Robert Koirala
In these expository notes, we give a proof of regularity of Anosov splitting for Anosov diffeomorphisms in a torus. We also generalize the idea to higher dimensions and to Anosov flows.
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Anosov Endomorphisms on the 2-torus: Regularity of foliations and rigidity
本文研究二维环面上Anosov自同态的光滑共轭，通过叶层正则性给出充分条件，并刻画了特殊自同态与其线性化的共轭关系。
Dominated splitting from constant periodic data and global rigidity of Anosov automorphisms
本文研究双曲系统上 GL(d,R) 上循环的支配分裂判据，并据此证明一般环面 Anosov 自同构及邻近某些非线性 Anosov 微分同胚的全局周期数据刚性。
Hölder regularity for metric Anosov flows
本文定义$C^\alpha$度量Anosov流，给出其正则性判据，证明这类流的局部乘积结构、拓扑共轭、马尔可夫编码的Hölder正则性，并将结果应用于负曲率流形的测地流。
Regularity of Non-stationary Stable Foliations of Toral Anosov Maps
本文研究满足共同锥条件的二维及高维环面Anosov映射序列，证明范数一致有界时非平稳稳定叶层的正则性结论，推广经典结果并给出反例，为Sturmian哈密顿算子谱维数猜想提供研究基础。
Oseledets regularity functions for Anosov flows
Oseledets正则性函数量化了与动力系统相关的增长沿其李雅普诺夫丛与相应的均匀指数增长性的偏差。这些函数的精确规律性程度是未知的。证明了对于闭光滑黎曼流形上保体积Anosov流的每个不变的李雅普诺夫丛，对于某些$p > 0$，相应的Oseledets正则函数都在$L^p(m)$中，其中$m$是由体积形式定义的概率测度.我们证明了一个类似的结果，本质上有界的体积保持Anosov流上的上循环。
We show a fibre-preserving self-diffeomorphism which has hyperbolic splittings along the fibres on a compact principal torus bundle is topologically conjugate to a map that is linear in the fibres.
Smooth Stable Foliations of Anosov Diffeomorphisms
本文研究 Anosov 微分同胚余维一稳定叶状结构的刚性，证明其正则性略高于 $C^2$ 时即可达到与微分同胚相同的光滑性。
Lyapunov exponents and regularity of invariant foliations for partially hyperbolic diffeomophisms on $\mathbb T^3$
本文简要回顾了三维流形上部分双曲型不变叶理的度量行为的一些最新结果，并提出了一个猜想来刻画$\mathbb T^3$上Anosov（DA）的保守部分双曲同伦的原子行为.另一方面，我们证明了，如果一个不变的叶理（稳定，中心或不稳定）的保守DA $\mathbb T^3$上是$C^1$和横向绝对连续的有界雅可比矩阵的李雅普诺夫指数在这个方向上的定义无处不在和常数。如果中心叶理是这个叶状体，那么DA叶理形态平滑地与线性Anosov共轭，特别是Anosov。主定理的另一个推论是它不存在一个保守的Mañé例子。
Given any smooth Anosov map we construct a Banach space on which the associated transfer operator is quasi-compact. The peculiarity of such a space is that in the case of expanding maps it reduces exactly to the usual space of functions of bounded variation which has proven particularly successful in studying the statistical properties of piecewise expanding maps. Our approach is based on a new method of studying the absolute continuity of foliations which provides new information that could prove useful in treating hyperbolic systems with singularities.
We provide a link between Anosov representations introduced by Labourie and dominated splitting of linear cocycles. This allows us to obtain equivalent characterizations for Anosov representations and to recover recent results due to Guéritaud-Guichard-Kassel-Wienhard and Kapovich-Leeb-Porti by different methods. We also give characterizations in terms of multicones and cone-types inspired in the work of Avila-Bochi-Yoccoz and Bochi-Gourmelon. Finally we provide a new proof of the higher rank Morse Lemma of Kapovich-Leeb-Porti.
The original proof has a gap, and need extra hypothesis that the strong stable and strong unstable filiation both to be $C^1$. The argument is like the following: with the regularity, one can show that the weak-stable and weak-unstable foliation both to be $C^{1+Lip}$, and then following the same argument as in the paper one can conclude the proof. But this extra hypothesis seems implying that the flow has a contact structure. Then it will be only a result which improves the Foulon's proof on contract structure for $C^\infty$ regularity to $C^2$, not so interest.
On Anosov diffeomorphisms with asymptotically conformal periodic data
我们考虑传递Anosov同构，其中每个周期轨道只有一个正的和一个负的李雅普诺夫指数。我们建立了这类系统的各种性质，包括强拼挤，Anosov分裂的C^{1+β}光滑性，可测不变共形结构和分布的C ^{1+β}光滑性。我们将这些结果应用到具有二维稳定和不稳定分布的保体积的自同构以及返回映射在周期点的可对角化导数。我们表明，这样的一个复盖的一个非同构光滑共轭的一个Anosov自同构的环面。作为推论，我们获得了局部刚性这样的超同态。我们还建立了三维Anosov同态的一个局部刚性结果。
Rigidity of center Lyapunov exponents for Anosov diffeomorphisms on 3-torus
该研究针对3-环面上具三子束部分双曲分裂的Anosov微分同胚，证明若两微分同胚共轭且共轭保持强稳定叶状结构，则对应周期点的中心Lyapunov指数相等，给出已有结果的逆命题，并将结论推广至由Anosov导出的部分双曲微分同胚。
Ahlfors regularity of Patterson-Sullivan measures of Anosov groups and applications
本文研究了Anosov群的Patterson-Sullivan测度的Ahlfors正则性，证明其与线性形式对称性相关，并探讨了其在Teichmüller空间和流形谱性质中的应用。
Regularity of Manhattan manifolds and exact dimensionality for relatively Anosov groups
研究相对阿诺索夫群的帕特森 - 沙利文测度，利用动态方法，证明其相对于特定视觉度量是精确维数的，曼哈顿流形是\(C^1\) - 正则的，增长指标有相应性质，扩展了阿诺索夫表示的相关结论。
Let $Γ$ be a one-ended, torsion-free hyperbolic group and let $G$ be a semisimple Lie group with finite center. Using the canonical JSJ splitting due to Sela, we define amalgam Anosov representations of $Γ$ into $G$ and prove that they form a domain of discontinuity for the action of $\mathrm{Out}(Γ)$. In the appendix, we prove, using projective Anosov Schottky groups, that if the restriction of the representation to every Fuchsian or rigid vertex group of the JSJ splitting of $Γ$ is Anosov, with respect to a fixed pair of opposite parabolic subgroups, then $ρ$ is amalgam Anosov.
Dominated splitting and zero volume for incompressible three-flows
我们证明了C^2类不可压缩3-流存在一个开的稠密子集，使得如果这个集合中的一个流对于线性Poincaré流有一个正体积正则不变量子集，且该子集具有支配分裂，那么它一定是Anosov流.有了这个结果，我们就能推广Bochi-Mañé和Newhouse的关于奇点流的二分法。也就是说，我们得到了3-流形上C^1不可压缩流的剩余子集：（i）所有李雅普诺夫指数都为零或流是Anosov，（ii）流是Anosov或椭圆周期点在流形上是稠密的。
Smooth dependence on parameters of solution of cohomology equations over Anosov systems and applications to cohomology equations on diffeomorphism groups
本文研究Anosov同态上的上同调方程的解对参数的依赖性。我们表明，解决方案依赖于参数的数据一样顺利。作为结果，我们证明了最优正则性结果的值在同构群方程的解决方案。这些结果的动机应用到刚性理论，动力系统和几何。特别是，在上下文中的同胚群，我们表明：让$f$是一个传递Anosov同胚的一个紧凑的流形$M$。设对紧致流形有$η\in C^{\reg}(M,\Diff^r(N))$，$N$，$k,r \in \N$，$r \geq 1$和$0 < α\leq \Lip$ .我们表明，如果存在一个$φ\in C^{\reg}(M,\Diff^1(N))$解决\begin{equation*} φ_{f(x)} = η_x \circ φ_x \end{equation*}，那么实际上$φ\in C^{\reg}(M,\Diff^r(N))$。
Rigidity of Lyapunov exponents for derived from Anosov diffeomorphisms
本文研究同伦于线性 Anosov 自同构的保体积部分双曲微分同胚，证明其 Lyapunov 指数和受线性化界约束，涵盖 DA 与非一致双曲系统。
A diffeomorphism $f:\mathbb{R}^2\to\mathbb{R}^2$ in the plane is Anosov if it has a hyperbolic splitting at every point of the plane. The two known topological conjugacy classes of such diffeomorphisms are linear hyperbolic automorphisms and translations (the existence of Anosov structures for plane translations was originally shown by W. White). P. Mendes conjectured that these are the only topological conjugacy classes for Anosov diffeomorphisms in the plane. We prove that this claim holds when the Anosov diffeomorphism is the time-one map of a flow, via a theorem about foliations invariant …
Anosov Diffeomorphisms of Finite-Type Surfaces
本文证明满足条件A的有限拓扑型完备曲面同胚于二维环面，且孤立平面端非周期，部分回答了此类曲面是否紧致的问题。
Smooth conjugacy of Anosov diffeomorphisms on higher dimensional tori
设$L$是$\mathbb T^d$，$d\ge3$的双曲自同构.研究了$L$的小$C^1$ -邻域$\mathcal U$中的光滑共轭问题.主要结果建立了具有相同周期特征值的两个Anosov系统共轭性的$C^{1+ν}$正则性。我们假设这些系统是$C^1$ -接近于具有简单实谱的不可约线性双曲自同构$L$，并且它们在某些中间叶理上满足自然传递性假设.我们详细介绍了在$\mathbb T^4$上的两个Anosov系统的de la Llave的例子，具有相同的常数周期特征值数据，只有Hölder共轭。我们表明，这些例子用尽所有可能的方法来扰动$C^{1+ν}$共轭类，而不改变周期性的特征值数据。此外，我们推广这些例子，以大多数的可约toral自同构，以及某些产品的$\mathbb T^4$ $C^1$ -接近原来的例子。
Vortices over Riemann surfaces and dominated splittings
本文研究负欧拉示性数闭黎曼曲面上的涡旋方程解所诱导的流，证明其总具有支配分裂，并利用分数次全纯微分构造出新的 Anosov 流例子。
Conformal hyperbolic distances and holonomies of Anosov diffeomorphisms
本文研究了实值Anosov微分同胚的传递性，通过共形双曲距离定义稳定/不稳定曲线的长度，证明其全局同伦性质。
Regularity of weak foliations for thermostats
设$M$是具有黎曼度量$g$的闭定向曲面。我们考虑由磁场$Ω$和具有外场${\bf e}$的恒温器的影响下的粒子运动所确定的流动$ϕ$。我们证明了如果$ϕ$是Anosov，那么它具有弱稳定和不稳定的$C^{1,1}$类叶理当且仅当外场${\bf e}$具有全局势$U$，$g_{1}:=e^{-2U}g$具有常曲率，并且$e^{-U}Ω$是$g_1$的面积形式的常倍数.我们还给出了一个弱叶理是$C^{1,1}$类的充分必要条件，我们表明，{ \it组合}的恒温器和磁场的效果可以产生一个Anosov流与弱稳定叶理类$C^{\infty}$和弱不稳定叶理这是{ \it不} $C^{1,1}$。最后，我们研究Anosov恒温器依赖于二次的速度，我们的特点是光滑弱叶理。特别是，我们表明，准fuchsian流定义的Ghys在\cite{Ghy1}可以出现在这种方式。
We consider classes of partially hyperbolic diffeomorphism $f:M\to M$ with splitting $TM=E^s\oplus E^c\oplus E^u$ and $\dim E^c=2$. These classes include for instance (perturbations of) the product of Anosov and conservative surface diffeomorphisms, skew products of surface diffeomorphisms over Anosov, partially hyperbolic symplectomorphisms on manifolds of dimension four with bidimensional center foliation whose center leaves are all compact. We prove that accessibility holds in these classes for $C^1$ open and $C^r$ dense subsets and moreover they are stably ergodic.
Pollicott-Ruelle resolvent and Sobolev regularity
本文研究Anosov流的Pollicott–Ruelle预解式，在一般向量丛提升情形下确定其在各向异性Sobolev空间中亚纯延拓的正则性阈值，并联系关联衰减所需正则性。
In this paper, we introduce a Lorentzian Anosov family (LAfamily) up to a sequence of distributions of null vectors. We prove for each p Mi, where Mi is a Lorentzian manifold for i Z the tangent space Mi at p has a unique splitting and this splitting varies continuously on a sequence via the distance function created by a unique torsion-free semi-Riemannian connection. We present three examples of LA-families. Also, we define Lorentzian shadowing property of type I and II and prove some results related to this property.
We consider families of diffeomorphisms with dominated splittings and preserving a Borel probability measure, and we study the regularity of the Lyapunov exponents associated to the invariant bundles with respect to the parameter. We obtain that the regularity is at least the sum of the regularities of the two invariant bundles (for regularities in $[0,1]$), and under suitable conditions we obtain formulas for the derivatives. Similar results are obtained for families of flows, and for the case when the invariant measure depends on the map. We also obtain several applications. Near the time on…
Let $f\colon\mathbb{T}^d\to\mathbb{T}^d$ be an Anosov diffeomorphism whose linearization $A\in{\rm GL}(d,\mathbb{Z})$ is irreducible. Assume that $f$ is also absolutely partially hyperbolic where a weak stable subbundle is considered as the center subbundle. We show that if the strong stable and unstable subbundles are jointly integrable, then $f$ is dynamically coherent and all foliations match corresponding linear foliation under the conjugacy to the linearization $A$. Moreover, $f$ admits the finest dominated splitting in weak stable subbundle with dimensions matching those for $A$, and it …
We develop a paradifferential approach for studying non-smooth hyperbolic dynamics and related non-linear PDE from a microlocal point of view. As an application, we describe the microlocal regularity, i.e the $H^s$ wave-front set for all $s$, of the unstable bundle $E_u$ for an Anosov flow. We also recover rigidity results of Hurder-Katok and Hasselblatt in the Sobolev class rather than Hölder: there is $s_0>0$ such that if $E_u$ has $H^s$ regularity for $s>s_0$ then it is smooth (with $s_0=2$ for volume preserving $3$-dimensional Anosov flows). In the appendix by Guedes Bonthonneau, it is als…
We study the regularity of a conjugacy between an Anosov automorphism $L$ of a nilmanifold $N/Γ$ and a volume-preserving, $C^1$-small perturbation $f$. We say that $L$ is locally Lyapunov spectrum rigid if this conjugacy is $C^{1+}$ whenever $f$ is $C^{1+}$ and has the same volume Lyapunov spectrum as $L$. For $L$ with simple spectrum, we show that local Lyapunov spectrum rigidity is equivalent to $L$ satisfying both an irreducibility condition and an ordering condition on its Lyapunov exponents.
Analiticity of the Lyapunov exponents of perturbed toral automorphisms
本文研究了环面上解析 Anosov 微分同胚在解析扰动下的动力系统，证明了不变子空间的分裂可通过部分共轭扩展至扰动系统，并得出非退化 Lyapunov 指数是扰动的解析函数。
Weakly hyperbolic actions of Kazhdan groups on tori
本文研究了弱双曲群作用的遍历性和刚性性质。受经典的Anosov同态定理的启发，我们得到了两个主要结果：第一，闭流形上的所有C^2保体积弱双曲作用是遍历的。这一结果将Anosov关于Anosov群同态遍历性的经典定理推广到更广泛的弱双曲群作用类。为了说明第二个主要结果，假设A表示高秩格（如SL（n，Z），n>2）在环面上的C^2保体积双同态的弱双曲作用。如果A被环面的万有覆叠空间R^n的作用覆盖，则A在环面的第一同调上诱导弱双曲表示。这一结果可以被看作是曼宁的贡献Anosov Mannos的弗兰克斯/曼宁分类的一个模拟。托丽。我们得出的结论与Zimmer的程序分类遍历体积保持行动的高秩格。这两个结果都依赖于Rauch和Talyor的一个新的正则性结果，该结果将相对于绝对连续叶理的叶子切向测量的索博列夫函数类与全局索博列夫函数类联系起来。
Genericity of non-uniform hyperbolicity in dimension 3
对于3-流形M的一般保守单同态，Oseledets分裂是全局控制分裂。此外，要么所有的李雅普诺夫指数消失几乎处处，否则系统是非一致双曲和遍历。这是马涅-博奇的一个著名结果的三维版本，该结果指出，一般保守曲面的超纯是Anosov或所有李雅普诺夫指数几乎处处消失。这一结果的启发和答案的积极三维猜想阿维拉和博奇。证明了中心维数为1的正测度部分双曲集有强同宿交。这意味着Cr一般地对任何r，一个双同态不包含具有正测度和中心维数为1的真部分双曲集。
Extremal distributions of partially hyperbolic systems: the Lipschitz threshold
研究部分双曲系统极值分布的正则性临界现象，证明当极值分布为Lipschitz时自动为C^∞，并应用于刚性问题，如研究保守设定下的联合积分性与u-Gibbs测度的刚性结果。
$C^1$-Generic Symplectic Diffeomorphisms: Partial Hyperbolicity and Zero Center Lyapunov Exponents
我们证明了，如果$f$是一个$C^1$ -一般的辛同构，然后Oseledets分裂沿几乎每一个轨道是平凡的或部分双曲。此外，如果$f$不是Anosov，则中心束中的所有指数都消失。这完全建立了R. 1983年，在ICM。主要的技术创新是一个概率方法的建设扰动，使用随机游动。
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