Shaobo Gan, Yi Shi
In this paper, we construct a partially hyperbolic skew-product diffeomorphism $f$ on $\mathbb{T}^3$, such that $f$ is accessible and chain transitive, but not transitive.
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Uniqueness of SRB measures for transitive diffeomorphisms on surfaces
我们利用与双曲周期点相关的遍历同宿类给出了SRB测度的遍历分量的描述。对于传递曲面的超同态，我们证明了至多存在一个SRB测度。
In this paper, we study transitive partially hyperbolic diffeomorphisms with one-dimensional topologically neutral center, meaning that the length of the iterate of small center segments remains small. Such systems are dynamically coherent. We show that there exists a continuous metric along the center foliation which is invariant under the dynamics. As an application, we classify the transitive partially hyperbolic diffeomorphisms on 3-manifolds with topologically neutral center.
Transitive partially hyperbolic diffeomorphisms in dimension three
该研究证明了三维闭合流形上C^{1+α}保守部分双曲微分同胚的传递性与遍历性等价，并给出了可访问类的精确分类。
We construct a family of partially hyperbolic skew-product diffeomorphisms on $\mathbb{T}^3$ that are robustly transitive and admitting two physical measures with intermingled basins. In particularly, all these diffeomorphisms are not topologically mixing. Moreover, for every such example, it exhibits a dichotomy under perturbation: every perturbation of such example either has a unique physical measure and is robustly topologically mixing, or has two physical measures with intermingled basins.
We present an example of a $\mathcal{C}^1$-robustly transitive skew-product with non-trivial, non-hyperbolic action on homology. The example is conservative, ergodic, non-uniformly hyperbolic and its fiber directions cannot be decomposed into two dominated expanded/contracted bundles.
Characterization of Ergodic Measures of Maximal Entropy for Topologically Transitive Partially Hyperbolic Diffeomorphisms with compact center leaves
该文研究三维环面上具紧一维中心叶的拓扑传递部分双曲微分同胚，给出零Lyapunov指数的最大熵遍历测度的数量上界，并完整刻画其支撑集。
Intrinsic ergodicity for certain nonhyperbolic robustly transitive systems
我们证明了最初由Mañé描述的一类鲁棒传递的自同构是内在遍历的。更确切地说，我们得到一个开集的同胚未能一致双曲，但仍然有恒定的熵和同构的最大熵的唯一措施。
Transitivity of Infinite-Dimensional Extensions of Anosov Diffeomorphisms
本文考虑了实向量空间R^ω上的下流形的Anosov同态的扩张。基于Nitica和Pollicott证明的有限维类似定理，我们的主要结果是，任何满足明显障碍的Holder上循环都会引发拓扑传递扩展（事实上是拓扑弱混合）。我们展示了如何为任何Anosov同构构造满足这些条件的上圈，然后观察到，与有限维情况不同，满足障碍的上圈是C^0-稳定传递的，关于几个上圈空间和度量，不可能有无限维稳定传递的上圈。
Manifolds with higher homotopy which do not support Anosov diffeomorphisms
我们证明了具有非平凡高同伦群的各类闭流形不支持（传递）Anosov同构。特别是，我们证明了有限产品的领域，其中至少有一个是偶数维不支持传递Anosov同构。
On uniformly quasiconformal Anosov diffeomorphisms with two dimensional distributions
本研究证明具有二维不稳定分布的传递一致$u$-拟共形Anosov微分同胚有全局稳定和乐，由此去除Kalinin-Sadovskaya定理的额外假设，推导出该类微分同胚与下环面仿射Anosov微分同胚$C^\infty$共轭。
Transitive centralizers and fibered partially hyperbolic systems
本文证明了光滑微分同胚中心化子的刚性结果，分类了具有传递中心化子的微分同胚，并在幂零流形上 Anosov 微分同胚等距扩张的扰动及 3 维部分双曲系统中给出了中心化子的 Lie 群结构与分类。
Diffeomorphisms with various $C^1$ stable properties
设$M$是光滑紧致流形，$Λ$是紧致不变集.本文证明了对每一个鲁棒传递集$Λ$，$f|_Λ$满足$C^1-$泛稳定的可阴影性质（分别为，$C^1-$类属稳定传递规范性质或$C^1-$类属稳定重心性质）当且仅当$Λ$是双曲基集。特别地，$f|_Λ$满足$C^1-$稳定的可遮蔽属性（分别地，$C^1-$稳定传递规范性质或$C^1-$稳定重心性质）当且仅当$Λ$是双曲基集。对于保体积情形，类似的结果也是有效的。
Partial Hyperbolicity for Symplectic Diffeomorphisms
证明了紧致流形上的鲁棒传递和稳定遍历辛同胚都允许控制分裂。事实上，这些双同态是部分双曲的。
We build an example of a non-transitive, dynamically coherent partially hyperbolic diffeomorphism $f$ on a closed $3$-manifold with exponential growth in its fundamental group such that $f^n$ is not isotopic to the identity for all $n\neq 0$. This example contradicts a conjecture in \cite{HHU}. The main idea is to consider a well-understood time-$t$ map of a non-transitive Anosov flow and then carefully compose with a Dehn twist.
Constraints On Dynamics Preserving Certain Hyperbolic Sets
我们建立了两个结果下，双曲集的拓扑约束环境动力学。首先，如果一个集合对于某个双同态是一个紧的、传递的、扩张的余维为1的双曲吸引子，那么它对于任何双同态是一个传递的、扩张的余维为1的吸引子（或收缩的排斥子）的并集，使得它是双曲的。其次，如果一个集合对于一个曲面双同态是一个非游荡的、局部极大的、紧的双曲集，那么对于任何它是双曲的双同态，它都是局部极大的。
C^k-Robust transitivity for surfaces with boundary
本文证明了带边界曲面上的C^1-鲁棒传递同态是不存在的，并给出了一类带边界曲面上的C^k-鲁棒传递同态，其中k大于或等于2.这类双同态是帕利斯关于有边界曲面的同宿切线和一致双曲性猜想的一个版本在C^2-拓扑中不成立的例子。这表明，无边界曲面上的伪Anosov仿射的爆破，成为有边界曲面上的C^2-鲁棒拓扑混合仿射。
We consider the space of $C^1$-diffeomorphims equipped with the $C^1$-topology on a three dimensional closed manifold. It is known that there are open sets in which $C^1$-generic diffeomorphisms display uncountably many chain recurrences classes, while only countably many of them may contain periodic orbits. The classes without periodic orbits, called aperiodic classes, are the main subject of this paper. The aim of the paper is to show that aperiodic classes of $C^1$-generic diffeomorphisms can exhibit a variety of topological properties. More specifically, there are $C^1$-generic diffeomorph…
Contribution to the ergodic theory of robustly transitive maps
在这篇文章中，我们打算在没有边界的紧致流形M上的鲁棒传递局部仿射集RT的遍历性质的理解作出贡献。证明了存在RT的C^1剩余子集R_0，使得R_0中的任意f都有M的具有稠密预轨道的剩余子集。此外，一般地，在没有分裂的局部同态空间中，所有点都有稠密的预轨道，有无数个遍历扩展的不变测度，具有完全支持，并表现出指数衰减的相关性。特别地，这些结果对[Lizana-Pujals'12]中考虑的一类重要的鲁棒传递映射成立。
We show that time-one maps of transitive Anosov flows of compact manifolds are accumulated by diffeomorphisms robustly satisfying the following dichotomy: either all of the measures of maximal entropy are non-hyperbolic, or there are exactly two ergodic measures of maximal entropy, one with a positive central exponent and the other with a negative central exponent. We establish this dichotomy for certain partially hyperbolic diffeomorphisms isotopic to the identity whenever both of their strong foliations are minimal. Our proof builds on the approach developed by Margulis for Anosov flows wher…
We prove that if $f$ is a partially hyperbolic diffeomorphism on 3-torus $\mathbb{T}^3$ and isotopic to Anosov, then $f$ is chain transitive.
We characterize which 3-dimensional Seifert manifolds admit transitive partially hyperbolic diffeomorphisms. In particular, a circle bundle over a higher-genus surface admits a transitive partially hyperbolic diffeomorphism if and only if it admits an Anosov flow.
We show that there is no transitive Anosov diffeomorphism with the global product structure, which is homotopic to a product of pseudo-Anosov diffeomorphisms, on a product of two closed surfaces each of which has genus greater than or equal to two.
It is known that transitive Anosov diffeomorphisms have a unique measure of maximal entropy (MME). Here we discuss the converse question. Under suitable hypothesis on Lyapunov exponents on the set of periodic points and the structure of the MME we get transitivity of $C^1-$Anosov diffeomorphisms.
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.
A note on minimality of foliations for partially hyperbolic diffeomorphisms
结果表明，在三维闭流形上的鲁棒传递、部分双曲微分同胚中，强稳定或不稳定的叶理是最小的。在这篇文章中，我们证明了“几乎所有”的稳定和不稳定的叶理叶是密集的整个流形。
Symbolic extensions for partially hyperbolic diffeomorphisms
我们证明了每一个具有1维中心丛的部分双曲双同态都有一个主符号扩张。在另一方面，我们表明，有没有符号扩展$C^1$ -generically之间包含非双曲鲁棒传递集与中心不可分解束的维数至少为2的非双曲型同态。
Phase transitions for transitive local diffeomorphism with break points on the circle and Holder continuous potentials
本文研究了具有圈上断点的传递局部微分同胚在Hölder连续势函数下的相变行为，证明了Hölder连续势函数最多有两个相变，并给出了谱间隙性质的稠密集及多标度分析应用。
Let $f$ be a $C^2$ partially hyperbolic diffeomorphisms of ${\mathbb T}^3$ (not necessarily volume preserving or transitive) isotopic to a linear Anosov diffeomorphism $A$ with eigenvalues $$λ_{s}<1<λ_{c}<λ_{u}.$$ Under the assumption that the set $$\{x: \,\mid\log \det(Tf\mid_{E^{cu}(x)})\mid \leq \log λ_{u} \}$$ has zero volume inside any unstable leaf of $f$ where $E^{cu} = E^c\oplus E^u$ is the center unstable bundle, we prove that the stable foliation of $f$ is $C^1$ robustly minimal, i.e., the stable foliation of any diffeomorphism $C^1$ sufficiently close to $f$ is minimal. In particula…
Stably ergodic diffeomorphisms which are not partially hyperbolic
本文证明了一类不含双曲不变子丛的保守单同态的稳定遍历性。此外，还讨论了非保守$C^1$扰动下的SRB测度的唯一性。这个类包含严格的非部分双曲的鲁棒传递的Bonatti-Viana同构，所以我们回答他们的问题，这种系统的稳定遍历性。
Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms
证明了紧致流形的任何一个非纯同态都可以在拓扑C1中被另一个非纯同态所近似，该非纯同态具有同宿分支（同宿切线或异维循环），或者被一个本质上是双曲的非纯同态所近似（它具有有限个带开放稠密吸引吸引区域的传递双曲吸引子）。
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…
The (measure-theoretical) entropy of a diffeomorphism along an expanding invariant foliation is the rate of complexity generated by the diffeomorphism along the leaves of the foliation. We prove that this number varies upper semi-continuously with the diffeomorphism ($\C^1$ topology), the invariant measure (weak* topology) and the foliation itself in a suitable sense. This has several important consequences. For one thing, it implies that the set of Gibbs $u$-states of $\C^{1+}$ partially hyperbolic diffeomorphisms is an upper semi-continuous function of the map in the $\C^1$ topology. Another…
In this paper, we prove that any group of diffeomorphisms acting on the 2-sphere and properly extending the conformal group of Möbius transformations must be at least 4-transitive or, more precisely, arc 4-transitive. In addition, we show that any such group must always contain an element of positive topological entropy. We also provide an elementary characterization, in terms of transitivity, of the Möbius transformations within the full group of sphere diffeomorphisms.
Zero Lyapunov Exponents in Transitive Skew-products of Iterated Function Systems
研究遍历斜积的性质，证明其存在非超bolic测度，并展示这些测度的支撑全集和弱*极限。
For diffeomorphisms or for non-singular flows, there are many results relating properties persistent under C1 perturbations and global structures for the dynamics ( such as hyperbolicity, partial hyperbolicity, dominated splitting). However, a difículty appears when a robust property of a flow holds on a set containing recurrent orbits accumulating a singular point. In [BdL] with Christan Bonatti we propose a a general procedure for adapting the usual hyperbolic structures to the singularities. In this paper, using this tool, we recover the results in [BDP] for flows, showing that robustly cha…
A Criterion For Ergodicity of Non-uniformly hyperbolic Diffeomorphisms
在这篇工作中，我们提出了一个新的准则遍历的一些不变流形的李雅普诺夫指数和一般位置的条件，涉及到的遍历性。一方面，我们得到了传递曲面仿射的SRB-测度的唯一性。另一方面，利用最近的结果blender的存在性，我们给出了一个积极的答案，在$C^1$拓扑，Pugh-Shub猜想的上下文中的部分双曲保守的双同态与二维中心丛。
Transverse foliations on the torus $\T^2$ and partially hyperbolic diffeomorphisms on 3-manifolds
本文证明环面上任意两个横截 $C^1$ 叶状结构可由非零伦微分同胚环路保持横截，并据此将 Dehn 扭转构造部分双曲微分同胚的方法推广到任意非传递 Anosov 流。
We show that various classes of products of manifolds do not support transitive Anosov diffeomorphisms. Exploiting the Ruelle-Sullivan cohomology class, we prove that the product of a negatively curved manifold with a rational homology sphere does not support transitive Anosov diffeomorphisms. We extend this result to products of finitely many negatively curved manifolds of dimensions at least three with a rational homology sphere that has vanishing simplicial volume. As an application of this study, we obtain new examples of manifolds that do not support transitive Anosov diffeomorphisms, inc…
Nonuniform hyperbolicity for C^1-generic diffeomorphisms
本文研究了非保守C^1-一般代数同态的遍历论。首先，我们表明，同宿类的任意同态表现出遍历措施，其支持与同宿类。第二，我们证明了C^1-一般双同态的一般（对于弱拓扑）遍历测度是非一致双曲的：它们没有零李雅普诺夫指数。第三，我们推广了Sigmund关于双曲基集的一个定理：任何C^1-一般双同态f的每一个孤立的传递集L具有许多遍历双曲测度，这些测度的支集与整个集L重合。此外，还证实了R. Mané在1982年的文章中，我们证明了其Oseledets分裂是受控的双曲测度满足Pesin稳定流形定理，即使双曲同态仅为C^1。
We show that the robustly transitive diffeomorphisms constructed by Bonatti and Viana have unique equilibrium states for natural classes of potentials. In particular, we characterize the SRB measure as the unique equilibrium state for a suitable geometric potential. The techniques developed are applicable to a wide class of DA diffeomorphisms, and persist under $C^1$ perturbations of the map. These results are an application of general machinery developed by the first and last named authors.
Quasi-isometric center action in dimension 3
研究了3维中保持中心叶理且拟等距作用的传递部分双曲微分同胚，证明其在有限提升和迭代下仅为斜积或离散化Anosov流。
In the present paper we contribute to the thermodynamic formalism of partially hyperbolic attractors for local diffeomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. These include the case of attractors for Axiom A endomorphisms and partially hyperbolic endomorphisms derived from Anosov. We prove these attractors have finitely many SRB measures, that these are hyperbolic, and that the SRB measure is unique provided the dynamics is transitive. Moreover, we show that the SRB measures…
In this article we prove that for a diffeomorphism on a compact Riemannian manifold, if there is a nontrival homoclinic class that is not uniformly hyperbolic or the diffeomorphism is a $C^{1+α}$ and there is a hyperbolic ergodic measure whose support is not uniformly hyperbolic, then we find a type of strongly distributional chaos which is stronger than usual distributional chaos and Li-Yorke chaos in the set of irregular orbits that are not uniformly hyperbolic. Meanwhile, we prove that various fractal sets are strongly distributional chaotic, such as irregular sets, level sets, several recu…
On sufficient conditions for the transitivity of homeomorphisms
本文推导了具有跟踪性质的同胚拓扑传递的充要条件（含稠密不变子集的重心性质），对比Anosov微分同胚的相关条件，描述$C^1$内部、讨论例子并给出应用，为动力学传递性研究提供新依据。
Codimension one partially hyperbolic diffeomorphisms with a uniformly compact center foliation
考虑光滑紧致流形M上的部分双曲C1-齐次同态f，其中心叶理为一致紧的f-不变。证明了如果不稳定丛是一维定向丛，则中心叶理的完整性处处为零，中心叶理的商空间是环面，f在其上诱导双曲自同构，特别地，f是中心传递的.我们实际上得到了进一步有趣的结果，而不受限制的不稳定，稳定和中心尺寸：我们证明了一种谱分解的链递归集的商动力学，我们建立了一个完整的不稳定的叶子（Margulis措施）不变的措施家庭的存在。
Partially hyperbolic diffeomorphisms that are center fixing
本文证明：若传递且动力学一致的偏双曲微分同胚具有一维中心叶状结构，并保持每个中心叶片不变，则其必为离散化 Anosov 流。
We construct examples of robustly transitive and stably ergodic partially hyperbolic diffeomorphisms $f$ on compact $3$-manifolds with fundamental groups of exponential growth such that $f^n$ is not homotopic to identity for all $n>0$. These provide counterexamples to a classification conjecture of Pujals.
$C^1$-generic dynamics: tame and wild behaviour
本文综述了$C^1$ -一般反同态的极大传递集的最新结果。
Partial hyperbolicity and specification
研究了部分双曲动力系统的规范性质。特别地，我们证明了如果一个部分双曲复同态有两个不同指标的鞍，并且其中一个鞍的稳定流形与强稳定叶重合，则它不满足规范性质.作为应用，我们证明了在三维闭流形上的鲁棒非双曲可迁非纯同态集合中存在$C^1$ -开稠密子集$\mathcal P$，使得$\mathcal P$中的非纯同态不满足规范性质.
In this paper we prove that for an ergodic hyperbolic measure $ω$ of a $C^{1+α}$ diffeomorphism $f$ on a Riemannian manifold $M$, there is an $ω$-full measured set $\widetildeΛ$ such that for every invariant probability $μ\in \mathcal{M}_{inv}(\widetildeΛ,f)$, the metric entropy of $μ$ is equal to the topological entropy of saturated set $G_μ$ consisting of generic points of $μ$: $$h_μ(f)=h_{\top}(f,G_μ).$$ Moreover, for every nonempty, compact and connected subset $K$ of $\mathcal{M}_{inv}(\widetildeΛ,f)$ with the same hyperbolic rate, we compute the topological entropy of saturated set $G_K$…
Some results on the integrability of the center bundle for partially hyperbolic diffeomorphisms
对中心维数为1的部分双曲型单同态，证明了关于其中心丛可积性的两个结果。一方面，我们证明了如果f的非徘徊集是整个流形，并且流形是三维的，那么周期点的不存在意味着中心丛的唯一可积性。在对边，我们证明了任何周期点p的大周期n有一个f n不变的中心流形，处处相切的中心丛。我们还得到，作为最后一个结果的后果，有一个开放的和稠密的子集C1鲁棒传递和部分双曲的自同态与中心维数为1，使强稳定或强不稳定的叶理是最小的。这将BDU中关于三维流形的一个结果推广到任意维。
Partially hyperbolic diffeomorphisms homotopic to the identity in dimension three
本文证明了三维中同伦于恒等的保守部分双曲微分同胚在环境流形基本群非虚可解时具有可达性，从而证实了该同伦类下的Hertz-Hertz-Ures遍历性猜想。
Full flexibility of entropies among ergodic measures for partially hyperbolic diffeomorphisms
本文研究具有一维中心方向的部分双曲微分同胚，在极小叶状结构和blender-horseshoes假设下证明了熵与中心Lyapunov指数的联合灵活性，并给出广泛适用例子。
In the late nineties, Smale proposed a list of problems for the next century and, among these, it was conjectured that for every $r\ge 1$ a $C^r$-generic diffeomorphism has trivial centralizer. Our contribution here is to prove the triviality of $C^r$-centralizers on hyperbolic basic sets. In particular, $C^r$-generic transitive Anosov diffeomorphisms have a trivial $C^1$-centralizer. These results follow from a more general criterium for expansive homeomorphisms with the gluing orbit property. We also construct a linear Anosov diffeomorphism on $\mathbb T^3$ with discrete, non-trivial central…
Minimality of one invariant lamination for partially hyperbolic attractors
证明了具有一维中心丛的强部分双曲吸引子的两个不变层中至少有一个是极小的。这一结果推广了[7]中关于鲁棒传递非同态的极小叶理的结果。
We show generic $C^\infty$ hyperbolic flows (Axiom A and no cycles, but not transitive Anosov) commute with no $C^\infty$-diffeomorphism other than a time-t map of the flow itself. Kinematic expansivity, a substantial weakening of expansivity, implies that $C^0$ flows have quasi-discrete $C^0$-centralizer, and additional conditions broader than transitivity then give discrete $C^0$-centralizer. We also prove centralizer-rigidity: a diffeomorphism commuting with a generic hyperbolic flow is determined by its values on any open set.
Periodic measures and partially hyperbolic homoclinic classes
本文证明鲁棒环产生的非双曲遍历测度在 C^1 开稠条件下可由周期测度逼近，并刻画一维中心部分双曲系统中遍历测度闭包的凸集结构。
Rich phase transitions in step skew-products
我们提出的例子，部分双曲和拓扑传递的局部非同态定义为斜产品在一个马蹄表现出丰富的相变的拓扑压力。这种相变是由中心李雅普诺夫指数的频谱中的间隙引起的。它与两个正熵的平衡态共存有关。双曲同态混合了不同类型的双曲行为。然而，在某种意义上，膨胀行为并不占主导地位，这表明存在一个具有非正中心指数的最大熵测度。
We consider group-valued cocycles over a partially hyperbolic diffeomorphism which is accessible volume-preserving and center bunched. We study cocycles with values in the group of invertible continuous linear operators on a Banach space. We describe properties of holonomies for fiber bunched cocycles and establish their Holder regularity. We also study cohomology of cocycles and its connection with holonomies. We obtain a result on regularity of a measurable conjugacy, as well as a necessary and sufficient condition for existence of a contionuous conjugacy between two cocycles.
We answer a question of Burns and Wilkinson, showing that there are open families of volume-preserving partially hyperbolic diffeomorphisms which are accessible and center bunched and neither dynamically coherent nor Anosov. We also show in the volume-preserving setting that any diffeomorphism which is partially hyperbolic and Anosov may be isotoped to a diffeomorphism which is partially hyperbolic and not Anosov.
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