G. Everest, V. Stangoe, T. Ward
Analogues of the prime number theorem and Merten's theorem are well-known for dynamical systems with hyperbolic behaviour. In this paper a 3-adic extension of the circle doubling map is studied. The map has a 3-adic eigendirection in which it behaves like an isometry, and the loss of hyperbolicity leads to weaker asymptotic results on orbit counting than those obtained for hyperbolic maps.
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Isometric orbit equivalence for probability-measure preserving actions
本文通过图化研究有限生成群的保测作用，引入等距轨道等价概念，证明了其对特定群具有刚性，并构造了自由群的非平凡示例，表明混合性在该等价下非不变量。
We study the functorial and growth properties of closed orbits for maps. By viewing an arbitrary sequence as the orbit-counting function for a map, iterates and Cartesian products of maps define new transformations between integer sequences. An orbit monoid is associated to any integer sequence, giving a dynamical interpretation of the Euler transform.
Counting functions over periodic orbits of a skew-product map
本文研究与有限生成有理半群相关的斜积映射，并证明沿其周期轨道计算的遍历和所对应计数函数之间的可比较性结果。
There are well-known analogs of the prime number theorem and Mertens' theorem for dynamical systems with hyperbolic behaviour. Here we consider the same question for the simplest non-hyperbolic algebraic systems. The asymptotic behaviour of the orbit-counting function is governed by a rotation on an associated compact group, and in simple examples we exhibit uncountably many different asymptotic growth rates for the orbit-counting function. Mertens' Theorem also holds in this setting, with an explicit rational leading coefficient obtained from arithmetic properties of the non-hyperbolic eigend…
Counting points by height in semigroup orbits
针对极化动力系统半群轨道中高度有界点的计数问题，本文利用 Wiener-Ikerhara Tauberian 定理改进已有估计，并得到射影直线上一般半群的精确渐近式。
Given a set of endomorphisms on $\mathbb{P}^N$, we establish an upper bound on the number of points of bounded height in the associated monoid orbits. Moreover, we give a more refined estimate with an associated lower bound when the monoid is free. Finally, we show that most sets of rational functions in one variable satisfy these more refined bounds.
We study a parametric family of piecewise rotations of the torus, in the limit in which the rotation number approaches the rational value 1/4. There is a region of positive measure where the discontinuity set becomes dense in the limit; we prove that in this region the area occupied by stable periodic orbits remains positive. The main device is the construction of an induced map on a domain with vanishing measure; this map is the product of two involutions, and each involution preserves all its atoms. Dynamically, the composition of these involutions represents linking together two sector maps…
In this paper, we study the distribution of integral points on parametric families of affine homogeneous varieties. By the work of Borel and Harish-Chandra, the set of integral points on each such variety consists of finitely many orbits of arithmetic groups, and we establish an asymptotic formula (on average) for the number of the orbits indexed by their Siegel weights. In particular, we deduce asymptotic formulas for the number of inequivalent integral representations by decomposable forms and by norm forms in division algebras, and for the weighted number of equivalence classes of integral …
Given a discrete group $Γ$ of isometries of a negatively curved manifold $\widetilde M$, a nontrivial conjugacy class $\mathfrak K$ in $Γ$ and $x_0\in\widetilde M$, we give asymptotic counting results, as $t\to +\infty$, on the number of orbit points $γx_0$ at distance at most $t$ from $x_0$, when $γ$ is restricted to be in $\mathfrak K$, as well as related equidistribution results. These results generalise and extend work of Huber on cocompact hyperbolic lattices in dimension $2$. We also study the growth of given conjugacy classes in finitely generated groups endowed with a word metric.
Effective counting for discrete lattice orbits in the plane via Eisenstein series
该研究利用Eisenstein级数证明了SL(2,R)非均匀格点平面轨道二次增长的有效误差界，改进了扇形与星形域的计数结果，可用于部分Veech曲面鞍连接和乐计数。
We study the asymptotic behaviour of the orbit-counting function and a dynamical Mertens' theorem for the full $G$-shift for a finitely-generated torsion-free nilpotent group $G$. Using bounds for the M{ö}bius function on the lattice of subgroups of finite index and known subgroup growth estimates, we find a single asymptotic of the shape \[ \sum_{|τ|\le N}\frac{1}{e^{h|τ|}}\sim CN^α(\log N)^β \] where $|τ|$ is the cardinality of the finite orbit $τ$. For the usual orbit-counting function we find upper and lower bounds together with numerical evidence to suggest that for actions of non-cyclic …
We realize the Apollonian group associated to an integral Apollonian circle packings, and some of its generalizations, as a group of automorphisms of an algebraic surface. Borrowing some results in the theory of orbit counting, we study the asymptotic of the growth of degrees of elements in the orbit of a curve on an algebraic surface with respect to a geometrically finite group of its automorphisms.
We address the problem of counting periodic orbits of vector fields on smooth closed manifolds. The space of non-constant periodic orbits is enlarged to a complete space by adding the ghost orbits, which are decorations of the zeros of vector fields. Associated with any compact and open subset $Γ$ of the moduli space of periodic and ghost orbits, we define an integer weight. When the vector field moves along a path, and $Γ$ deforms in a compact and open family, we show that the weight function stays constant. We also give a number of examples and computations, which illustrate the applications…
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.
Counting prime orbits in shrinking intervals for expanding Thurston maps
本文研究了扩张图映射中原始周期轨道在收缩区间内的计数问题，通过局部中心极限定理和渐近估计，揭示了非均匀扩展动力学下的细粒度行为。
For a geometrically finite group Gamma of G=SO(n,1), we survey recent developments on counting and equidistribution problems for orbits of Gamma in a homogeneous space H\G where H is trivial, symmetric or horospherical. Main applications are found in an affine sieve on orbits of thin groups as well as in sphere counting problems for sphere packings invariant under a geometrically finite group. In our sphere counting problems, spheres can be ordered with respect to a general conformal metric.
Counting periodic orbits on fractals weighted by their Lyapunov exponents
本文为分形上的矩阵值Gibbs测度定义压力并证明其最大化性质，进而利用该测度按Lyapunov指数加权计数分形上的周期轨道。
Let $X$ be a proper, geodesically complete Hadamard space, and $\ Γ<\mbox{Is}(X)$ a discrete subgroup of isometries of $X$ with the fixed point of a rank one isometry of $X$ in its infinite limit set. In this paper we prove that if $Γ$ has non-arithmetic length spectrum, then the Ricks' Bowen-Margulis measure -- which generalizes the well-known Bowen-Margulis measure in the CAT$(-1)$ setting -- is mixing. If in addition the Ricks' Bowen-Margulis measure is finite, then we also have equidistribution of $Γ$-orbit points in $X$, which in particular yields an asymptotic estimate for the orbit coun…
We prove the equidistribution of (weighted) periodic orbits of the geodesic ow on noncompact negatively curved manifolds toward equilibrium states in the narrow topology, i.e. in the dual of bounded continuous functions. We deduce an exact asymptotic counting for periodic orbits (weighted or not), which was previously known only for geometrically nite manifolds.
Mixing of the Mineyev flow, orbital counting and Poincaré series for strongly hyperbolic metrics
本研究结合遍历理论与符号动力学，为双曲群上的强双曲度量建立了轨道计数结果，并证明了Mineyev拓扑流的混合性及Poincaré级数的解析域。
The main result of this article is that if a $3$-manifold $M$ supports an Anosov flow, then the number of conjugacy classes in the fundamental group of $M$ grows exponentially fast with the length of the shortest orbit representative, hereby answering a question raised by Plante and Thurston in 1972. In fact we show that, when the flow is transitive, the exponential growth rate is exactly the topological entropy of the flow. We also show that taking only the shortest orbit representatives in each conjugacy classes still yields Bowen's version of the measure of maximal entropy. These results ar…
In this paper we study the action of the fundamental group of a finite metric graph on its universal covering tree. We assume the graph is finite, connected and the degree of each vertex is at least three. Further, we assume an irrationality condition on the edge lengths. We obtain an asymptotic for the number of elements in a fixed conjugacy class for which the associated displacement of a given base vertex in the universal covering tree is at most $T$. Under a mild extra assumption we also obtain a polynomial error term.
The automorphisms of a two-generator free group acting on the space of orientation-preserving isometric actions of on hyperbolic 3-space defines a dynamical system. Those actions which preserve a hyperbolic plane but not an orientation on that plane is an invariant subsystem, which reduces to an action on R^3 by polynomial automorphisms preserving the cubic polynomial and an area form on the level surfaces. We describe the dynamical decomposition of this action. The domain of discontinuity of this action corresponds to geometric structures: either complete hyperbolic structures on the 2-holed …
Displacement convexity of Invariant Measures and curvature bounds of isometric actions
该研究针对紧致李群的真等距作用，刻画了不变概率测度空间上内能泛函的位移凸性，建立了其与加权商空间Bakry–Émery曲率条件的等价关系，推广了von Renesse–Sturm定理。
Let G be the identity component of SO(n,1), acting linearly on a finite dimensional real vector space V. Consider a vector w_0 in V such that the stabilizer of w_0 is a symmetric subgroup of G or the stabilizer of the line Rw_0 is a parabolic subgroup of G. For any non-elementary discrete subgroup Gamma of G with w_0Gamma discrete, we compute an asymptotic formula for the number of points in w_0Gamma of norm at most T, provided that the Bowen-Margulis-Sullivan measure on the associated hyperbolic manifold and the Gamma skinning size of w_0 are finite. The main ergodic ingredient in our approac…
Let G:=SO(n,1)^\circ and Γbe a geometrically finite Zariski dense subgroup with critical exponent delta bigger than (n-1)/2. Under a spectral gap hypothesis on L^2(Γ\ G), which is always satisfied for delta>(n-1)/2 for n=2,3 and for delta>n-2 for n>= 4, we obtain an {\it effective} archimedean counting result for a discrete orbit of Γin a homogeneous space H\G, where H is the trivial group, an affine symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family {B_T} of compact subsets in H\G, there exists η>0 such that #[e]\G\cap B_T=M(B_…
Counting geodesics on prime-order $k$-differentials
本文研究了当 $k$ 为素数且亏格大于 $2$ 时 $k$-微分上测地线计数的弱渐近性，通过分类完整覆盖的 $GL^+(2,\mathbb{R})$-轨道闭包并应用推广的 Eskin-Mirzakhani-Mohammadi 定理，证明其轨道闭包通常为平移曲面层分支或超椭圆轨迹。
A short note on the orbit growth of sofic shifts
本文通过Artin-Mazur zeta函数及最小右解析表示，简短证明了sofic移位素轨道与Mertens轨道计数函数的渐近行为。
Counting orbits of certain infinitely generated non-sharp discontinuous groups for the anti-de Sitter space
本文构造了三维反德西特空间中无限生成的非锐利不连续群，给出其判别准则，并研究轨道计数增长的上、下界，进而证明此类流形上拉普拉斯算子可有无穷多个 L² 特征值，且轨道计数可超过任意给定的递增函数。
We consider asymptotic orbit-counting problems for certain expansive actions by commuting automorphisms of compact groups. A dichotomy is found between systems with asymptotically more periodic orbits than the topological entropy predicts, and those for which there is no excess of periodic orbits.
Given a connected, oriented, complete, finite area hyperbolic surface $X$ of genus $g$ with $n$ punctures, Mirzakhani showed that the number of multi-geodesics on $X$ of total hyperbolic length $\leq L$ in the mapping class group orbit of a given simple or filling closed multi-curve is asymptotic as $L \to \infty$ to a polynomial in $L$ of degree $6g-6+2n$. We establish asymptotics of the same kind for countings of multi-geodesics in mapping class group orbits of simple or filling closed multi-curves that keep track of the hyperbolic lengths of individual components, proving and generalizing a…
Anosov groups: local mixing, counting, and equidistribution
本文研究了连通半单实代数群中 Anosov 子群的矩阵系数渐近行为，引入了 Burger-Robin 测度的高秩类比，并应用于仿射对称空间中离散轨道的双扇区计数。
We prove a quantitative estimate with a power saving error term for the number of filling closed geodesics of a given topological type and length $\leq L$ on an arbitrary closed, orientable, negatively curved surface. More generally, we prove estimates of the same kind for the number of free homotopy classes of filling closed curves of a given topological type on a closed, orientable surface whose geometric intersection number with respect to a given filling geodesic current is $\leq L$. The proofs rely on recent progress made on the study of the effective dynamics of the mapping class group o…
We establish a precise asymptotic formula for the number of homotopy classes of periodic orbits for the geodesic flow on rank one manifolds of nonpositive curvature. This extends a celebrated result of G. A. Margulis to the nonuniformly hyperbolic case and strengthens previous results by G. Knieper. We also establish some useful properties of the measure of maximal entropy.
Let $Σ$ be a hyperbolic surface. We study the set of curves on $Σ$ of a given type, i.e. in the mapping class group orbit of some fixed but otherwise arbitrary $γ_0$. For example, in the particular case that $Σ$ is a once-punctured torus, we prove that the cardinality of the set of curves of type $γ_0$ and of at most length $L$ is asymptotic to $L^2$ times a constant.
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