Remarks on shrinking target properties

Two remarks on the Shrinking Target Property

We show that a translation of vector V on the torus has the monotone shrinking target property if and only if the vector V is of constant type. Then, using reparametrizations of linear flows, we show that there exist area preserving real analytic maps of the three torus that are mixing of all orders and do not enjoy the monotone shrinking target property.

自相似地毯上的矩形收缩目标

Rectangular Shrinking Targets on Self-Similar Carpets

本文研究自相似地毯上矩形收缩目标集的Hausdorff维数，发现其依赖于目标中心，并推广了Wang和Wu的定理。

Shrinking-Targets for Non-Autonomous Systems

In the present work we establish a Bowen-type formula for the Hausdorff dimension of shrinking-target sets for non-autonomous conformal iterated function systems in arbitrary dimensions and satisfying certain conditions. In the case of dimension 1 we also investigate non-linear perturbations of linear systems and obtain sufficient conditions under which the perturbed systems satisfy the conditions in our hypotheses

Shrinking targets problems for flows on homogeneous spaces

We study shrinking targets problems for discrete time flows on a homogenous space $Γ\backslash G$ with $G$ a semisimple group and $Γ$ an irreducible lattice. Our results apply to both diagonalizable and unipotent flows, and apply to very general families of shrinking targets. As a special case, we establish logarithm laws for cusp excursions of unipotent flows answering a question of Athreya and Margulis.

A shrinking-target problem in the space of unimodular lattices in the three dimensional Euclidean space

In this paper, we study the shrinking-target problem with target at infinity induced by the injectivity radius function under the action of a regular diagonalizable flow on $\operatorname{SL}_3(\mathbb R)/\operatorname{SL}_3(\mathbb Z)$. In particular, we establish an explicit formula for the Hausdorff dimension of the subset of points $p$ whose orbit approaches the cusp infinitely often with a rate $γ\geq0$.

Discretely shrinking targets in moduli space

We consider the discrete shrinking target problem for Teichmüller geodesic flow on the moduli space of abelian or quadratic differentials and prove that the discrete geodesic trajectory of almost every differential will hit a shrinking family of targets infinitely often provided the measures of the targets are not summable. This result applies to any ergodic $\mathrm{SL}(2,\mathbb{R})$--invariant measure and any nested family of spherical targets. Under stronger conditions on the targets, we moreover prove that almost every differential will eventually always hit the targets. As an application…

A Shrinking Target Problem with Target at Infinity in Rank One Homogeneous Spaces

In this paper, we give a definition of Diophantine points of type $γ$ for $γ\geq0$ in a homogeneous space $G/Γ$, and compute the Hausdorff dimension of the subset of points which are not Diophantine of type $γ$ when $G$ is a semisimple Lie group of real rank one. We also deduce a Jarník-Besicovitch Theorem on Diophantine approximation in Heisenberg groups.

Przytycki-Urbański 分形上收缩目标集的豪斯多夫维数

Hausdorff dimension of shrinking targets on Przytycki-Urbański fractals

本研究针对仿射迭代函数系统，计算点反复进入几何球收缩目标集的豪斯多夫维数，通过分析 Przytycki-Urbański 系统并扩展伯努利卷积理论，克服了仿射映射带来的几何-动力复杂性。

Shrinking targets in parametrised families

We consider certain parametrised families of piecewise expanding maps on the interval, and estimate and sometimes calculate the Hausdorff dimension of the set of parameters for which the orbit of a fixed point has a certain shrinking target property. This generalises several similar results for $β$-transformations to more general non-linear families. The proofs are based on a result by Schnellmann on typicality in parametrised families.

Shrinking targets in fast mixing flows and the geodesic flow on negatively curved manifolds

We show that in a rapidly mixing flow with an invariant measure, the time which is needed to hit a given section is related to a sort of conditional dimension of the measure at the section. The result is applied to the geodesic flow of compact variable negative sectional curvature manifolds, establishing a logarithm law for such kind of flow.

Skew products, quantitative recurrence, shrinking targets and decay of correlations

We consider toral extensions of hyperbolic dynamical systems. We prove that its quantitative recurrence (also with respect to given observables) and hitting time scale behavior depend on the arithmetical properties of the extension. By this we show that those systems have a polynomial decay of correlations with respect to $C^{r}$ observables, and give estimations for its exponent, which depend on $r$ and on the arithmetical properties of the system. We also show examples of systems of this kind having not the shrinking target property, and having a trivial limit distribution of return time sta…

收缩目标与递归：简要综述

Shrinking Targets versus Recurrence: a brief survey

本文综述了保测动力系统中收缩目标集与递归集在混合条件下的满测度结果，探讨了满测度命题的定量强化及理论关键差异。

Shrinking Targets for Countable Markov Maps

Let $T$ be an expanding Markov map with a countable number of inverse branches and a repeller $Λ$ contained within the unit interval. Given $α\in \R_+$ we consider the set of points $x \in Λ$ for which $T^n(x)$ hits a shrinking ball of radius $e^{-nα}$ around $y$ for infinitely many iterates $n$. Let $s(α)$ denote the infimal value of $s$ for which the pressure of the potential $-s\log|T'|$ is below $s α$. Building on previous work of Hill, Velani and Urbański we show that for all points $y$ contained within the limit set of the associated iterated function system the Hausdorff dimension of th…

On shrinking targets and self-returning points

We consider the set $\mathcal{R}_\mathrm{io}$ of points returning infinitely many times to a sequence of shrinking targets around themselves. Under additional assumptions we improve Boshernitzan's pioneering result on the speed of recurrence. In the case of the doubling map as well as some linear maps on the $d$ dimensional torus, we even obtain a dichotomy condition for $\mathcal{R}_\mathrm{io}$ to have measure zero or one. Moreover, we study the set of points eventually always returning and prove an analogue of Boshernitzan's result in similar generality.

On Shrinking Targets for Piecewise Expanding Interval Maps

For a map $T \colon [0,1] \to [0,1]$ with an invariant measure $μ$, we study, for a $μ$-typical $x$, the set of points $y$ such that the inequality $|T^n x - y| < r_n$ is satisfied for infinitely many $n$. We give a formula for the Hausdorff dimension of this set, under the assumption that $T$ is piecewise expanding and $μ_ϕ$ is a Gibbs measure. In some cases we also show that the set has a large intersection property.

收缩目标与回归：豪斯多夫维数理论综述

Shrinking targets versus recurrence: A survey on the Hausdorff dimension theory

该综述对比动力系统中收缩目标集与回归集的豪斯多夫维数，涉及多类动力系统，还涵盖相关测度结果及坏逼近轨道等内容。

Shrinking Targets for Semisimple Groups

We study the shrinking target problem for actions of semisimple groups on homogeneous spaces, with applications to logarithm laws and Diophantine approximation.

Quantitative shrinking target properties for rotations and interval exchanges

This paper presents quantitative shrinking target results for rotations and interval exchange transformations. To do this a quantitative version of a unique ergodicity criterion of Boshernitzan is established.

A quantitative shrinking target result on Sturmian sequences for rotations

Let $R_α$ be an irrational rotation of the circle, and code the orbit of any point $x$ by whether $R_α^i(x)$ belongs to $[0,α)$ or $[α,1)$ -- this produces a Sturmian sequence. A point is undetermined at step $j$ if its coding up to time $j$ does not determine its coding at time $j+1$. We prove a pair of results on the asymptotic frequency of a point being undetermined, for full measure sets of $α$ and $x$.

方砖铺砌曲面上的收缩目标

Shrinking targets on square-tiled surfaces

本文研究方砖铺砌曲面上的收缩目标问题，证明正则方砖铺砌曲面的Veech群子群具有丢番图性质，推广了Finkelshtein在平坦环面上的结果。

收缩目标集的拓扑熵与 Hausdorff 维数

Topological entropy and Hausdorff dimension of shrinking target sets

本文针对具有指数 specification 性质和 Lipschitz 连续性的动力系统，建立收缩目标集拓扑熵与 Hausdorff 维数的上下界，并在环面双曲、扩张及若干符号系统中证明二者重合。

On circle rotations and the shrinking target properties

We generalize the monotone shrinking target property (MSTP) to the s-exponent monotone shrinking target property (sMSTP) and give a necessary and sufficient condition for a circle rotation to have sMSTP. Using another variant of MSTP, we obtain a new, very short, proof of a known result, which concerns the behavior of irrational rotations and implies a logarithm law similar to D. Sullivan's logarithm law for geodesics.

Logarithm laws and shrinking target properties

We survey some of the recent developments in the study of logarithm laws and shrinking target properties for various families of dynamical systems. We discuss connections to geometry, diophantine approximation, and probability theory.

On the refined shrinking target property of rotations

We discuss the shrinking target property of irrational rotations. We obtain the condition of an irrational $θ$ and monotone increasing $φ(n)$ such that $$ \liminf_{n \to \infty} n φ(n) \| nθ- s \| = 0 \text{ for almost every } s. $$ We also consider the class of irrationals for which the limit inferior is 0 for every monotone increasing $φ(n)$ such that $\sum_n 1/(nφ(n))$ diverges.

Recurrence to shrinking targets on typical self-affine fractals

We explore the problem of finding the Hausdorff dimension of the set of points that recur to shrinking targets on a self-affine fractal. To be exact, we study the dimension of a certain related symbolic recurrence set. In many cases this set is equivalent to the recurring set on the fractal.

Shrinking targets for IETs: Extending a theorem of Kurzweil

This paper proves shrinking target results for IETs. Let {a_1\geq a_2 \geq...} be a sequence of positive real numbers with divergent sum. Then for almost every IET T, the limsup of B(T^ix,a_i) has full Lebesgue measure (where B(z, e) is the open ball around z of radius e). Related results are established including the analogous result for geodesic flows on a translation surface.

环面上$\mathbb{Z}^m$作用的矩形收缩目标：良好与不良逼近系统

Rectangular shrinking targets for $\mathbb{Z}^m$ actions on tori: well and badly approximable systems

本文研究了无理旋转的收缩目标性质，采用新方法证明了其加权有效类比，推广了Kim和Shapira的结果，并在更广泛的$S$-arithmetic设定下成立。

A shrinking target theorem for ergodic transformations of the unit interval

We show that for any ergodic Lebesgue measure preserving transformation $f: [0,1) \rightarrow [0,1)$ and any decreasing sequence $\{b_i\}_{i=1}^{\infty}$ of positive real numbers with divergent sum, the set $$\underset{n=1}{\overset{\infty}{\cap}} \, \underset{i=n}{\overset{\infty}{\cup}}\, f^{-i}(B (R_α^{i} x,b_i))$$ has full Lebesgue measure for almost every $x \in [0,1)$ and almost every $α\in [0,1)$. Here $B(x,r)$ is the ball of radius $r$ centered at $x \in [0,1)$ and $R_α: [0,1) \rightarrow [0,1)$ is rotation by $α\in [0,1)$. As a corollary, we provide partial answer to a question asked …

On the Hausdorff measure of shrinking target sets on self-conformal sets

In this article, we study the Hausdorff measure of shrinking target sets on self-conformal sets. The Hausdorff dimension of the sets we are interested in here was established by Hill and Velani in 1995. However, until recently, little more was known about the Hausdorff measure of these particular sets. In this paper we provide a complete characterisation of the Hausdorff measure of these sets, obtaining a dichotomy for the Hausdorff measure which is determined by the convergence or divergence of a sum depending on the radii of our "shrinking targets". Our main result complements earlier work o…

Shrinking targets for non-autonomous dynamical systems corresponding to Cantor series expansions

We provide a closed formula of Bowen type for the Hausdorff dimension of a very general shrinking target scheme generated by the non-autonomous dynamical system on the interval $[0,1)$, viewed as $\mathbb{R}/\mathbb{Z}$, corresponding to a given method of Cantor series expansion. We also examine a wide class of examples utilizing our theorem. In particular, we provide a Diophantine approximation interpretation of our scheme.

Shrinking targets and eventually always hitting points for interval maps

We study shrinking target problems and the set $\mathcal{E}_{\text{ah}}$ of eventually always hitting points. These are the points whose first $n$ iterates will never have empty intersection with the $n$-th target for sufficiently large $n$. We derive necessary and sufficient conditions on the shrinking rate of the targets for $\mathcal{E}_{\text{ah}}$ to be of full or zero measure especially for some interval maps including the doubling map, some quadratic maps and the Manneville-Pomeau map. We also obtain results for the Gauss map and correspondingly for the maximal digits in continued fract…

关于Dirichlet不可改进数与收缩目标问题

On Dirichlet non-improvable numbers and shrinking target problems

本文在收缩目标框架下推广Dirichlet不可改进集，研究连分数动力系统中满足特定不等式无穷多次的点的Hausdorff维数，将不可改进性与部分商乘积增长相联系。

β-动力系统中收缩目标集的Hausdorff测度二分律

Dichotomy laws for the Hausdorff measure of shrinking target sets in $β$-dynamical systems

本文研究β-动力系统中收缩目标集的Hausdorff测度，证明其测度为零或满取决于特定级数的收敛与发散，从而给出完备的测度论刻画。

Higher dimensional shrinking target problem in beta dynamical systems

We consider the two dimensional shrinking target problem in the beta dynamical system for general $β>1$ and with the general error of approximations. Let $f, g$ be two positive continuous functions. For any $x_0,y_0\in[0,1]$, define the shrinking target set $$ E(T_β, f,g):=\left\{(x,y)\in [0,1]^2: \begin{array}{ll} |T_β^{n}x-x_{0}|<e^{-S_nf(x)}\\ [1ex] |T_β^{n}y-y_{0}|< e^{-S_ng(y)} \end{array} \ {\text{for infinitely many}} \ n\in \N \right\}, $$ where $S_nf(x)=\sum_{j=0}^{n-1}f(T_β^jx)$ is the Birkhoff sum. We calculate the Hausdorff dimension of this set and prove that it is the solution to…

重叠迭代函数系统的定量回复性与收缩目标问题

Quantitative recurrence and the shrinking target problem for overlapping iterated function systems

本文研究重叠迭代函数系统动力学的定量回复性与收缩目标问题，证明关于勒贝格测度的几乎必然结果，并在测度为零时给出豪斯多夫维数界，且方法可处理非球形奇异几何目标。

一般仿射迭代函数系统中的路径相关收缩目标

Path-dependent shrinking targets in generic affine iterated function systems

该研究针对矩阵范数≤1/2的一般仿射迭代函数系统，在仅施加完全可约性的温和条件下，计算了路径相关收缩目标集的豪斯多夫维数，其维数公式为某压力函数的零点。

Central limit theorems for the shrinking target problem

Suppose $B_i:= B(p,r_i)$ are nested balls of radius $r_i$ about a point $p$ in a dynamical system $(T,X,μ)$. The question of whether $T^i x\in B_i$ infinitely often (i. o.) for $μ$ a.e.\ $x$ is often called the shrinking target problem. In many dynamical settings it has been shown that if $E_n:=\sum_{i=1}^n μ(B_i)$ diverges then there is a quantitative rate of entry and $\lim_{n\to \infty} \frac{1}{E_n} \sum_{j=1}^{n} 1_{B_i} (T^i x) \to 1$ for $μ$ a.e. $x\in X$. This is a self-norming type of strong law of large numbers. We establish self-norming central limit theorems (CLT) of the form $\lim…

An inhomogeneous Dirichlet theorem via shrinking targets

We give an integrability criterion on a real-valued non-increasing function $ψ$ guaranteeing that for almost all (or almost no) pairs $(A, \textbf{b})$, where $A$ is a real $m\times n$ matrix and $\textbf{b} \in \mathbb{R}^m$, the system $\|A \textbf{q}+\textbf{b}-\textbf{p}\|^m< ψ({T})$, $\|\textbf{q}\|^n<{T}$ is solvable in $\textbf{p} \in \mathbb{Z}^m$, $\textbf{q} \in \mathbb{Z}^n$ for all sufficiently large $T$. The proof consists of a reduction to a shrinking target problem on the space of grids in $\mathbb{R}^{m+n}$. We also comment on the homogeneous counterpart to this problem, whose …

非自治系统的收缩目标与复发问题

The shrinking target and recurrence problem for non-autonomous systems

本文研究了非自治系统中的收缩目标和复发问题，建立了零一准则，并通过统一框架分析了自治与非自治动态系统的行为。

快速混合系统中最终始终击中各点的零一律

Zero-one laws for eventually always hitting points in in rapidly mixing systems

该研究聚焦收缩目标系统中的最终始终击中点集合，针对伯努利概型、高斯映射这类目标平移近乎完全独立的系统，给出目标收缩速率的严格条件，判定该集合为零集或余零集。

仿射映射的平移表面：缩小目标与迪奥菲antine性质

Affine mappings of translation surfaces: shrinking targets and Diophantine properties

该研究证明了平移表面的仿射同胚群轨道可以迪奥菲antine精度逼近表面点，并利用谱间隙理论和均值遍历定理进行证明。

Shrinking target equidistribution of horocycles in cusps

Consider a shrinking neighborhood of a cusp of the unit tangent bundle of a noncompact hyperbolic surface of finite area, and let the neighborhood shrink into the cusp at a rate of $T^{-1}$ as $T \rightarrow \infty$. We show that a closed horocycle whose length $\ell$ goes to infinity or even a segment of that horocycle becomes equidistributed on the shrinking neighborhood when normalized by the rate $T^{-1}$ provided that $T/\ell \rightarrow 0$ and, for any $δ>0$, the segment remains larger than $\max\left\{T^{-1/6},\left(T/\ell\right)^{1/2}\right\}\left(T/\ell\right)^{-δ}$. We also have an e…

庞加莱回归与收缩目标问题的极限定律

Limit Laws for Poincaré Recurrence and the Shrinking Target Problem

针对满足指数多重去相关和短返回条件的保测系统，建立了庞加莱回归的分布律，并验证了公理A系统的强不变原理。

On shrinking targets for Z^m actions on tori

Let A be an n by m matrix with real entries. Consider the set Bad_A of x \in [0,1)^n for which there exists a constant c(x)>0 such that for any q \in Z^m the distance between x and the point {Aq} is at least c(x) |q|^{-m/n}. It is shown that the intersection of Bad_A with any suitably regular fractal set is of maximal Hausdorff dimension. The linear form systems investigated in this paper are natural extensions of irrational rotations of the circle. Even in the latter one-dimensional case, the results obtained are new.

Inhomogeneous potentials, Hausdorff dimension and shrinking targets

Generalising a construction of Falconer, we consider classes of $G_δ$-subsets of $\mathbb{R}^d$ with the property that sets belonging to the class have large Hausdorff dimension and the class is closed under countable intersections. We relate these classes to some inhomogeneous potentials and energies, thereby providing some useful tools to determine if a set belongs to one of the classes. As applications of this theory, we calculate, or at least estimate, the Hausdorff dimension of randomly generated limsup-sets, and sets that appear in the setting of shrinking targets in dynamical systems. F…

自共形集上最终总是命中点集的豪斯多夫维数

Hausdorff dimension of the set of eventually always hitting points on a self-conformal set

本文在自共形迭代函数系统框架下研究最终总是命中集与收缩目标集之交的豪斯多夫维数，给出其上界和下界，并得到两者交集的维数结果。

环面矩阵变换的修正收缩目标问题

Modified shrinking target problem for Matrix Transformations of Tori

本文研究环面矩阵变换的修正收缩目标问题，计算乘积型逼近集及以 Lipschitz 移动超矩形为目标的分形集的 Hausdorff 维数，统一收缩目标与定量回归性质。

收缩目标与递推：定量理论

Shrinking targets versus recurrence: the quantitative theory

该研究针对[0,1]上的扩张分段线性映射，推导了递推计数函数的渐近公式并给出误差项，同时将结果推广至高维，丰富了动力系统中收缩目标与递推的定量理论。

通过平移 Farey 序列的收缩靶标 horosphere 等分布

Shrinking target horospherical equidistribution via translated Farey sequences

本文研究SL(d,Z)\SL(d,R)上对角流的收缩靶标horosphere等分布问题，通过重整化技术与推广的平移Farey序列等分布结果，将已有二维情形推广到一般d≥2及平移horosphere情形。

关于线性扩张与双曲环面自同态的收缩靶问题

On shrinking targets for linear expanding and hyperbolic toral endomorphisms

本文研究环面上由整数矩阵诱导的自映射的收缩靶问题，完整给出任意扩张矩阵下靶集 Hausdorff 维数，并对 2×2 双曲矩阵计算维数，同时用反例否定了一般维数下的自然公式。

Dynamical Borel-Cantelli Lemma for Recurrence under Lipschitz Twists

In the study of some dynamical systems the limsup set of a sequence of measurable sets is often of interest. The shrinking targets and recurrence are two of the most commonly studied problems that concern limsup sets. However, the zero-one laws for the shrinking targets and recurrence are usually treated separately and proved differently. In this paper, we introduce a generalized definition that can specialize into the shrinking targets and recurrence; our approach gives a unified proof of the zero-one laws for the two problems.

半单代数群齐性空间中的收缩目标问题

A shrinking target problem in homogeneous spaces of semisimple algebraic groups

本文利用表示论方法研究半单代数群齐性空间中目标在无穷远的收缩目标问题，定义 $ψ$-Diophantine 元素集并给出其补集的 Hausdorff 维数公式，同时关联旗簇与 Grassmann 簇上的有理逼近。

内函数前向复合的收缩目标与回复行为

Shrinking targets and recurrent behaviour for forward compositions of inner functions

本文将内函数迭代的庞加莱回复定理与行为二分法推广到非自治前向复合情形，给出收缩条件，运用Borel–Cantelli引理等工具证明结果，并验证条件的最优性。

β-动力系统中的收缩平行六面体目标

Shrinking parallelepiped targets in beta-dynamical systems

本文研究了β-动力系统中带旋转的收缩平行六面体目标，通过确定其最优覆盖来求得极限集的Hausdorff维数，并举例说明了超矩形旋转对该维数的影响。