Lucas Backes
A twisted cocycle taking values on a Lie Group $G$ is a cocycle that, in each step, is twisted by an automorphism of $G$. In the case when $G=GL(d,\mathbb{R})$, we prove that if two Hölder continuous twisted cocycles satisfying the so called fiber-bunching condition have the same periodic data then they are cohomologous.
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A Note on the Continuity of Oseledets Subspaces for Fiber-Bunched Cocycles
我们证明，限制到子集的纤维束元素的空间的$GL(2,\mathbb{R})$值上循环Oseledets子空间不断变化，在措施，相对于上循环。
Rigidity of Fiber Bunched Cocycles
本文证明了纤维丛矩阵值H的一个刚性定理双曲同胚上的旧上循环更精确地说，我们证明了两个这样的上循环是上同调的当且仅当它们具有共轭周期数据。
We prove a new characterization of uniform hyperbolicity for fiber-bunched cocycles. Specifically, we show that the existence of a uniform gap between the Lyapunov exponents of a fiber-bunched $SL(2,\mathbb{R})$-cocycle defined over a subshift of finite type or an Anosov diffeomorphism implies uniform hyperbolicity. In addition, we construct an $α$-Hölder cocycle which has uniform gap between the Lyapunov exponents, however it is not uniformly hyperbolic.
We provide conditions which imply the continuity of the Lyapunov exponents for non-uniformly fiber-bunched cocycles in $SL(2,\mathbb{R})$. The main theorem is an extension of the result of Backes, Brown and Butler and gives a partial answer to a conjecture of Marcelo Viana.
We prove that the set of fiber-bunched $SL(2,\mathbb{R})$-valued Hölder cocycles with nonvanishing Lyapunov exponents over a volume preserving, accessible and center-bunched partially hyperbolic diffeomorphism is open. Moreover, we present an example showing that this is no longer true if we do not assume acessibility in the base dynamics.
We give examples of locally constant $SL(2,\mathbb{R})$-cocycles over a Bernoulli shift which are discontinuity points for Lyapunov exponents in the Hölder topology and are arbitrarily close to satisfying the fiber bunching inequality. Backes, Brown, and the author have shown that the Lyapunov exponents vary continuously when restricted to the space of fiber bunched Hölder continuous cocycles. Our examples give evidence that this theorem is optimal within certain families of Hölder cocycles.
Cohomology of fiber bunched cocycles over hyperbolic systems
考虑Anosov同态上的Holder连续纤维束GL（d，R）-值余圈.我们证明了两个这样的上圈是Holder连续上同调的，如果它们具有相等的周期数据，并证明了共轭周期数据的上圈的一个结果。得到了任意常数上圈与其小扰动之间的上同调的一个推论。纤维聚束条件意味着上循环的非共形性由基底的膨胀和收缩控制。我们表明，这个条件可以建立在周期性的数据。上圈的一些重要例子来自于非同态的微分及其对不变子丛的限制。讨论了当Anosov同态光滑共轭于一个C^1-小扰动时，我们的结果在这个问题上的应用。我们还建立了纤维束上圈和一致拟共形上圈之间的可测共轭的Holder连续性。我们的主要结果也适用于值在GL（d，R）的闭子群中的上循环，双曲集上的上循环和有限型移位，以及非平凡向量丛上的线性上循环。
Coexistence of measures with simple Lyapunov spectrum for fiber-bunched cocycles
该研究针对可逆双曲传递移位上的Hölder连续纤维束上循环，在Grassmann流形强不可约条件下，证明了具有全支撑与乘积结构的遍历测度的简单Lyapunov谱具有共存性。
We prove that generic fiber-bunched and Hölder continuous linear cocycles over a non-uniformly hyperbolic system endowed with a u-Gibbs measure have simple Lyapunov spectrum. This gives an affirmative answer to a conjecture proposed by Viana in the context of fiber-bunched cocycles.
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 consider Holder continuous cocycles over hyperbolic dynamical systems with values in the group of invertible bounded linear operators on a Banach space. We show that two fiber bunched cocycles are Holder continuously cohomologous if and only if they have Holder conjugate periodic data. The fiber bunching condition means that non-conformality of the cocycle is dominated by the expansion and contraction in the base system. We show that this condition can be established based on the periodic data of a cocycle. We also establish Holder continuity of a measurable conjugacy between a fiber bunche…
Linear cocycles over Lorenz-like flows
证明了Lorenz-like流上典型纤维束线性上圈的李雅普诺夫指数重数为1：例外上圈集具有无穷余维，即它局部包含在具有任意高余维的闭子流形的有限并中.
Multifractal formalism of Lyapunov exponents for fiber-bunched linear cocycles
本文提出了一种用于纤维束线性cocycles的Lyapunov指数多标度形式化方法，通过建立相对变分原理和广义奇异值函数的变分原理，解决了拓扑熵与度量熵的关系问题，并应用于动态定义的线性cocycles。
Statistical properties of equilibrium states for fiber-bunched matrix cocycles and applications
本文证明典型纤维束化矩阵余循环在特定参数范围内存在唯一Gibbs平衡态且具有混合性，并由此肯定回答了Gatzouras和Peres关于双曲排斥子的猜想。
On regularity of conjugacy between linear cocycles over partially hyperbolic systems
本文研究部分双曲系统上线性上循环之间可测共轭的正则性，证明了常值上循环与其扰动之间共轭的连续性，并推广了不变子丛、Riemann度量及共形结构的正则性结果。
Simplicity over singular hyperbolicity
证明了奇异双曲流上典型的Hölder连续纤维束线性上圈的李雅普诺夫指数重数为1：例外上圈的子空间具有无穷余维.
We study singular value potentials of Hölder continuous $GL_2(\mathbb{R})$-cocycles over hyperbolic systems whose canonical holonomies converge and are Hölder continuous. Such cocycles include locally constant $GL_2(\mathbb{R})$-cocycles as well as fiber-bunched $GL_2(\mathbb{R})$-cocycles. We show that singular value potentials of irreducible such cocycles have unique equilibrium states. Among the reducible cocycles, we provide a characterization for cocycles whose singular value potentials have more than one equilibrium states.
Generic simple cocycles over Markov maps
Avila和Viana给出了Markov映射上线性上圈的李雅普诺夫指数重数为1的一个明确的充分条件.在这里，我们证明了在几何扰动下，这个充分准则在Markov映射上的所有纤维束线性上圈空间中是通用的：例外上圈的集合具有无穷余维，即它局部包含在具有任意高余维的闭子流形的有限并中。
Cocycles with one exponent over partially hyperbolic systems
本文考虑部分双曲同态上的Holder连续线性上圈。对于具有一个李雅普诺夫指数的纤维丛上圈，我们证明了可测不变共形结构和子丛的连续性。此外，我们建立了一个连续版本的齐默的顺从约化定理。对于双曲系统上的上循环，我们也得到了多项式增长估计的范数和拟共形偏差的周期数据。
In traditional Ergodic Optimization, one seeks to maximize Birkhoff averages. The most useful tool in this area is the celebrated Mañé Lemma, in its various forms. In this paper, we prove a non-commutative Mañé Lemma, suited to the problem of maximization of Lyapunov exponents of linear cocycles or, more generally, vector bundle automorphisms. More precisely, we provide conditions that ensure the existence of an extremal norm, that is, a Finsler norm with respect to which no vector can be expanded in a single iterate by a factor bigger than the maximal asymptotic expansion rate. These conditio…
We show that typical (in the sense of Bonatti-Viana) Hölder and fiber-bunched $GL_d(\mathbb{R})$-valued cocycles over a subshift of finite type are uniformly quasi-multiplicative with respect to all singular value potentials. We prove the continuity of the singular value pressure and its corresponding (necessarily unique) equilibrium state for such cocycles, and apply this result to repellers. Moreover, we show that the pointwise Lyapunov spectrum is closed and convex, and establish partial multifractal analysis on the level sets of pointwise Lyapunov exponents for such cocycles.
We prove that every smooth diffeomorphism group valued cocycle over certain abelian Anosov actions on tori (and more generally on infranilmanifolds), is a smooth coboundary on a finite cover, if the cocycle is center bunched and trivial at a fixed point. For smooth cocycles which are not trivial at a fixed point, we have smooth reduction of cocycles to constant ones, when lifted to the universal cover. These results on cocycle trivialisation apply, via the existing global rigidity results, to maximal Cartan actions by Anosov diffeomorphisms (with at least one transitive), on any compact smooth…
From a Gap in the Lyapunov Spectrum to Dominated Splittings
本文证明在强聚束条件下，周期测度李雅普诺夫谱的一致渐近间隙蕴含控制分裂，将周期间隙判据推广至任意维 Hölder 余环。
Nondegeneracy of the spectrum of the twisted cocycle for interval exchange transformations
本文证明了与区间交换变换相关的扭曲谱上循环顶部 Lyapunov 指数为正，通过联系亏格趋于无穷的仿射不变子流形族的指数极限，并应用于偏差估计及谱测度局部维数公式。
On the Lyapunov spectrum of the twisted cocycle for substitutions
本文基于Rajabzadeh和Safaee发现的扭曲上同调不变截面，证明代换动力系统存在零李雅普诺夫指数，简化双符号代换奇异谱证明并得到最大指数正性结果，附录关联谱测度局部维数与李雅普诺夫指数。
Twisted cocycle for interval exchange transformations: Invariant structures and Lyapunov spectrum
本文研究了区间交换变换的扭曲上循环，通过不变子丛分解和不变辛形式证明了零 Lyapunov 指数的存在性与谱对称性，并揭示了旋转型置换的谱退化现象。
On invariant holonomies between centers
针对C^{1+θ}、θ-聚束的部分双曲微分同胚，证明中心叶间稳定/不稳定完整群的C^1性及导数连续性，推广了Pugh-Shub-Wilkinson等学者的相关研究成果。
Metric dimension of $C^{\ast}$-algebras of cocycle twisted transformation groupoids: Growth and dynamical complexity
研究通过分层cLip-范数在扭变变换群胚C*-代数上构造CQMS结构，证明了当群为多项式增长时度量维数有上界，指数增长时维数无穷，揭示了群增长二分法在余圈扭变下的保持性。
Suppose $G$ is a higher-rank connected semisimple Lie group with finite center and without compact factors. Let $\mathbb{G}=G$ or $\mathbb{G}=G\ltimes V$, where $V$ is a finite dimensional vector space $V$. For any unitary representation $(π,\mathcal{H})$ of $\GG$, we study the twisted cohomological equation $π(a)f-λf=g$ for partially hyperbolic element $a\in \mathbb{G}$ and $λ\in U(1)$, as well as the twisted cocycle equation $π(a_1)f-λ_1f=π(a_2)g-λ_2 g$ for commuting partially hyperbolic elements $a_1,\,a_2\in \mathbb{G}$. We characterize the obstructions to solving these equations, construc…
On bounded cocycles of isometries over a minimal dynamics
我们证明了W. H. Gottschalk和G.A.赫德伦德：由极小动力学上的希尔伯特空间的（仿射）等距的上圈诱导的斜作用具有连续不变截面当且仅当该上圈是有界的。等价地，相应的扭上同调方程有连续解当且仅当上圈有界。我们将其解释为连续函数空间中的Bruhat-Tits中心引理。当纤维是适当的CAT（0）空间时，我们的结果也成立。其中一个应用涉及矩阵上循环。利用$\mathrm{GL} (n,\mathbb{R})$在正定矩阵（非正曲）空间上的作用，证明了极小动力学上的有界线性上圈与取值于正交群的上圈是上同调的。
Let $\mathbb{G}$ be a higher-rank connected semisimple Lie group with finite center and without compact factors. In any unitary representation $(π, \mathcal{H})$ of $\mathbb{G}$ without non-trivial $\mathbb{G}$-fixed vectors, we study the twisted cohomological equation $(X+m)f=g$, where $m\in\mathbb{R}$ and $X$ is in a $\mathbb{R}$-split Cartan subalgebra of $\text{Lie}(\mathbb{G})$. We characterize the obstructions to solving the cohomological equation, construct smooth solutions of the cohomological equation and obtain tame Sobolev estimates for $f$. We also study common solution to (the inf…
The Twisted Top
本文描述了一种新的顶--扭曲顶，它是通过在带电重顶的Lie-Poisson括号上附加一个上循环而得到的，从而打破了它的半直积结构。扭曲的顶部有一个可积的情况，对应于拉格朗日（对称）顶部。我们给出了一个规范的描述扭曲的顶部的欧拉角。我们还表明，通过数值计算的最大李雅普诺夫指数的Kovalevskaya情况下的扭曲的顶部是混沌的。
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