Howard Masur
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.
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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.
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…
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。
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.
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。
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…
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).
For a Z-cover of a translation surface, which is a lattice surface, and which admits infinite strips, we prove that almost every direction for the straightline flow is ergodic.
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…
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。
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).
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.
Super-fast spreading of billiard orbits in rational polygons and geodesics on translation surfaces
本文证明有理多边形台球轨道与平移曲面测地线具有超快扩散特性，该最佳时间定量多数性质意味着几乎所有方向上的一致性。
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…
In this paper, we study topological properties of the right action by translation of the Weyl Chamber flow on the space of Weyl chambers. We obtain a necessary and sufficient condition for topological mixing. (1)
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。
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…
We analyze the cutting sequences associated to geodesic flow on a large class of translation surfaces, including Bouw-Moller surfaces. We give a combinatorial rule that relates a cutting sequence corresponding to a given trajectory, to the cutting sequence corresponding to the image of that trajectory under the parabolic element of the Veech group. This extends previous work for regular polygon surfaces to a larger class of translation surfaces. We find that the combinatorial rule is the same as for regular polygon surfaces in about half of the cases, and different in the other half.
Let M be a translation surface. We show that certain deformations of M supported on the set of all cylinders in a given direction remain in the GL(2,R)-orbit closure of M. Applications are given concerning complete periodicity, field of definition, and the number of of parallel cylinders which may be found on a translation surface in a given orbit closure. The proof uses Eskin-Mirzakhani-Mohammadi's recent theorem on orbit closures of translation surfaces, as well as results of Minsky-Weiss and Smillie-Weiss on the dynamics of horocycle flow.
The description of almost periodic or quasiperiodic structures has a long tradition in mathematical physics, in particular since the discovery of quasicrystals in the early 80's. Frequently, the modelling of such structures leads to different types of dynamical systems which include, depending on the concept of quasiperiodicity being considered, skew products over quasiperiodic or almost-periodic base flows, mathematical quasicrystals or maps of the real line with almost-periodic displacement. An important problem in this context is to know whether the considered system is semiconjugate to a r…
Characterization of Translation Bounded Measure Dynamical Systems and Positive Measure Dynamical Systems within the Spatial Processes
本文针对局部紧阿贝尔群上的空间过程与测度动力系统，借助C*代数盖尔范德理论，刻画了空间过程等价于平移有界测度动力系统及正测度动力系统的条件。
The orbit closure of any translation surface under the horocycle flow in almost any direction equals its $SL_2(\mathbb{R})$ orbit closure. This result gives rise to new characterizations of lattice surfaces in terms of the hororcycle flow.
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。
We classify the locally finite ergodic invariant measures of certain infinite interval exchange transformations (IETs). These transformations naturally arise from return maps of the straight-line flow on certain translation surfaces, and the study of the invariant measures for these IETs is equivalent to the study of invariant measures for the straight-line flow in some direction on these translation surfaces. For the surfaces and directions for which our methods apply, we can characterize the locally finite ergodic invariant measures of the straight-line flow in a set of directions of Hausdor…
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。
Certain topological dynamical systems are considered that arise from actions of $σ$-compact locally compact Abelian groups on compact spaces of translation bounded measures. Such a measure dynamical system is shown to have pure point dynamical spectrum if and only if its diffraction spectrum is pure point.
We study some special classes of piecewise continuous maps on a finite smooth partition of a compact manifold and look for invariant measures for such maps. We show that in the simplest one-dimensional case (so-called interval translation maps) a Borel probability non-atomic invariant measure exists for any map. We use this result to demonstrate that any interval translation map endowed with such a measure is metrically equivalent to an interval exchange map. Finally, we study the general case of piecewise continuous maps and prove a simple result on existence of an invariant measure provided …
Unique ergodicity of branched covers of translation surfaces
研究平移曲面沿狭缝分支循环覆盖的垂直流唯一遍历性，提出几何判据并证明对几乎所有狭缝端点成立。
The flow in a fixed direction on a translation surface S determines a decomposition of S into closed invariant sets, each of which is either periodic or minimal. We study this decomposition for translation surfaces in the hyperelliptic connected components $\mathcal{H}^{hyp}(2g-2)$ and $\mathcal{H}^{hyp}(g-1,g-1)$ of the corresponding strata of the moduli space of translation surfaces. Specifically, we characterize the pairs of nonnegative integers (p,m) for which there exists a translation surface in $\mathcal{H}^{hyp}(2g-2)$ or $\mathcal{H}^{hyp}(g-1,g-1)$ with precisely p periodic component…
Dynamical systems on an infinite translation surface with the lattice property are studied. The geodesic flow on this surface is found to be recurrent in all but countably many rational directions. Hyperbolic elements of the affine automorphism group are found to be nonrecurrent, and a precise formula regarding their action on cylinders is proven. A deformation of the surface in the space of translation surfaces is found, which "behaves nicely" with the geodesic flow and action of the affine automorphism group.
On the Ergodicity of Flat Surfaces of Finite Area
本文证明有限面积平坦曲面的遍历定理，通过 Veech 群重整化平移流，并给出基于度量变形控制的遍历性判据，改进并推广了 Cheung 和 Eskin 的结果。
Finite codimension stability of invariant surfaces
本文利用仿微分演算和上同调方程方法，证明了平移曲面上平坦测地流的不变曲面在有限可微类中具有有限余维稳定性。
The goal of this note is to study the action of the backward Rauzy-Veech algorithm on the translation surfaces with horizontal saddle connections. In particular, we prove that the orbit of a translation surface via the aforementioned algorithm is $\infty$-complete if and only if the surface does not posses horizontal saddle connections. Moreover, we show that appearance of all saddle connections as sides of the polygonal representations of the translations surface along the backward Rauzy-Veech induction orbit is equivalent to the minimality of the horizontal translation flow.
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…
In the moduli space $H_g$ of normalized translation surfaces of genus $g$, consider, for a small parameter $ρ>0$, those translation surfaces which have two non-parallel saddle-connections of length $\leq ρ$. We prove that this subset of $H_g$ has measure $o(ρ^2)$ w.r.t. any probability measure on $H_g$ which is invariant under the natural action of $SL(2,R)$. This implies that any such probability measure is regular, a property which is important in relation with the recent fundamental work of Eskin-Kontsevich-Zorich on the Lyapunov exponents of the KZ-cocycle.
We prove that on the typical translation surface the flow in almost every pair of directions are not isomorphic to each other and are in fact disjoint. It was not known if there were any translation surfaces other than torus covers with this property. We provide an application to the convergence of `circle averages' for the flow (away from a sequence of radii of density 0) for such surfaces. Even the density of a sequence of 'circles' was only known in a few special examples.
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。
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。
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…
In this paper, we compare two definitions of Rauzy classes. The first one was introduced by Rauzy and was in particular used by Veech to prove the ergodicity of the Teichmüller flow. The second one is more recent and uses a "labeling" of the underlying intervals, and was used in the proof of some recent major results about the Teichmüller flow. The Rauzy diagrams obtained from the second definition are coverings of the initial ones. In this paper, we give a formula that gives the degree of this covering. This formula is related to moduli spaces of "framed" translation surfaces, which correspon…
We consider the flow in direction $θ$ on a translation surface and we study the asymptotic behavior for $r\to 0$ of the time needed by orbits to hit the $r$-neighborhood of a prescribed point, or more precisely the exponent of the corresponding power law, which is known as hitting time. For flat tori the limsup of hitting time is equal to the diophantine type of the direction $θ$. In higher genus, we consider a generalized geometric notion of diophantine type of a direction $θ$ and we seek for relations with hitting time. For genus two surfaces with just one conical singularity we prove that t…
Dilation surfaces are generalizations of translation surfaces where the transition maps of the atlas are translations and homotheties with a positive ratio. In contrast with translation surfaces, the directional flow on dilation surfaces may contain trajectories accumulating on a limit cycle. Such a limit cycle is called hyperbolic because it induces a nontrivial homothety. It has been conjectured that a dilation surface with no actual hyperbolic closed geodesic is in fact a translation surface. Assuming that a dilation surface contains a horizon saddle connection, we prove that the directions…
In this paper the authors find examples of translation surfaces that have infinitely generated Veech groups, satisfy the topological dichotomy property that for every direction either the flow in that direction is completely periodic or minimal, and yet have minimal but non uniquely ergodic directions.
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