Pierre Arnoux, Thomas A. Schmidt
We show that each of Veech's original examples of translation surfaces with ``optimal dynamics'' whose trace field is of degree greater than two has non-periodic directions of vanishing SAF-invariant. Furthermore, we give explicit examples of pseudo-Anosov diffeomorphisms whose contracting direction has zero SAF-invariant.
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For a non-arithmetic Veech surface, it is known that the set points having finite orbit under the Veech group, called the set of periodic points, is finite. However, few examples of these periodic point sets have been computed. In what follows, we compute new examples of periodic point sets for a family of non-hyperelliptic Veech surfaces with unbounded complexity and consider applications to classifying the holomorphic sections of certain surface bundles as well as applications to the study of infinitely generated Veech groups.
We announce a classification of genus 2 Veech surfaces in the stratum with a single double zero. Furthermore, we classify all completely periodic translation surfaces in genus 2.
We study periodic wind-tree models, billiards in the plane endowed with $\mathbb{Z}^2$-periodically located identical connected symmetric right-angled obstacles. We exhibit effective asymptotic formulas for the number of (isotopy classes of) periodic billiard trajectories (up to $\mathbb{Z}^2$-translations) on Veech wind-tree billiards, that is, wind-tree billiards whose underlying compact translation surfaces are Veech surfaces. We show that the error term depends on spectral properties of the Veech group and give explicit estimates in the case when obstacles are squares of side length $1/2$.
Over the course of studying billiard dynamics, several questions were raised. One of the questions was, which surfaces satisfy the following property (which is called Veech's dichotomy): Any direction is either completely periodic or uniquely ergodic. In an important paper Veech gave a sufficient condition for this dichotomy. He showed that if the stabilizer of a translation surface is a lattice in $SL_2(\R)$, then the surface satisfies Veech's dichotomy. Later, Smillie and Weiss proved that this condition is not necessary. They constructed a translation surface which satisfies Veech's dichoto…
In this note we are interested in the dynamics of the linear flow on infinite periodic $\mathbb{Z}^d$-covers of Veech surfaces. An elementary remark allows us to show that the kernel of some natural representations of the Veech group acting on homology is "big". In particular, the same is true for the Veech group of the infinite surface, answering a question of Pascal Hubert. We give some applications to the dynamics on wind-tree models where the underlying compact translation surface is a Veech surface.
The logarithmic asymptotics for the growth of the number of periodic orbits, such that the norm of the corresponding renormalization matrix does not exceed a given constant, is computed for the Teichmueller flow on Veech's moduli space of zippered rectangles. The rate is equal to the entropy of the flow with respect to the absolutely continuous invariant measure.
Computing Periodic Points on Veech Surfaces
提出计算非正方形平铺 Veech 曲面周期点的算法，并应用于亏格 3 最小分层中的 Prym 本征形式，证明低判别式下除 Prym 对合固定点外无周期点。
An Abelian differential gives rise to a flat structure (translation surface) on the underlying Riemann surface. In some directions the directional flow on the flat surface may contain a periodic region that is made up of maximal cylinders filled by parallel geodesics of the same length. The growth rate of the number of such regions counted with weights, as a function of the length, is quadratic with a coefficient, called Siegel-Veech constant, that is shared by almost all translation surfaces in the ambient stratum. We evaluate various Siegel-Veech constants associated to the geometry of confi…
We prove that a ``bouillabaisse'' surface (translation surface which has two transverse parabolic elements) has totally real trace field. As a corollary, non trivial Veech groups which have no parabolic elements do exist. The proof follows Veech's viewpoint on Thurston's construction of pseudo-Anosov diffeomorphisms.
Permutation of periodic points of Veech surfaces in $\mathcal{H}(2)$
研究$\mathcal{H}(2)$中Veech曲面Weierstrass点的置换群，依据判别式分类证明了其仿射群诱导的置换群分别同构于Dih4、Dih5和Dih6，且Dehn multitwists亦产生相同群。
This paper deals with Prym eigenforms which are introduced previously by McMullen. We prove several results on the directional flow on those surfaces, related to complete periodicity (introduced by Calta). More precisely we show that any homological direction is algebraically periodic, and any direction of a regular closed geodesic is a completely periodic direction. As a consequence we draw that the limit set of the Veech group of every Prym eigenform in some Prym loci of genus 3,4, and 5 is either empty, one point, or the full circle at infinity. We also construct new examples of translation…
We prove the existence of Veech groups having a critical exponent strictly greater than any elementary Fuchsian group (i.e. $>\frac{1}{2}$) but strictly smaller than any lattice (i.e. $<1$). More precisely, every affine covering of a primitive L-shaped Veech surface $X$ ramified over the singularity and a non-periodic connection point $P\in X$ has such a Veech group. Hubert and Schmidt showed that these Veech groups are infinitely generated and of the first kind. We use a result of Roblin and Tapie which connects the critical exponent of the Veech group of the covering with the Cheeger constan…
Periodic Points of Prym Eigenforms
本文证明亏格 2、3、4 最小层中 Prym 特征形式的周期点恰为 Prym 对合固定点，回答了 Apisa–Wright 的问题，并给出 Möller 亏格 2 分类的几何证明。
Ergodic measures for periodic type $\mathbb{Z}^m$-skew-products over Interval Exchange Transformations
本文研究了区间交换变换上周期型$\mathbb{Z}^m$-斜积的遍历不变Radon测度分类问题，通过引入adic编码的符号语言并应用相关符号结果，证明了此类测度恰为由$\mathbb{R}^m$参数化的Maharam测度，并给出了其显式形式与弱*-连续性。
There is a natural action of SL(2,R) on the moduli space of translation surfaces, and this yields an action of the unipotent subgroup $U = {\begin{pmatrix} 1 & * 0 & 1 \end{pmatrix}}$. We classify the U-invariant ergodic measures on certain special submanifolds of the moduli space. (Each submanifold is the SL(2,R)-orbit of the set of branched covers of a fixed Veech surface.) For the U-action on these submanifolds, this is an analogue of Ratner's Theorem on unipotent flows. The result yields an asymptotic estimate of the number of periodic trajectories for billiards in a certain family of non-…
Trace field degrees of Abelian differentials
研究证明了阿贝尔微分的迹域次数可以覆盖所有偶数范围，揭示了伪阿诺索夫伸缩因子在模空间层中的普遍性。
Trace field degrees in the Torelli group
研究证明闭曲面Torelli群中伪Anosov映射的迹数域次数范围，并通过实例验证了Thurston的断言。
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.
In this paper, we investigate the closure of a large class of Teichmüller discs in the stratum Q(1,1,1,1) or equivalently, in a GL^+_2(R)-invariant locus L of translation surfaces of genus three. We describe a systematic way to prove that the GL^+_2(R)-orbit closure of a translation surface in L is the whole of L. The strategy of the proof is an analysis of completely periodic directions on such a surface and an iterated application of Ratner's theorem to unipotent subgroups acting on an ``adequate'' splitting. This analysis applies for example to all Teichmueller discs stabilized obtained by …
Comparing Periodic Point Invariants for Parameterized Families of Maps
本文比较紧流形参数化映射族的周期点不变量，指出纤维Fuller迹比迭代的纤维Reidemeister迹集合更敏感，解决了Malkiewich和Ponto的相关猜想。
The present paper is devoted to studying the Hill-type formula and Krein-type trace formula for ODE, which is a continuous work of our previous work for Hamiltonian systems \cite{HOW}. Hill-type formula and Krein-type trace formula are given by Hill at 1877 and Krein in 1950's separately. Recently, we find that there is a closed relationship between them \cite{HOW}. In this paper, we will obtain the Hill-type formula for the $S$-periodic orbits of the first order ODEs. Such a kind of orbits is considered naturally to study the symmetric periodic and quasi-periodic solutions. By some similar id…
In this paper, we consider a family of Sturm-Liouville operators on the $ω$-periodic domain. The bifurcation with respect to the parameter region is studied, and the elliptic regions are estimated by trace formula. At last, these results are used to study the linear stability of the elliptic equilibrium point along $z$-axis in Robe's restricted three-body problem.
Straight-line trajectories on the Mucube
研究Mucube上的直线流周期性和漂移轨道，通过其刚体对称性描述，并确定周期方向集，最终证明周期和ergodic方向集的密度。
A Periodic Orbit Trace Formula for Quantum Scrambling: The Role of the Normally Hyperbolic Invariant Manifold
本文推导了局部微canonical OTOC的 semiclassical 扩展，揭示了正常双曲不变流形上不稳定周期轨道对量子混动率的影响，并提出了局部不稳定性指数Λ(J)。
In this paper we show that any Bohr-zero non-periodic set $B$ of traceless integer valued matrices, denoted by $Λ$, intersects non-trivially the conjugacy class of any matrix from $Λ$. As a corollary, we obtain that the family of characteristic polynomials of $B$ contains all characteristic polynomials of matrices from $Λ$. The main ingredient used in this paper is an equidistribution result for an $SL_d(\mathbf{Z})$ random walk on a finite-dimensional torus deduced from Bourgain-Furman-Lindenstrauss-Mozes work.
Gutzwiller's famous semiclassical trace formula plays an important role in theoretical and experimental quantum mechanics with tremendous success. We review the physical derivation of this deep periodic orbit theory in terms of the phase space formulation with an view towards the Hamiltonian dynamical systems. The Maslov phase appearing in the trace formula is clarified by Meinrenken as Conley-Zhender index for periodic orbits of Hamiltonian systems. We also survey and compare various versions of Maslov indices to establish this fact. A refinement and improvement to Conley-Zehnder's index theo…
In the present paper, we build up trace formulas for both the linear Hamiltonian systems and Sturm-Liouville systems. The formula connects the monodromy matrix of a symmetric periodic orbit with the infinite sum of eigenvalues of the Hessian of the action functional. A natural application is to study the non-degeneracy of linear Hamiltonian systems. Precisely, by the trace formula, we can give an estimation for the upper bound such that the non-degeneracy preserves. Moreover, we could estimate the relative Morse index by the trace formula. Consequently, a series of new stability criteria for t…
Silent Orbits and Cancellations in the Wave Trace
该研究构造了含多条等长异Maslov指标周期弹球轨道的凸平面区域，通过微局部参数化子求解曲率射流方程组实现波迹抵消，证明波迹可任意光滑，揭示逆谱问题的局限及拉普拉斯谱与长度谱的本质差异。
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