Kariane Calta, Kevin Wortman
We study the action of the horocycle flow on the moduli space of abelian differentials in genus two. In particular, we exhibit a classification of a specific class of probability measures that are invariant and ergodic under the horocycle flow on the stratum H(1,1).
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We provide a new proof that the horocycle flow preserving the Margulis measure on a variable negative curvature surface is standard. This was first proved by Ratner. The main purpose of this note is to provide a simplified case of the arguments for Kakutani equivalence of unipotent flows on homogeneous spaces, which have similar but more complicated structures, as well as illustrate the versatility of the method by applying it to a non-homogeneous flow.
Let $G$ be a connected semisimple Lie group with finite centre, and let $M= Γ\backslash G$ be a compact homogeneous manifold. Under a spectral gap assumption, we show that smooth time-changes of any unipotent flow on $M$ have polynomial decay of correlations. Our result applies also in the case where $M$ is a finite volume, non-compact quotient under some additional assumptions on the generator of the time-change. This generalizes a result by Forni and Ulcigrai (JMD, 2012) for smooth time-changes of horocycle flows on compact surfaces.
Let $d \geq 2$, and let $Γ\subset PSL(2, \mathbb{R})^d$ be an irreducible, cocompact lattice. We prove a sharp estimate up to a logarithmic factor on the rate of equidistribution of coordinate horocycle flows on $Γ\backslash PSL(2, \mathbb{R})^d$.
We show that the horocycle flows of open tight hyperbolic surfaces do not admit minimal sets.
We show that the horocycle flow associated with a foliation on a compact manifold by hyperbolic surfaces is minimal under certain conditions.
We prove effective equidistribution of non-closed horocycles in the unit tangent bundle of infinite-volume geometrically finite hyperbolic surfaces.
We prove bounds for twisted ergodic averages for horocycle flows of hyperbolic surfaces, both in the compact and in the non-compact finite area case. From these bounds we derive effective equidistribution results for horocycle maps. As an application of our main theorems in the compact case we further improve on a result of A. Venkatesh, recently already improved by J. Tanis and P. Vishe, on a sparse equidistribution problem for classical horocycle flows proposed by N. Shah and G. Margulis, and in the general non-compact, finite area case we prove bounds on Fourier coefficients of cups forms w…
We derive, from the work of M. Ratner on joinings of time-changes of horocycle flows and from the result of the authors on its cohomology, the property of orthogonality of powers for non-trivial smooth time-changes of horocycle flows on compact quotients. Such a property is known to imply P. Sarnak's Möbius orthogonality conjecture, already known for horocycle flows by the work of J. Bourgain, P. Sarnak and T. Ziegler.
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.
We show that the Hausdorff dimension of any proper Teichmüller horocycle flow orbit closure on any $\mathrm{SL}{(2,\mathbf{R})}$-invariant subvariety of Abelian or quadratic differentials is bounded away from the dimension of the subvariety in terms of the polynomial mixing rate of the Teichmüller horocycle flow on the subvariety. The proof is based on abstract methods for measurable flows adapted from work of Bourgain and Katz on sparse ergodic theorems.
A celebrated result of Ratner from the eighties says that two horocycle flows on hyperbolic surfaces of finite area are either the same up to algebraic change of coordinates, or they have no non-trivial joinings. Recently, Mohammadi and Oh extended Ratner's theorem to horocycle flows on hyperbolic surfaces of infinite area but finite genus. In this paper, we present the first joining classification result of a horocycle flow on a hyperbolic surface of infinite genus: a $\mathbb{Z}$ or $\mathbb{Z}^2$-cover of a general compact hyperbolic surface. We also discuss several applications.
On rigidity properties of time-changes of unipotent flows
本文研究半单线性群有限体积商上单幂流时间变换的可测同构，在谱间隙条件下证明其要么产生平凡联结极限，要么为仿射同构，推广了Ratner的horocycle流结果。
We study expansive properties for the geodesic and horocycle flows on compact Riemann surfaces of constant negative curvature. It is well-known that the geodesic flow is expansive in the sense of Bowen-Walters and the horocycle flow is positive and negative separating in the sense of Gura. In this paper, we give a new proof of the expansiveness in the sense of Bowen-Walters for the geodesic flow and show that the horocycle flow is positive and negative kinematic expensive in the sense of Artigue as well as expansive in the sense of Katok-Hasselblatt but not expensive in the sense of Bowen-Walt…
We consider a minimal compact lamination by hyperbolic surfaces. We prove that if it admits a leaf whose holonomy covering is not topologically trivial, then the horocycle flow on its unitary tangent bundle is minimal.
We consider smooth time-changes of the classical horocycle flows on the unit tangent bundle of a compact hyperbolic surface and prove sharp bounds on the rate of equidistribution and the rate of mixing. We then derive results on the spectrum of smooth time-changes and show that the spectrum is absolutely continuous with respect to the Lebesgue measure on the real line and that the maximal spectral type is equivalent to Lebesgue.
Mixing asymptotics for time-changes of horocycle flows
本文研究了光滑时间变换的水平流的混合性质，通过改进的剪切混合方法，证明了多项式上界与Ratner的最优速率一致，并在存在谱间隙时建立了精确的多项式渐近性。
A point is called generic for a flow preserving an infinite ergodic invariant Radon measure, if its orbit satisfies the conclusion of the ratio ergodic theorem for every pair of continuous functions with compact support and non-zero integrals. The generic points for horocycle flows on hyperbolic surfaces of finite genus are understood, but there are no results in infinite genus. We give such a result, by characterizing the generic points for $\Z^d$--covers.
In this paper, we provide a model for cross sections to the geodesic and horocycle flows on $\operatorname{SL}(2, \mathbb{R})/G_q$ using an extension of a heuristic of P. Arnoux and A. Nogueira. Our starting point is a continued fraction algorithm related to the group $G_q$, and a cross section to the horocycle flow on $\operatorname{SL}(2, \mathbb{R})/G_q$ from a previous paper. As an application, we get the natural extension and invariant measure for a symmetric $G_q$-Farey interval map resulting from projectivizing the aforementioned continued fraction algorithm.
In this paper we compute all the smooth solutions to the Hamilton-Jacobi equation associated with the horocycle flow. This can be seen as the Euler-Lagrange flow (restricted to the energy level set $E^{-1}(\frac 12)$) defined by the Tonelli Lagrangian $L:T\mathbb H\rightarrow \mathbb R$ given by (hyperbolic) kinetic energy plus the standard magnetic potential. The method we use is to look at Lagrangian graphs that are contained in the level set $\{H=\frac 12\}$, where $H:T^*\mathbb H\rightarrow \mathbb R$ denotes the Hamiltonian dual to $L$.
We study the dynamics of the geodesic and horocycle flows of the unit tangent bundle $(\hat M, T^1\mathcal{F})$ of a compact minimal lamination $(M,\mathcal F)$ by negatively curved surfaces. We give conditions under which the action of the affine group generated by the joint action of these flows is minimal, and examples where this action is not minimal. In the first case, we prove that if $\mathcal F$ has a leaf which is not simply connected, the horocyle flow is topologically transitive.
A classic result due to Furstenberg is the strict ergodicity of the horocycle flow for a compact hyperbolic surface. Strict ergodicity is unique ergodicity with respect to a measure of full support, and therefore implies minimality. The horocycle flow has been previously studied on minimal foliations by hyperbolic surfaces on closed manifolds, where it is known not to be minimal in general. In this paper, we prove that for the special case of Riemannian foliations, strict ergodicity of the horocycle flow still holds. This in particular proves that this flow is minimal, which establishes a conj…
Horocycle flows on abelian covers of surfaces of negative curvature
本文研究双曲曲面的阿贝尔覆盖上回旋流的渐进行为，通过傅里叶分解和谱理论方法，证明了正规函数的测度积分渐进行为，并在常曲率情况下恢复了Ledrappier和Sarig的结果，同时得到回旋测度平均的幂偏差估计。
We show that the horocyclic flow of an orientable compact higher genus surface without conjugate points and with continuous Green bundles is uniquely ergodic. The result applies to nonflat nonpositively curved surfaces and generalizes a classical result of Furstenberg and Marcus in negative curvature. The proof relies on the definition of a uniformly expanding parametrization on the quotient by the strips of the surface.
We construct a Poincaré map $\mathcal{P}_h$ for the positive horocycle flow on the modular surface $PSL(2,\mathbb{Z})\backslash \mathbb{H}$, and begin a systematic study of its dynamical properties. In particular we give a complete characterisation of the periodic orbits of $\mathcal{P}_h$, and show that they are equidistributed with respect to the invariant measure of $\mathcal{P}_h$ and that they can be organised in a tree by using the Stern-Brocot tree of rational numbers. In addition we introduce a time-reparameterisation of $\mathcal{P}_h$ which gives an insight into the dynamics of the n…
Topology of horocycles on geometrically finite nonpositively curved surfaces
该研究针对有限生成基本群的秩1非正曲率曲面，探究horocycle的拓扑性质，完成半horocycle分类、确定horocycle流非游荡集并刻画存在极小集的曲面，推进了相关动力系统与几何拓扑研究。
We study dynamics of the horocycle flow on strata of translation surfaces, introduce new invariants for ergodic measures, and analyze the interaction of the horocycle flow and real Rel surgeries. We use this analysis to complete and extend results of Calta and Wortman classifying horocycle-invariant measures in the eigenform loci. We classify the orbit-closures and prove that every orbit is equidistributed in its orbit-closure. We also prove equidistribution statements regarding limits of sequences of measures, some of which have applications to counting problems.
Extreme events for horocycle flows
本文研究了常负曲率曲面上极环流尖点游历的极端事件，通过研究收缩庞加莱截面的命中时间分布，推广了模曲面的极值定律，并证明极限定律可用 Farey 序列间隙分布的 Hall 公式表达。
Let $M = Γ\backslash \text{SL}(2,\mathbb{R})$ be a compact quotient of $\text{SL}(2,\mathbb{R})$ equipped with the normalized Haar measure $\text{vol}$, and let $\{h_t\}_{t \in \mathbb{R}}$ denote the horocycle flow on $M$. Given $p \in M$ and $W \in \mathfrak{sl}_2(\mathbb{R}) \setminus \{0\}$ not parallel to the generator of the horocycle flow, let $γ_{p}^W$ denote the probability measure uniformly distributed along the arc $s \mapsto p \exp(sW)$ for $0\leq s \leq 1$. We establish quantitative estimates for the rate of convergence of $[(h_t)_{\ast} γ_{p}^W](f)$ to $\text{vol}(f)$ for suffici…
Using zippered rectangle coordinates we parametrize a Poincaré section for horocycle flow on the space of genus 2 translation surfaces with one singular cone point of angle $6π$. In addition, we bound the return time under horocycle flow to this Poincaré section by examining a subset of surfaces where a certain sum of parameters is large.
Horocycle flow on flat projective bundles: topological remarks and applications
本文研究双曲曲面上平坦射影丛的叶状 horocycle 流，在 Conze-Guivarc'h 条件下用近端动力学刻画其拓扑与遍历性质，并证明凸余紧时限制流具有唯一保守遍历 Radon 测度。
We formulate and prove a finite version of Vinogradov's bilinear sum inequality. We use it together with Ratner's joinings theorems to prove that the Mobius function is disjoint from discrete horocycle flows on $Γ\backslash SL_2(\mathbb{R}).
We prove (under the condition of A. G. Kushnirenko) that all time changes for the horocycle flow have purely absolutely continuous spectrum in the orthocomplement of the constant functions. This provides an answer to a question of A. Katok and J.-P. Thouvenot on the spectral nature of time changes for horocycle flows. Our proofs rely on positive commutator methods for self-adjoint operators.
Abelian covers of hyperbolic surfaces: equidistribution of spectra and infinite volume mixing asymptotics for horocycle flows
本文研究紧双曲曲面的 Abel 覆盖，建立 $\mathbb{Z}^d$ 覆盖上 horocycle 流相关函数的渐近展开以证明强 Krickeberg 混合，并证明有限 Abel 覆盖序列中 Casimir 算子谱测度弱收敛于绝对连续测度。
\noindent In [1] L. Polterovich and Z. Rudnick considered the behavior of a one-parameter subgroup of a Lie group under the influence of a sequence of kicks. Among others they raise the following problem: {\it is the horocycle flow stably quasi-mixing on $SL(2,\mathbb{R})/Γ$?} Equivalently it can be reformulated in terms of boundedness of the sequences of products $ P_n(t) = Φ_n H(t)Φ_{n-1} H(t) \, ... \, Φ_1 H(t) $ where $H(t) = \begin{pmatrix} 1 & t 0 & 1 \end{pmatrix}$ and $Φ=\{Φ_n\} \subset SL(2,\mathbb{R})$. We solve this problem positively and as a consequence obtain the following applic…
We apply a method inspired by Ratner's work on quantitative mixing for the geodesic flow (Ergod. Theory Dyn. Syst., 1987) and developed by Burger (Duke Math. J., 1990) to study ergodic integrals for horocycle flows. We derive an explicit asymptotic expansion for horocycle averages, recovering a celebrated result by Flaminio and Forni (Duke Math. J., 2003), and we show that the coefficients in the asymptotic expansion are Hölder continuous with respect to the base point. Furthermore, we provide short and streamlined proofs of the spatial limit theorems of Bufetov and Forni (Ann. Sci. Éc. Norm. …
Extreme events and impact statistics for unipotent actions on the space of lattices
本文将二维格空间上horocycle流的极端值律推广至n维格空间上秩为k的幂幺作用，基于收缩截面分析命中时间与影响统计，并推导了不同k值下的极限律与尾部渐近性质。
The Ratner property, a quantitative form of divergence of nearby trajectories, is a central feature in the study of parabolic homogeneous flows. Discovered by Marina Ratner and used in her 1980th seminal works on horocycle flows, it pushed forward the disjointness theory of such systems. In this paper, exploiting a recent variation of the Ratner property, we prove new disjointness phenomena for smooth parabolic flows beyond the homogeneous world. In particular, we establish a general disjointness criterion based on the switchable Ratner property. We then apply this new criterion to study disjo…
This article is a first step towards the understanding of the dynamics of the horocycle flow on foliated manifolds by hyperbolic surfaces. This is motivated by a question formulated by M. Martinez and A. Verjovsky on the minimality of this flow assuming that the natural affine foliation is minimal too. We have tried to offer a simple presentation, which allows us to update and shed light on the classical theorem proved by G. A. Hedlund in 1936 on the minimality of the horocycle flow on compact hyperbolic surfaces. Firstly, we extend this result to the product of PSL(2,R) and a Lie group G, whi…
Let $M$ be a closed oriented surface endowed with a Riemannian metric $g$ and let $Ω$ be a 2-form. We show that the magnetic flow of the pair $(g,Ω)$ has zero asymptotic Maslov index and zero Liouville action if and only $g$ has constant Gaussian curvature, $Ω$ is a constant multiple of the area form of $g$ and the magnetic flow is a horocycle flow. This characterization of horocycle flows implies that if the magnetic flow of a pair $(g,Ω)$ is $C^1$-conjugate to the horocycle flow of a hyperbolic metric $\bar{g}$ then there exists a constant $a>0$, such that $ag$ and $\bar{g}$ are isometric an…
We introduce two properties: strong R-property and $C(q)$-property, describing a special way of divergence of nearby trajectories for an abstract measure preserving system. We show that systems satisfying the strong R-property are disjoint (in the sense of Furstenberg) with systems satisfying the $C(q)$-property. Moreover, we show that if $u_t$ is a unipotent flow on $G/Γ$ with $Γ$ irreducible, then $u_t$ satisfies the $C(q)$-property provided that $u_t$ is not of the form $h_t\times\operatorname{id}$, where $h_t$ is the classical horocycle flow. Finally, we show that the strong R-property hol…
We prove that the orbit of a non-periodic point at prime values of the horocycle flow in the modular surface is dense in a set of positive measure. For some special orbits we also prove that they are dense in the whole space (assuming the Ramanujan/Selberg conjectures for GL(2)/Q). In the process, we derive an effective version of Dani's Theorem for the (discrete) horocycle flow.
On the Space of Ergodic Measures for the Horocycle Flow on Strata of Abelian Differentials
本文研究平移曲面分层$\mathcal{H}(2)$上的水平流，证明存在一列支撑在周期轨道上的遍历测度弱收敛到非遍历不变测度，并由此得出某些点的水平流轨道不向任何不变测度等分布。
The main results of this paper are limit theorems for horocycle flows on compact surfaces of constant negative curvature. One of the main objects of the paper is a special family of horocycle-invariant finitely-additive Hoelder measures on rectifiable arcs. An asymptotic formula for ergodic integrals for horocycle flows is obtained in terms of the finitely-additive measures, and limit theorems follow as a corollary of the asymptotic formula. The objects and results of this paper are similar to those in [15], [16], [4] and [5] for translation flows on flat surfaces. The arguments are based on t…
On the unit tangent bundle of a hyperbolic surface, we study the density of positive orbits $(h^s v)_{s\ge 0}$ under the horocyclic flow. More precisely, given a full orbit $(h^sv)_{s\in \R}$, we prove that under a weak assumption on the vector $v$, both half-orbits $(h^sv)_{s\ge 0}$ and $(h^s v)_{s\le 0}$ are simultaneously dense or not in the nonwandering set $\mathcal{E}$ of the horocyclic flow. We give also a counter-example to this result when this assumption is not satisfied.
Limits of geodesic push-forwards of horocycle invariant measures
本文证明SL(2,R)作用中horocycle与geodesic子群遍历平均的条件收敛，改进测地推进弱收敛，并给出Birkhoff一般性与Oseledets正则性较弱版本的简短证明。
Horocycle flow at product of two primes
本文证明了在特定算术格或 $SL(2,\mathbb{Z})$ 下，非周期点的极限环轨道在具有两个素因子的整数处采样时等分布。
Let $(h_t)_{t\in \mathbb{R}}$ be the horocycle flow acting on $(M,μ)=(Γ\backslash \text{SL}(2,\mathbb{R}),μ)$, where $Γ$ is a co-compact lattice in $\text{SL}(2,\mathbb{R})$ and $μ$ is the homogeneous probability measure locally given by the Haar measure on $\text{SL}(2,\mathbb{R})$. Let $τ\in W^6(M)$ be a strictly positive function and let $μ^τ$ be the measure equivalent to $μ$ with density $τ$. We consider the time changed flow $(h_t^τ)_{t\in \mathbb{R}}$ and we show that there exists $γ=γ(M,τ)>0$ and a constant $C>0$ such that for any $ f_0, f_1, f_2\in W^6(M)$ and for all $0=t_0<t_1<t_2$, …
Let M be an invariant subvariety in the moduli space of translation surfaces. We contribute to the study of the dynamical properties of the horocycle flow on M. In the context of dynamics on the moduli space of translation surfaces, we introduce the notion of a 'weak classification of horocycle invariant measures' and we study its consequences. Among them, we prove genericity of orbits and related uniform equidistribution results, asymptotic equidistribution of sequences of pushed measures, and counting of saddle connection holonomies. As an example, we show that invariant varieties of rank on…
For a given finite index subgroup H of SL(2,Z), we use a process developed by Fisher and Schmidt to lift a Poincaré section of the horocycle flow on SL(2,R)/SL(2,Z) found by Athreya and Cheung to the finite cover SL(2,R)/H of SL(2,R)/SL(2,Z). We then use the properties of this section to prove the existence of the limiting gap distribution of various subsets of Farey fractions. Additionally, to each of these subsets of fractions, we extend solutions by Xiong and Zaharescu, and independently Boca, to a Diophantine approximation problem of Erdős, Szüsz, and Turán.
We consider flows, called $W^{\rm u}$ flows, whose orbits are the unstable manifolds of a codimension one Anosov flow. Under some regularity assumptions, we give a short proof of the strong mixing property of $W^{\rm u}$ flows and we show that $W^{\rm u}$ flows have purely absolutely continuous spectrum in the orthocomplement of the constant functions. As an application, we obtain that time changes of the classical horocycle flows for compact surfaces of constant negative curvature have purely absolutely continuous spectrum in the orthocomplement of the constant functions for time changes in a…
We prove analogs of the logarithm laws of Sullivan and Kleinbock-Margulis in the context of unipotent flows. In particular, we prove results for horospherical actions on homogeneous spaces $G/Γ$. We describe some relations with multi-dimensional diophantine approximation.
We study the dynamics of unipotent flows on frame bundles of hyperbolic manifolds of infinite volume. We prove that they are topologi-cally transitive, and that the natural invariant measure, the so-called " Burger-Roblin measure ", is ergodic, as soon as the geodesic flow admits a finite measure of maximal entropy, and this entropy is strictly greater than the codi-mension of the unipotent flow inside the maximal unipotent flow. The latter result generalises a Theorem of Mohammadi and Oh.
Tremors and horocycle dynamics on the moduli space of translation surfaces
本文在平移曲面层上引入震颤形变，并借此构造出 horocycle 流轨道的新行为，包括支撑严格包含于轨道的通用测度、非通用轨道以及 Hausdorff 维数非整数的轨道闭包。
We construct a Poincaré section for the horocycle flow on the modular surface $SL(2, \R)/SL(2, \Z)$, and study the associated first return map, which coincides with a transformation (the {\it BCZ map}) defined by Boca-Cobeli-Zaharescu. We classify ergodic invariant measures for this map and prove equidistribution of periodic orbits. As corollaries, we obtain results on the average depth of cusp excursions and on the distribution of gaps for Farey sequences and slopes of lattice vectors.
We prove logarithm laws for unipotent flows on non-compact finite-volume hyperbolic manifolds. Our method depends on the estimate of norms of certain incomplete Eisenstein series.
The analogous statement to Oppenheim conjecture over a local field of positive characteristic is proved. The dynamical argument is most involved in the case of characteristic 3.
The Farey sequence is the sequence of all rational numbers in the real unit interval, stratified by increasing denominators. A classical result by Hall says that its normalized gap distribution is the same as the distribution of the random variable 1/(2 zeta(2) xy) on a certain unit triangle. In this paper we weight the denominators by an arbitrary piecewise-smooth continuous function, and we characterize the resulting gap distribution as that of a multiple of the above variable, defined on a certain unit pentagon. Our characterization refines previous results by Boca, Cobeli and Zaharescu, bu…
We adapt to $C^r$ Anosov flows on compact manifolds a construction for $C^r$ discrete-time hyperbolic dynamics ($r>1$), obtaining anisotropic Banach or Hilbert spaces on which the resolvent of the generator of weighted transfer operators for the flow is quasi-compact. We apply this to study the ergodic integrals of the horocycle flows $h_ρ$ of $C^r$ codimension one mixing Anosov flows. In dimension three, for any suitably bunched $C^3$ contact Anosov flow with orientable strong-stable distribution, we establish power-law convergence of the ergodic average. We thereby implement the program of G…
An upper bound for the mean Lorentz force of magnetic flows without conjugate points in terms of the geodesic curvature of horocycles
本文研究亏格至少二的闭黎曼曲面上无共轭点的磁流，证明在 Mañé 临界值之上，洛伦兹力积分受水平圆测地曲率约束，并由此推出基于面积和高斯曲率的显式上界。
Let $u_{X}^{t}$ be a unipotent flow on $X=SO(n,1)/Γ$, $u_{Y}^{t}$ be a unipotent flow on $Y=G/Γ^{\prime}$. Let $\tilde{u}_{X}^{t}$, $\tilde{u}_{Y}^{t}$ be time-changes of $u_{X}^{t}$, $u_{Y}^{t}$ respectively. We show the disjointness (in the sense of Furstenberg) between $u_{X}^{t}$ and $\tilde{u}_{Y}^{t}$ (or $\tilde{u}_{X}^{t}$ and $u_{Y}^{t}$) in certain situations. Our method refines the works of Ratner and extends a recent work of Dong, Kanigowski and Wei.
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