Alexander I. Bufetov
The aim of this paper is to obtain an asymptotic expansion for ergodic integrals of translation flows on flat surfaces of higher genus (Theorem 1) and to give a limit theorem for these flows (Theorem 2).
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
The aim of this paper is to obtain an asymptotic expansion for ergodic integrals of translation flows on flat surfaces of higher genus (Theorem 1) and to give a limit theorem for these flows (Theorem 2).
Bufetov, Bufetov-Forni and Bufetov-Solomyak have recently proved limit theorems for translation flows, horocycle flows and tiling flows, respectively. We present here analogous results for skew translations of a torus.
A generalization of the Brouwer plane translation theorem
该研究通过证明含拓扑链回归点的平面保向同胚必有不动点，推广了Brouwer平面平移定理。
In this article, we prove a limit theorem for an observable on the torus with a singularity of type $x^{-a}$, where $0<a<1$.
We consider how quickly a typical point returns to neighborhoods of itself under the flow in a typical direction on a translation surface.
We give an example of a real analytic reparametrization of a minimal translation flow on $\mathbb{T}^{5}$ that has a Lebesgue spectrum with infinite multiplicity.
Through the method of brick decomposition and the operations on essential topological lines, we generalize the line translation theorem of Beguin, Crovisier, Le Roux [BCL] in the case where the property of preserving a finite measure with total support is replaced by the intersection property.
On Mixing Flows on Finite Area Translation Surfaces
本文构造了有限面积无限亏格翻译表面，证明其垂直流强混合，回答了Lindsey和Treviño的问题。
We introduce a twisted cohomology cocycle over the Teichmueller flow and prove a "spectral gap" for its Lyapunov spectrum with respect to the Masur-Veech measures. We then derive Hoelder estimates on spectral measures and bounds on the speed of weak mixing for almost all translation flows in every stratum of Abelian differentials on Riemann surfaces, as well as bounds on the deviation of ergodic averages for product translation flows on the product of a translation surface with a circle.
Twisted cohomological equations for translation flows
通过调和分析证明扭曲上同调方程解的存在性，导数损失至多3+，进而证明乘积平移流及时间-τ映射的稳定性。
For substitution systems and translation flows, a new cocycle, which we call {\em spectral cocycle}, is introduced, whose Lyapunov exponents govern the local dimension of the spectral measure for higher-level cylindrical functions. The construction relies on the symbolic representation of translation flows and the formalism of matrix Riesz products.
In this paper we prove that translation structures for which the corresponding vertical translation flows is weakly mixing and disjoint with its inverse, form a $G_δ$-dense set in every non-hyperelliptic connected component of the moduli space $\mathcal M$. This is in contrast to hyperelliptic case, where for every translation structure the associated vertical flow is isomorphic to its inverse. To prove the main result, we study limits of the off-diagonal 3-joinings of special representations of vertical translation flows. Moreover, we construct a locally defined continuous embedding of the mo…
Bromberg and Ulcigrai constructed piecewise smooth functions f on the torus such that the set of angles alpha for which the Birkhoff sums of f with respect to the irrational translation by alpha satisfies a temporal distributional limit theorem along the orbit of a.e. x has Hausdorff dimension one. We show that the Lebesgue measure of this set of angles is equal to zero.
We show that a typical interval exchange transformation is either weakly mixing or it is an irrational rotation. We also conclude that a typical translation flow on a surface of genus $g \geq 2$ (with prescribed singularity types) is weakly mixing.
Linear Flows on Translation Prisms
本文研究平移棱柱上的线性流，利用Furstenberg和Veech思想建立平移曲面弱混合性与棱柱流遍历性的联系，得到唯一遍历性结果，并构造Pisot扩张因子方向上的显式特征函数及非遍历棱柱流和台球流例子。
We consider straight line flows on a translation surface that are minimal but not uniquely ergodic. We give bounds for the number of generic invariant probability measures.
We discuss some classical and recent results and open problems on the statistical behavior of ergodic sums above toral translations, and their applications to Diophantine approximations and to ergodic properties of systems related to quasi-periodic dynamics such as skew products, cylindrical cascades and special flows.
We study a diophantine property for translation surfaces, defined in term of saddle connections and inspired by the classical theorem of Khinchin. We prove that the same dichotomy holds as in Khinchin' result, then we deduce a sharp estimation on how fast the typical Teichmuller geodesic wanders towards infinity in the moduli space of translation surfaces. Finally we prove some stronger result in genus one.
This preliminary report contains a sketch of the proof of the following result: a slowly divergent Teichmuller geodesic satisfying a certain logarithmic law is determined by a uniquely ergodic measured foliation.
We show that generic infinite group extensions of geodesic flows on square tiled translation surfaces are ergodic in almost every direction, subject to certain natural constraints. Recently K. Frcaczek and C. Ulcigrai have shown that certain concrete staircases, covers of square-tiled surfaces, are not ergodic in almost every direction. In contrast we show the almost sure ergodicity of other concrete staircases. An appendix provides a combinatorial approach for the study of square-tiled surfaces.
On asymptotic expansions of ergodic integrals for $\Z^d$-extensions of translation flows
本文研究紧自相似平移流的$\Z^d$-覆盖，建立了遍历积分的渐近展开，并由此得到具有最优速率的弱有理遍历性，同时应用于自相似阶梯流和Ehrenfest风树模型。
The paper is devoted to generic translation flows corresponding to Abelian differentials with one zero of order two on flat surfaces of genus two. These flows are weakly mixing by the Avila-Forni theorem. Our main result gives first quantitative estimates on their spectrum, establishing the Hoelder property for the spectral measures of Lipschitz functions. The proof proceeds via uniform estimates of twisted Birkhoff integrals in the symbolic framework of random Markov compacta and arguments of Diophantine nature in the spirit of Salem, Erdos and Kahane.
Effective Unique Ergodicity and Weak Mixing of Translation Flows
本文介绍基于Hodge理论的上同调方法研究平移流的有效唯一遍历性与弱混合，首次用扭上同调Hodge理论给出Veech弱混合准则的新证明，并提供相关遍历理论的入门内容与研究问题。
Hölder regularity for the spectrum of translation flows
本文结合Forni的思想与符号方法，证明了任意亏格平坦曲面上一般平移流及随机Markov紧统上流的谱测度具有Hölder正则性。
We consider infinite staircase translation surfaces with varying step sizes. For typical step sizes we show that the translation flow is uniquely ergodic in almost every direction. Our result also hold for typical configurations of the Ehrenfest wind-tree model endowed with the Hausdorff topology. In contrast, we show that the translation flow on a periodic translation surface can not be uniquely ergodic in any direction.
Maharam-Pollicott-Ruelle resonances and self-similar translation flows on abelian covers
研究紧致平移曲面\(\mathbb{Z}^d\)覆盖上自相似平移流关于马哈拉姆测度的遍历性质，基于扭曲转移算子开发重整化方法，得出马哈拉姆 - 波利科特 - 吕埃勒共振，还给出相关应用成果。
Heisenberg Translation Flows
本文研究预量子化产生的平移曲面圆丛上的海森堡平移流，证明其相对混合性，并应用于波函数空间遍历理论及区间交换变换上的仿射斜积。
Let $d >1$. In this paper we show that for an irreducible permutation $π$ which is not a rotation, the set of $[λ]\in \mathbb{P}_+^{d-1}$ such that the interval exchange transformation $f([λ],π)$ is not weakly mixing does not have full Hausdorff dimension. We also obtain an analogous statement for translation flows. In particular, it strengthens the result of almost sure weak mixing proved by G. Forni and the first author. We adapt here the probabilistic argument developed in their paper in order to get some large deviation results. We then show how the latter can be converted into estimates o…
In this note we show that the transfer operator of a Rauzy-Veech-Zorich renormalization map acting on a space of quasi-Hölder functions is quasicompact and derive certain statistical recurrence properties for this map and its associated Teichmüller flow. We establish Borel-Cantelli lemmas, Extreme Value statistics and return time statistics for the map and flow. Previous results have established quasicompactness in Hölder or analytic function spaces, for example the work of M. Pollicott and T. Morita. The quasi-Hölder function space is particularly useful for investigating return time statisti…
We consider suspension flows over uniquely ergodic skew-translations on a $d$-dimensional torus $\mathbb{T}^d$, for $d \geq 2$. We prove that there exists a set $\mathscr{R}$ of smooth functions, which is dense in the space $\mathscr{C}(\mathbb{T}^d)$ of continuous functions, such that every roof function in $\mathscr{R}$ which is not cohomologous to a constant induces a mixing suspension flow. We also construct a dense set of mixing examples which is explicitly described in terms of their Fourier coefficients. In the language of nilflows on nilmanifolds, our result implies that, for every uni…
We consider smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces and show that, when the genus of the surface is two, almost every such locally Hamiltonian flow with two non degenerate isomorphic saddle has singular spectrum. More in general, singularity of the spectrum holds for special flows over a full measure set of interval exchange transformations with a hyperelliptic permutation (of any number of exchanged intervals), under a roof with symmetric logarithmic singularities. The result is proved using a criterion for…
An asymptotic expansion is established for time averages of translation flows on flat surfaces. This result, which extends earlier work of A.Zorich and G.Forni, yields limit theorems for translation flows. The argument, close in spirit to that of G.Forni, uses the approximation of ergodic integrals by holonomy-invariant Hoelder cocycles on trajectories of the flows. The space of holonomy-invariant Hoelder cocycles is finite-dimensional, and is given by an explicit construction. First, a symbolic representation for a uniquely ergodic translation flow is obtained following S.Ito and A.M. Vershik…
For an orientation-preserving homeomorphism of the sphere, we prove that if a translation line does not accumulate in a fixed point, then it necessarily spirals towards a topological attractor. This is in analogy with the description of flow lines given by Poincaré-Bendixson theorem. We then apply this result to the study of invariant continua without fixed points, in particular to circloids and boundaries of simply connected open sets. Among the applications, we show that if the prime ends rotation number of such an open set $U$ vanishes, then either there is a fixed point in the boundary, or…
Flat surface models of ergodic systems
本文提出构造有限面积无限型平坦曲面的一般框架，刻画其垂直平移流的动力学行为范围，证明遍历性充分条件并给出有限型曲面无法实现的动力学现象实例。
Let $\mathbf{x}=(x,y)$. A projective 2-dimensional flow is a solution to a 2-dimensional projective translation equation (PrTE) $(1-z)ϕ(\mathbf{x})=ϕ(ϕ(\mathbf{x}z)(1-z)/z)$, $ϕ:\mathbb{C}^{2}\mapsto\mathbb{C}^{2}$. Previously we have found all solutions of the PrTE which are rational functions. The rational flow gives rise to a vector field $\varpi(x,y)\bullet\varrho(x,y)$ which is a pair of 2-homogenic rational functions. On the other hand, only very special pairs of 2-homogenic rational functions, as vector fields, give rise to rational flows. The main ingredient in the proof of the classif…
Uniformity of geodesic flow in non-integrable 3-manifolds
本文研究非可积有限多立方体平移三维流形上测地流的均匀分布问题，提出分裂方法处理含一个可积方向的笛卡尔积情形，并结合分裂覆盖论证将结果推广到三个方向均不可积且满足特殊条件的流形。
TASEP modeling was shown to offer a parsimonious explanation for the experimentally confirmed ability of a single upstream Open Reading Frames (uORFs) to upregulate downstream translation during the integrated stress response. As revealed by numerical simulations, the model predicts that reducing the density of scanning ribosomes upstream of certain uORFs increases the flow of ribosomes downstream. To gain a better insight into the mechanism which ensures the non-monotone relation between the upstream and downstream flows, in this work, we propose a phenomenological deterministic model approxi…
We derive results on the distribution of directions of saddle connections on translation surfaces using only the Birkhoff ergodic theorem applied to the geodesic flow on the moduli space of translation surfaces. Our techniques, together with an approximation argument, also give an alternative proof of a weak version of a classical theorem in multi-dimensional Diophantine approximation due to W. Schmidt \cite{SchmidtMetrical, SchmidtMetrical2}. The approximation argument allows us to deduce the Birkhoff genericity of almost all lattices in a certain submanifold of the space of unimodular lattic…
Indiscriminate covers of infinite translation surfaces are innocent, not devious
本文研究无限拓扑类型平移曲面上有限覆盖对直线流遍历性的影响，证明破坏遍历性的现象罕见，并引入随机覆盖概念，证明遍历性在随机覆盖下通常得以保持。
Continuing the work in \cite{ergodic-infinite}, we show that within each stratum of translation surfaces, there is a residual set of surfaces for which the geodesic flow in almost every direction is ergodic for almost-every periodic group extension produced using a technique referred to as \emph{cuts}.
We construct, over some minimal translations of the two torus, special flows under a differentiable ceiling function that combine the properties of mixing and rank one.
In this paper, we investigate the embeddings for topological flows. We prove an embedding theorem for discrete topological system. Our results apply to suspension flows via constant function, and for this case we show an embedding theorem for suspension flows and give a new proof of Gutman-Jin embedding theorem.
Bratteli diagrams, translation flows and their $C^*$-algebras
本文通过建立双无限 Bratteli 图路径空间与平移曲面之间的显式拓扑联系，将尾等价 C*-代数与曲面叶状结构及其 K-理论关联起来，并利用 Rauzy-Veech 归纳显式构造有限亏格情形的 Bratteli 图。
We give a criterion which allows to prove non-ergodicity for certain infinite periodic billiards and directional flows on Z-periodic translation surfaces. Our criterion applies in particular to a billiard in an infinite band with periodically spaced vertical barriers and to the Ehrenfest wind-tree model, which is a planar billiard with a $Z^2$-periodic array of rectangular obstacles. We prove that, in these two examples, both for a full measure set of parameters of the billiard tables and for tables with rational parameters, for almost every direction the corresponding billiard flow is not erg…
We obtain limit theorems (Stable Laws and Central Limit Theorems, both Gaussian and non-Gaussian) and thermodynamic properties for a class of non-uniformly hyperbolic flows: almost Anosov flows, constructed here. The proofs of the limit theorems for these flows are applications of corresponding theorems in an abstract functional analytic framework, which may be applicable to other classes of non-uniformly hyperbolic flows.
We construct conservative analytic flows of zero metric entropy which satisfy the classical central limit theorem.
Logarithmic laws and unique ergodicity
本文证明 Masur 对数律不蕴含平移流唯一遍历性，而涉及平坦 systole 的类似对数律蕴含唯一遍历性，说明平坦几何对遍历性质的控制优于双曲几何。
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。