且该最大熵测度是一个混合测度（与刘晓恺、王方合作）。在本报告中，我们将研究较之无焦点流形更为广泛的一类流形——无共轭点流形——的测地流的动力学性质。我们（与刘晓恺、王方合作）证明了一类非紧致的秩1无共轭点黎曼流形的测地流在非游荡集的一个自然的子集上满足 Anosov 封闭引理，具有局部乘积结构和传递性，更进一步，我们证明了测地流的不变测度的一些通有性质。报告人：王方题目： 流形的拓扑结构与正定拉格朗日系统的复杂性摘要：我们将简要介绍正定拉格朗日系统的 Mather 理论的一些基本结论，并讨论定义在具有复杂拓扑结构的紧流形上的拉格朗日系统的动力学复杂性。我们致力于阐述这样一个事实：复杂的空间拓扑结构将导致定义在这个空间的切丛上的拉格朗日系统出现复杂的动力学行为。在此基础上，我们会讨论如何利用同伦提升的方法研究更广义的极小测度，同伦提升的过程也是一个逐步揭示出拉格朗日系统的动力学复杂性的过程。报告人：朱玉峻题目：Entropy along unstable and stable foliations摘要：For a partially hyperbolic diffeomorphism (resp. endomorphism) on a closed manifold, we consider the entropy (including topological and metric versions) along the unstable (resp. stable) foliations. Several recent results on this topic are introduced. (These are joint works with Huyi Hu, Zhiming Li and Weisheng Wu).报告人：史逸题目：Spectrum rigidity and integrability for Anosov diffeomorphisms摘要：Let f be a partially hyperbolic DA-diffeomorphism on 3-torus. We show that the stable and unstable bundles of f is jointly integrable if and only if f is Anosov and has spectrum rigidity in the center bundle. This proves the Ergodic Conjecture on 3-torus. In higher dimensions, let f be an irreducible Anosov diffeomorphism on torus. If f is also absolutely partially hyperbolic and su-integrable, then it has spectrum rigidity in the center bundle. This talk is based on a series of work joint with S.Gan, A.Hammerlindl, A.Gogolev.报告人：周云华题目：Shadowing for nonuniformly hyperbolic maps in Hilbert spaces摘要：We prove that nonuniformly hyperbolic maps have shadowing property in Hilbert spaces and give some applications.报告人：甘少波题目：Centralizers of DA on T^3: rigidity vs triviality摘要：In this talk, we will first give the motivation for the study of the centralizers. Then, we will study the centralizers of derived-from-Anosov diffeomorphisms (DA) on T^3, i.e., partially hyperbolic diffeomorphisms which is isotopic to Anosov automorphisms. We will show that either the centralizers of a DA on T^3 is virtually trivial or the DA is smoothly conjugate to its linear part. This is a joint work with Yi Shi, Disheng Xu and Jinhua Zhang.报告人：李治平题目：Quasi-shadowing property for flows摘要：The pseudo-orbit shadowing theory is an important basic tool in the qualitative theory of dynamical systems. In differentiable dynamical systems, uniform hyperbolic and non-uniform hyperbolic properties imply pseudo-orbit shadowing property, but partially hyperbolic systems generally no longer have pseudo-orbit shadowing property. We say a partially hyperbolic flow $\varphi_{t}$ has the quasi-shadowing property if for any $(\delta, T)$-pseudo-orbit $g(t)$ of $\varphi_{t}$ there exist a sequence of points $\{y_{k}\}$ and a reparametrization $\alpha$ such that $\varphi_{\alpha(t) -\alpha(kT)}(y_k)$ trace $g(t)$ in which $y_{k}$ is obtained from $\varphi_{\alpha(kT)-\alpha((k-1)T)}(y_{k-1})$ by a motion along the central direction. In this talk we discuss the quasi-shadowing properties and limit quasi-shadowing properties of partially hyperbolic flows and quasi-partially hyperbolic strings of flows.报告人：孙文祥题目：Liapunov exponents in Liao perturbations摘要：Consider  a $C^1$ vector field together with an ergodic invariant probability that has $\ell$ nonzero Lyapunov exponents. Using orthonormal moving frames along a generic  orbit we construct  a linear system of  $\ell$ differential equations which is a linearized Liao standard system. We show that the Lyapunov exponentsof this linear system coincide with all the  nonzero exponents of the given vector field with respect to the given ergodic probability. Moreover, we prove that these Lyapunov exponents have a persistence property meaning that  a small perturbation to the linear system ( Liao perturbation)  preserves both sign and value of the nonzero Lyapunov exponents.报告人：吴万楼题目：Shrinking target for some dynamical systems摘要：In this talk, we will introduce the shrinking target problem and we will talk about the metric and dimensional results on the shrinking target for some dynamical systems.报告人：吴伟胜题目：Measure of maximal entropy for geodesic flows and closed geodesics摘要：We consider the geodesic flow on rank one manifolds without focal points, and show the construction and uniqueness of the measure of maximal entropy. As an application, we obtain Margulis type asymptotics of the number of the closed geodesics.报告人：臧运涛题目：Entropy and volume growth摘要：Let $f$ be a $C^2$ diffeomorphism on a compact manifold. Ledrappier and Young introduced entropies along unstable foliations for an ergodic measure $\mu$. We relate those entropies to covering numbers in order to give a new upper bound on the metric entropy of $\mu$ in terms of Lyapunov exponents and topological entropy or volume growth of sub-manifolds. We also talk about similar topics for partially hyperbolic systems.报告人：潘娟题目：Shadowing properties of weakly hyperbolic $\mathbb{Z}^d$-actions摘要：We consider the shadowing and quasi-shadowing properties for weakly hyperbolic $\mathbb{Z}^d$-actions.  A smooth $\mathbb{Z}^d$-action is non-uniformly hyperbolic (or partially  hyperbolic) if one of the generators is non-uniformly hyperbolic (or partially  hyperbolic). We show that non-uniformly hyperbolic $\mathbb{Z}^d$-actions enjoy the  shadowing property and partially hyperbolic $\mathbb{Z}^d$-actions enjoy the quasi-shadowing. as well as the $\mathcal{L}^p$, limit, and asymptotic quasi-shadowing properties. Furthermore, we also obtain a closing lemma for non-uniformly hyperbolic $\mathbb{Z}^d$-actions. A diffeomorphism is non-uniformly partially hyperbolic if it preserves an ergodic measure with a zero Lyapunov exponent. We prove that a $C^{1}$-smooth $\mathbb{Z}^d$-action has the quasi-shadowing property if one of the generators  is $C^{1+\alpha}$ non-uniformly partially hyperbolic. This is a joint work with W. Zhang，X. Ren and Y. Zhou.报告人：章梅荣题目：旋转p-Laplacian周期特征值的结构问题摘要：给定介于1与无穷之间的指标p，考虑d维欧式空间的1-周期运动，旋转p-Laplacian的1-周期特征值问题是指周期运动在p-次势能约束下的p-次动能的临界点和临界值（特征函数和特征值）。当p=2时，其特征值与维数无关且是简谐振动的周期特征值。对于一般的p， Manasevich-Mawhin在20多年前观测到2维空间中有两列比较明显的特征值，并试图说明是否这个问题仅有这两列特征值。在这个报告中，我们将看到，对于任何不为2的p，该问题一定包含有无穷多列不同的特征值。这一结果离旋转p-Laplacian的周期特征值的全貌还有很大的差距。我们的构造和证明是基于可积哈密顿系统的动力学分析。报告人：王昕晟题目：Local entropy via preimage structure摘要：In this talk, local entropy via the preimage structure for noninvertible maps is considered. For a topological dynamical system and its factor, several relative and local versions of preimage entropies with respect to a Borel cover from topological and measure-theoretic viewpoints are introduced and investigated. The relationships among these quantities are discussed, in particular, variational principles (or variational inequalities) are established. Moreover, it is shown that each local version of these relative preimage entropies coincides with its corresponding global version for a system with uniform separation of preimages. This is a joint work with Weisheng Wu and Yujun Zhu.报告人：陈晔星题目：Topological entropy of irregular sets for typical cocycles摘要：We consider Holder continuous Fiber-bunched cocycles on sub-shift with finite type. We prove that if the cocycle is typical, then the set of irregular points (i.e.,points that are not Lyapunov regular) carries full entropy.报告人：瞿聪聪题目：Exceptional sets for average conformal dynamical systems摘要：Let f: M \to M be a C^{1+\alpha} map/diffeomorphism of a compact Riemannian manifold M and \mu be an expanding/hyperbolic ergodic f-invariant Borel probability measure on M. Assume f is average conformal expanding/hyperbolic on the support set W of \mu and  W is locally maximal. For any subset A\subset W with small entropy or dimension,  we investigate the topological entropy and Hausdorff dimensions of the A-exceptional set and the limit A-exceptional set.报告人：田学廷题目：Some research on ergodic average摘要：In this talk we will discuss some topological properties of Ergodic Average from density, residual property, full topological entropy and distributional chaos of type 1 etc.报告人：梁超题目：Density and openness properties of non-uniformly hyperbolic diffeomorphism摘要：In this talk I will introduce some results on the $C^r$-topological properties（especially，density and openness） of the subset of nonuniformly hyperbolic diffeomorphisms in a certain class of partially hyperbolic symplectic systems (or volume-preserving systems).邀请人：杨大伟