Han Yu
In this paper, we study orbit closures under diagonal torus actions. We show that if $(x,y)\in\mathbb{T}^2$ is not contained in any rational lines, then its orbit under the $\times 2, \times 3, \times 5$ actions is dense in $\mathbb{T}^2.$
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We develop an algorithm that determines, for a given squarefree binary form $F$ with real coefficients, a smallest representative of its orbit under $\operatorname{SL}(2,\mathbb Z)$, either with respect to the Euclidean norm or with respect to the maximum norm of the coefficient vector. This is based on earlier work of Cremona and Stoll. We then generalize our approach so that it also applies to the problem of finding an integral representative of smallest height in the $\operatorname{PGL}(2,\mathbb Q)$ conjugacy class of an endomorphism of the projective line. Having a small model of such an …
Hyperplane absolute winning property of bounded orbits under diagonalizable flows on $\mathrm{SL}_3(\mathbb{C})/\mathrm{SL}_3(\mathcal{O}_{\mathbb{K}})$
本文将 An-Guan-Kleinbock 关于实对角化流有界轨道的赢性质推广到复虚二次域情形，通过证明复数 Minkowski 线性形式定理并构造 Schmidt 博弈，建立了超平面绝对赢集与有界轨道的对应关系。
We prove a number field analogue of W. M. Schmidt's conjecture on the intersection of weighted badly approximable vectors and use this to prove an instance of a conjecture of An, Guan and Kleinbock. Namely, let $G := SL_2(\mathbb{R}) \times \dots \times SL_2(\mathbb{R}) $ and $Γ$ be a lattice in $G$. We show that the set of points on $G/Γ$ whose forward orbits under a one parameter Ad-semisimple subsemigroup of $G$ are bounded, form a hyperplane absolute winning set.
We prove that the set of points that have bounded orbits under certain diagonalizable flows is a hyperplane absolute winning subset of $SL_{n}(\mathbb{R})/SL_{n}(\mathbb{Z})$.
Density of almost squares of horospherical orbits in non-uniform quotients of $\operatorname{SL}_2(\mathbb{R})\times\operatorname{SL}_2(\mathbb{R})$
本文证明在SL2(R)×SL2(R)的非均匀格商空间中，具有稠密轨道的标准轨形子群在几乎平方点处的轨道仍稠密，通过有效追踪周期轨道并利用Weyl不等式及齐次动力学工具实现。
Let $D$ be a non-empty effective divisor on $\mathbb{P}^1$. We show that when ordered by height, any set of $(D,S)$-integral points on $\mathbb{P}^1$ of bounded degree has relative density zero. We then apply this to arithmetic dynamics: let $φ(z)\in \overline{\mathbb{Q}}(z)$ be a rational function of degree at least two whose second iterate $φ^2(z)$ is not a polynomial. We show that as we vary over points $P\in\mathbb{P}^1(\overline{\mathbb{Q}})$ of bounded degree, the number of algebraic integers in the forward orbit of $P$ is absolutely bounded and zero on average.
We prove that for any countably many one-parameter diagonalizable subgroups $F_n$ of $\rm{SL}_3(\mathbb{R})$, the set of $Λ\in\rm{SL}_3(\mathbb{R})/\rm{SL}_3(\mathbb{Z})$ such that all the orbits $F_nΛ$ are bounded has full Hausdorff dimension.
We first show that the subgroup of the abelian real group $\mathbb{R}$ generated by the coordinates of a point in $x = (x_1,\dots,x_n)\in\mathbb{R}^n$ completely classifies the $\mathsf{GL}(n,\mathbb Z)$-orbit of $x$. This yields a short proof of J.S.Dani's theorem: the $\mathsf{GL}(n,\mathbb Z)$-orbit of $x\in\mathbb{R}^n$ is dense iff $x_i/x_j\in \mathbb{R} \setminus \mathbb Q$ for some $i,j=1,\dots,n$. We then classify $\mathsf{GL}(n,\mathbb Z)$-orbits of rational affine subspaces $F$ of $\mathbb{R}^n$. We prove that the dimension of $F$ together with the volume of a special parallelotope a…
Given a maximal $\mathbb{Q}$-torus in $SL(N,\mathbb{Q})$, its orbit from identity coset in $SL(N,\mathbb{R})/SL(N,\mathbb{Z})$ naturally carries a possibly infinite Haar measure. We classify all possible limit measures of it when translated by a sequence of elements from $SL(N,\mathbb{R})$. This is a natural extension of Shapira and Zheng's work where only $\mathbb{Q}$-split tori are considered.
Effective equidistribution of polynomial trajectories in $\operatorname{SL}_3(\mathbb R)/\operatorname{SL}_3(\mathbb Z)$
本文针对SL₃(R)/SL₃(Z)中的固定矩阵多项式轨迹，给出保证其时间平均有效等分布的内在丢番图条件，结合多种定量方法完成证明，所得多项式测试的次数界最优。
Quantitative orbit equivalence for $\mathbb{Z}$-odometers
本文证明了任意两个 $\mathbb{Z}$-odometers 在 $L^1$ 意义下轨道等价，强化了先前结果并明确了这些系统的定量化轨道等价性图景。
Uniform distribution of saddle connection lengths in all $\mathsf{SL}(2,\mathbb{R})$ orbits
研究显示SL(2,R)轨道中平坦表面的鞍点连接长度序列在单位区间内呈现均匀分布特性。
We consider an autonomous differential system in $\mathbb{R}^n$ with a periodic orbit and we give a new method for computing the characteristic multipliers associated to it. Our method works when the periodic orbit is given by the transversal intersection of $n-1$ codimension one hypersurfaces and is an alternative to the use of the first order variational equations. We apply it to study the stability of the periodic orbits in several examples, including a periodic solution found by Steklov studying the rigid body dynamics.
Consider a smooth action of $\mathbb R^n$ on a connected manifold $M$, not necessarily compact, of dimension $m$ and rank $k$. Assume that $M$ is not a cylinder. Then there exists an orbit of the action of dimension $<(m+k)/2$. As a consequence, one shows that if there is a non-zero element of the ring of Pontrjagin classes of $M$ of degree $4\ell\geq 4$, then there exists an orbit of the action of dimension $\leq m-\ell-1$.
Let $\mathcal G_2$ denote the affine group $GL(2,\mathbb Z) \ltimes \mathbb Z^{2}$. For every point $x=(x_1,x_2) \in \R2$ let $\orb(x)=\{y\in\R2\mid y=γ(x)$ for some $γ\in \mathcal{G}_2 \}$. Let $G_{x}$ be the subgroup of the additive group $\mathbb R$ generated by $x_1,x_2, 1$. If $\rank(G_x)\in \{1,3\}$ then $\orb(x)=\{y\in\R2\mid G_y=G_x\}$. If $\rank(G_x)=2$, knowledge of $G_x$ is not sufficient in general to uniquely recover $\orb(x)$: rather, $G_x$ classifies precisely $\max(1,ϕ(d)/2)$ different orbits, where $d$ is the denominator of the smallest positive nonzero rational in $G_x$ and $…
We consider a class of smooth mixing flows $T^{α,γ}$ on $\mathbb{T}^2$ with one degenerated fixed point $x_0\in \mathbb{T}^2$ of power type $γ\in (-1,0)$. We prove that for a $G_δ$ dense set of $α\in \mathbb{T}$, a prime number theorem for $T^{α,γ}$ holds along a full upper density subsequence. In particular it follows that for every $x\in \mathbb{T}^2\setminus\{x_0\}$, the prime orbit $\mathbb{T}^2$. We also show that there exists a class of smooth weakly mixing flows on $\mathbb{T}^2$ for which a prime number theorem holds. In fact we show that there exists a dense set of smooth functions (i…
Let $\mathcal{W}\subset\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ be a surface given by the vanishing of a $(2,2,2)$-form. These surfaces admit three involutions coming from the three projections $\mathcal{W}\to\mathbb{P}^1\times\mathbb{P}^1$, so we call them $\textit{tri-involutive K3 (TIK3) surfaces}$. By analogy with the classical Markoff equation, we say that $\mathcal{W}$ is of $\textit{Markoff type (MK3)}$ if it is symmetric in its three coordinates and invariant under double sign changes. An MK3 surface admits a group of automorphisms $\mathcal{G}$ generated by the three involuti…
We give examples of $n \times n$ matrices $A$ and $B$ over the filed $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ such that for almost every column vector $x \in \mathbb{K}^n$, the orbit of $x$ under the action of the semigroup generated by $A$ and $B$ is dense in $\mathbb{K}^n$.
In this article we consider the action of affine group and time rescaling on planar quadratic differential systems. We construct a system of representatives of the orbits of systems with at least five invariant lines, including the line at infinity and including multiplicities. For each orbit we exhibit its configuration. We characterize in terms of algebraic invariants and comitants and also geometrically, using divisors of the complex projective plane, the class of quadratic differential systems with at least five invariant lines. These conditions are such that no matter how a system may be …
A note on the bounded orbit conjecture
本文证明：平面上无不动点、反定向且所有轨道有界的同胚必有无穷多个周期轨道，从而为有界轨道猜想提供了一个注记。
We study the topological dynamics of the horocycle flow $h_\mathbb{R}$ on a geometrically infinite hyperbolic surface S. Let u be a non-periodic vector for $h_\mathbb{R}$ in T^1 S. Suppose that the half-geodesic $u(\mathbb{R}^+)$ is almost minimizing and that the injectivity radius along $u(\mathbb{R}^+)$ has a finite inferior limit $Inj(u(\mathbb{R}^+))$. We prove that the closure of $h_\mathbb{R} u$ meets the geodesic orbit along un unbounded sequence of points $g_{t_n} u$. Moreover, if $Inj(u(\mathbb{R}^+)) = 0$, the whole half-orbit $g_{\mathbb{R}^+} u$ is contained in $h_\mathbb{R} u$. Wh…
Uniformly distributed orbits in $\mathbb{T}^d$ and singular substitution dynamical systems
本研究推广了代换动力系统谱奇异性的已有充分条件，通过研究环面自同态轨道的分布，得到相关轨道均匀分布的充要条件，并给出奇异代换作用的新例子。
In this note, we consider the orbits $\{pu(n^{1+γ})|n\in\mathbb N\}$ in $Γ\backslash\text{PSL}(2,\mathbb R)$, where $Γ$ is a non-uniform lattice in $\text{PSL}(2,\mathbb R)$ and $u(t)$ is the standard unipotent group in $\text{PSL}(2,\mathbb R)$. Under a Diophantine condition on the intial point $p$, we can prove that $\{pu(n^{1+γ})|n\in\mathbb N\}$ is equidistributed in $Γ\backslash\text{PSL}(2,\mathbb R)$ for small $γ>0$, which generalizes a result of Venkatesh (Ann.of Math. 2010). We will compute Hausdorff dimensions of subsets of non-Diophantine points in Appendix A, using results of latti…
We study periodic orbits on a nondegenerate dynamically convex starshaped hypersurface in $\mathbb{R}^{2n}$ along the lines of Long and Zhu, but using properties of the $S^1$-equivariant symplectic homology. We prove that there exist at least $n$ distinct simple periodic orbits on any nondegenerate starshaped hypersurface in $\mathbb{R}^{2n}$ satisfying the condition that the minimal Conley-Zehnder index is at least $n-1$. The condition is weaker than dynamical convexity.
Topological rigidity of closures of certain sparse unipotent orbits in finite-volume quotients of $\prod_{i=1}^k\operatorname{SL}_2(\mathbb R)$
本文提供了一个简单的证明，阐述了在 $G=\prod_{i=1}^k\operatorname{SL}_2(\mathbb R)$ 且 $Γ$ 为不可约格时，$G/Γ$ 中特定稀疏幂零轨道闭包的拓扑刚性。
We study an action of ${\rm Aut}(F_n)$ on $\mathbb{R}^{2^n-1}$ by trace maps, defined using the traces of $n$-tuples of matrices in $\mathrm{SL}(2,\mathbb{C})$ having real traces. We determine the finite orbits for this action. These orbits essentially come from (i) the finite subgroups of $\mathrm{SL}(2,\mathbb C)$, and (ii) a dense set of (rational) points in an embedded quotient of an $n$-torus.
We show that, if $f\colon \mathbb{B}^q\to \mathbb{B}^q$ is a holomorphic self-map of the unit ball in $\mathbb{C}^q$ and $ζ\in \partial \mathbb{B}^q$ is a boundary repelling fixed point with dilation $λ>1$, then there exists a backward orbit converging to $ζ$ with step $\log λ$. Morever, any two backward orbits converging to the same boundary repelling fixed point stay at finite distance. As a consequence there exists a unique canonical pre-model $(\mathbb{B}^k,\ell, τ)$ associated with $ζ$ where $1\leq k\leq q$, $τ$ is a hyperbolic automorphism of $\mathbb{B}^k$, and whose image $\ell(\mathbb…
Let $A\subseteq SL_n(\mathbb{R})$ group of diagonal matrices with positive diagonal, let ${\rm ST}_n\subseteq X_n:=SL_n(\mathbb{R})/SL_n(\mathbb{Z})$ be the set of stable lattices, and let ${\rm WR}_n\subseteq X_n$ be the set of well-rounded lattices. We prove that any $A$-orbit in $X_n$ intersects both ${\rm ST}_n$ and ${\rm WR}_n$.
Classification of horocycle orbit closures in $ \mathbb{Z} $-covers
本研究完整分类紧双曲曲面$\mathbb{Z}$-覆叠中的 horocycle 轨道闭包，通过分析覆叠中距离最小化测地射线的效率，证明非极大轨道闭包虽呈分形特征却具有整数 Hausdorff 维数。
We obtain bounds on the numbers of intersections between triangulations as the conformal structure of a surface varies along a Teichm{ü}ller geodesic contained in an $\mathrm{SL}\left(2,\mathbb{R}\right)$-orbit closure of rank 1 in the moduli space of Abelian differentials. For $0 \leq θ\leq 1$, we obtain an exponential bound on the number of closed geodesics in the orbit closure, of length at most $R$, that spend at least $θ$-fraction of their length in a region with short saddle connections.
Bounded orbits for 3 bodies in $\mathbb{R}^4$
本文研究了四维牛顿 3 体问题，证明能量函数在无界状态序列上不能趋于下确界，从而得出能量下确界即为其最小值，完善了关于 $\mathbb{R}^4$ 中 Lyapunov 稳定相对周期轨道存在性的研究。
On free minimal constant speedups violating continuous orbit equivalence in $p$-adic $\mathbb{Z}^d$-odometers of adding type
本文研究加法型p进ℤ^d-odometer的自由极小常加速，发现其与原odometer可能不连续轨道等价，否定了Johnson和McClendon的猜想，指出d≥2时这类加速的连续轨道等价现象罕见。
In this paper we prove the following theorem. Let $f:\mathbb{A}^2\rightarrow \mathbb{A}^2$ be a dominate polynomial endomorphisms defined over an algebraically closed field $k$ of characteristic $0$. If there are no nonconstant rational function $g:\mathbb{A}^2-rightarrow \mathbb{P}^1$ satisfying $g\circ f=g$, then there exists a point $p\in \mathbb{A}^2(k)$ whose orbit under $f$ is Zariski dense in $\mathbb{A}^2$. This result gives us a positive answer to a conjecture of Amerik, Bogomolov and Rovinsky ( and Zhang) for polynomial endomorphisms on the affine plane.
In this article, we give a growth rate about the number of periodic orbits in the Franks type theorem obtained by the authors \cite{LWY}. As applications, we prove the following two results: there exist infinitely many distinct non-contractible closed geodesics on $\mathbb{R}P^2$ endowed with a Riemannian metric such that its Gaussian curvature is positive, moreover, the number of non-contractible closed geodesics of length $\leq l$ grows at least like $l^2$; and there exist either two or infinitely many distinct non-contractible closed geodesics on Finsler $\mathbb{R}P^2$ with reversibility $…
We consider a (mathbb{Z}_2)-equivariant flow in (mathbb{R}^{4}) with an integral of motion and a hyperbolic equilibrium with a transverse homoclinic orbit (Gamma). We provide criteria for the existence of stable and unstable invariant manifolds of (Gamma). We prove that if these manifolds intersect transversely, creating a so-called super-homoclinic, then in any neighborhood of this super-homoclinic there exist infinitely many multi-pulse homoclinic loops. An application to a system of coupled nonlinear Schrödinger equations is considered.
Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear two dimensional representations of the fundamental group of the surface. For surfaces of genus at least two, such orbits correspond to homomorphisms with finite image. For genus one, they correspond to the finite or special dihedral representations. We also obtain an analogous result for bounded orbits in the moduli space.
Free Borel $\mathbb{R}^{d}$-flows are smoothly equivalent if there is a Borel bijection between the phase spaces that maps orbits onto orbits and is a $C^{\infty}$-smooth orientation preserving diffeomorphism between orbits. We show that all free non-tame Borel $\mathbb{R}^{d}$-flows are smoothly equivalent in every dimension $d \ge 2$. This answers a question of B. Miller and C. Rosendal.
The local topological dynamics of subgroups of ${\rm Diff} ({\mathbb C^n}, 0)$, with special emphasis on ${\rm Diff} ({\mathbb C^2}, 0)$, is discussed with a view towards integrability questions. It is proved in particular that a subgroup of ${\rm Diff} ({\mathbb C^2}, 0)$ possessing locally finite orbits is necessarily solvable. Other results and examples related to higher-dimensional generalizations of Mattei-Moussu's celebrated topological characterization of integrability are also provided. These examples also settle a fundamental question raised by the previous work of Camara-Scardua.
Two irrationally elliptic closed orbits of Reeb flows on the boundary of star-shaped domain in $\mathbb{R}^{2n}$
本文证明在 $\mathbb{R}^{2n}$ 中星形域边界上具有恰好 $n$ 个素闭轨道的动态凸 Reeb 流中，至少两个轨道是无理椭圆，支持了相关猜想。
Given an integer $κ\geq2$, we introduce a class of nearly integrable systems on $\mathbb{A}^3$, of the form $$ H_n(θ,r)=\frac12 \Vert r\Vert ^2+\tfrac{1}{n} U(θ_2,θ_3)+f_n(θ,r) $$ where $U\in C^κ(\mathbb{T}^2)$ is a generic potential function and $f_n$ a $C^{κ-1}$ additional perturbation such that $\Vert f_n\Vert_{C^{κ-1}(\mathbb{A}^3)}\leq \tfrac{1}{n}$, so that $H_n$ is a perturbation of the completely integrable system $h(r)=\frac12\Vert r\Vert ^2$. Let $Π:\mathbb{A}^3\to\mathbb{R}^3$ be the canonical projection. We prove that for each $δ>0$, there exists $n_0$ such that for $n\geq n_0$, th…
Reading analytic invariants of parabolic diffeomorphisms from their orbits
本文研究复平面抛物微分同胚芽的轨道反演问题，通过构造动态theta函数与分形theta函数，证明其可编码解析不变量，实现从单条轨道读取微分同胚的解析性质。
We show that the bilateral backward shift on $\ell^p(\mathbb{Z},ω)$ that has a projective orbit with a non-zero limit point is supercyclic. This phenomenon holds also for $Γ$-supercyclicity, which extends a result obtained for the first time by Chan and Seceleanu. Moreover, we show that if $K$ is a compact subset of $\ell^p(\mathbb{N},ω)$ such that its orbit under the unilateral backward shift $B$ on $\ell^p(\mathbb{N},ω)$ has a non-zero weak limit point, then $B$ is hypercyclic.
We study the $\mathrm{SL}_2(\mathbb{R})$-action on the moduli space of (triangulable) dilation tori with one boundary component. We prove that every orbit is either closed or dense, and that every orbit of the Teichmuller flow escapes to infinity.
We study equivalence relations $\mathcal R(Γ\curvearrowright G)$ that arise from left translation actions of countable groups on their profinite completions. Under the assumption that the action $Γ\curvearrowright G$ is free and has spectral gap, we describe precisely when $\mathcal R(Γ\curvearrowright G)$ is orbit equivalent or Borel reducible to another such equivalence relation $\mathcal R(Λ\curvearrowright H)$. As a consequence, we provide explicit uncountable families of free ergodic probability measure preserving (p.m.p.) profinite actions of $SL_2(\mathbb Z)$ and its non-amenable subgro…
A family of Hurwitz polynomial vector fields in $\mathbb{R}^3$ having non-constant periodic orbits
构造 $\mathbb{R}^3$ 中双参数 Hurwitz 多项式向量场族，证明在参数开集上每个成员具有鞍型和吸引型两个非恒定周期轨道，回答 Gasull 第 19 个问题并提供反例。
Linked orbits of homeomorphisms of the plane and Gambaudo-Kolev Theorem
本文证明对任意保定向平面同胚，其任一有界轨道都与某个不动点在Gambaudo意义下链接，从而支持Gambaudo-Kolev定理所描述的轨道与不动点之间的拓扑纠缠关系。
Minimizing laminations in regular covers, horospherical orbit closures, and circle-valued Lipschitz maps
本文揭示了紧双曲流形的$\mathbb{Z}$-覆盖中距离极小叶状结构与球面轨道闭包的联系，构造了具有指定性质的$\mathbb{Z}$-覆盖并明确描述了所有horocycle轨道闭包，同时证明双曲度量的微小扰动会导致其拓扑发生剧烈变化。
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