Jon Chaika, David Constantine
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.
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Limiting distribution of visits of sereval rotations to shrinking intervals
我们发现，给定单位圆上的$n$归一化区间，$d$随机旋转到这些区间的访问次数具有联合极限分布，因为轨迹的长度趋于无穷大。如果$d$趋于无穷大，则不同区间内的点数变得渐近独立，除非出现算术障碍。这是J.Marklof的早期结果的推广。
On Shrinking Targets for Piecewise Expanding Interval Maps
对于具有不变测度$μ$的映射$T \colon [0,1] \to [0,1]$，研究了对于$μ$ -典型$x$，使得不等式$|T^n x - y| < r_n$对无穷多个$n$都满足的点集$y$.在$T$是分段扩张的和$μ_ϕ$是Gibbs测度的假设下，我们给出了这个集合的豪斯多夫维数的一个公式.在某些情况下，我们还表明，该集有一个大的交集属性。
Distribution of periods of closed trajectories in exponentially shrinking intervals
本文研究了双曲流$ϕ_t$的闭轨线$γ$的个数的渐近性，该双曲流的原始周期$T_γ$位于指数收缩区间$(x - e^{-δx}, x + e^{-δx}),\:δ> 0,\: x \to + \infty.$。我们的结果适用于具有非格屋顶函数$f$的符号模型的双曲动力系统，假设与$f$相关的Ruelle算子满足强谱估计。特别是，我们的分析工程开放台球系统和测地线流流形上的常负曲率。
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。
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…
Skew products, quantitative recurrence, shrinking targets and decay of correlations
我们考虑双曲型动力系统的环面扩张。我们证明了它的定量递归（也相对于给定的可观测量）和首中时标度行为依赖于算术性质的扩展。由此我们证明了这些系统关于$C^{r}$可观测量的相关性具有多项式衰减，并给出了其指数的估计，它依赖于$r$和系统的算术性质.我们还展示了这种系统的例子，没有收缩的目标属性，并具有平凡的极限分布的返回时间统计。
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 …
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 $x$.
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。
Let $\mathcal{M}$ be a geometrically finite hyperbolic manifold. We present a very general theorem on the shrinking target problem for the geodesic flow, using its exponential mixing. This includes a strengthening of Sullivan's logarithm law for the excursion rate of the geodesic flow. More generally, we prove logarithm laws for the first hitting time for shrinking cusp neighborhoods, shrinking tubular neighborhoods of a closed geodesic, and shrinking metric balls, as well as give quantitative estimates for the time a generic geodesic spends in such shrinking targets.
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.
We describe the shrinking target problem for random iterated function systems which semi-conjugate to a random subshifts of finite type. We get the Hausdorff dimension of the set based on shrinking target problems with given targets. The main idea is an extension of ubiquity theorem which plays an important role to get the lower bound of the dimension. Our method can be used to deal with the sets with respect to an more general targets and the sets based on the quantitative Poincaré recurrence properties.
Shrinking targets for non-autonomous dynamical systems corresponding to Cantor series expansions
给出了区间$[0,1)$上的非自治动力系统生成的一个非常一般的收缩目标格式的豪斯多夫维数的Bowen型闭公式，称为$\mathbb{R}/\mathbb{Z}$，对应于给定的Cantor级数展开方法.我们还研究了广泛的一类例子，利用我们的定理。特别是，我们提供了一个丢番图逼近解释我们的计划。
On the escape rate for intermittent maps with holes shrinking around the indifferent fixed point
本文研究了间歇映射中围绕惰性固定点收缩孔的逃逸率，通过转移算子分析了开放系统的渐进行为。
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。
Shrinking Targets for Countable Markov Maps
设$T$是一个逆分支数可数且单位区间内含有排斥子$Λ$的扩展Markov映射。给定$α\in \R_+$，我们考虑点集$x \in Λ$，$T^n(x)$在$y$周围击中半径为$e^{-nα}$的收缩球，无限次迭代$n$。令$s(α)$表示$s$的下限值，其中电势$-s\log|T'|$的压力低于$s α$。建立在以前的工作希尔，Velani和Urbaidski，我们表明，对于所有点$y$包含在极限集的相关迭代函数系统的Hausdorff维数的收缩目标集是由$s(α)$。此外，当使用$\barΛ=[0,1]$时，对于所有$y \in [0,1]$都是如此。然而，给定$β\in (0,1)$，我们提供了一个扩展马尔可夫映射$T$的例子，其具有Hausdorff维数$β$的排斥子$Λ$，具有点$y\in \barΛ$，使得对于所有$α\in \R_+$，收缩目标集的维数为零。
We prove an abstract result establishing that one can obtain the convergence of Rare Events Point Processes counting the number of orbital visits to a sequence of shrinking target sets from the convergence of corresponding point processes for some induced system and matching shadowing shrinking sets inside the base of the inducing scheme. We apply this result to prove a dichotomy for two classes of non-uniformly hyperbolic interval maps: Misiurewicz quadratic maps and doubly intermittent maps. The dichotomy holds in the sense that the shrinking target sets may accumulate in any individual poin…
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。
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。
Central limit theorems for the shrinking target problem
假设$B_i:= B(p,r_i)$是一个动力系统$(T,X,μ)$中关于点$p$的半径为$r_i$的嵌套球。问题是$T^i x\in B_i$是否无限频繁（即。o.）对于$μ$ a.e.\ $x$通常被称为缩小目标问题。在许多动力学设置中，已经表明，如果$E_n:=\sum_{i=1}^n μ(B_i)$发散，则存在定量的进入率和$μ$ a.e.的$\lim_{n\to \infty} \frac{1}{E_n} \sum_{j=1}^{n} 1_{B_i} (T^i x) \to 1$。$x\in X$。这是一个自赋范类型的强大数定律。对一类双曲和非一致双曲动力系统建立了形式为$\lim_{n\to \infty} \frac{1}{a_n} \sum_{i=1}^{n} [1_{B_i} (T^i x)-μ(B_i)] \to N(0,1)$（in distribution）的自赋范中心极限定理（CLT），其规范常数为$a^2_n \sim E [\sum_{i=1}^n 1_{B_i} (T^i x)-μ(B_i)]^2$ .动力系统，我们的研究结果适用于包括光滑的扩展映射的间隔，Rychlik型映射，吉布斯-马尔可夫映射，合理的地图，并在更高的维度，分段扩展的地图。对于这样的中心极限定理的主要困难是证明非平稳方差具有概率极限。
Hitting times of shrinking targets: Transversality and an ergodic theorem
本文研究收缩目标集的命中时间，证明其遍历平均点态收敛性及与任意固定集合的几何独立性，通过整数-分形维数刻画。
This article provides optimal constants for two quantitative recurrence problems. First of all for recurrence of maps of the interval [0,1] that preserve the Lebesgue measure and on the other hand Lagrange spectrum of interval exchange transformations. Both results are based on a non-conventional packing problem in the plane with respect to the "pseudo-norm" N(x,y) = sqrt(|xy|).
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。
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。
In this note we show that the transfer operator of a Rauzy-Veech-Zorich renormalization map acting on a space of quasi-Hölder functions is quasicompact and derive certain statistical recurrence properties for this map and its associated Teichmüller flow. We establish Borel-Cantelli lemmas, Extreme Value statistics and return time statistics for the map and flow. Previous results have established quasicompactness in Hölder or analytic function spaces, for example the work of M. Pollicott and T. Morita. The quasi-Hölder function space is particularly useful for investigating return time statisti…
We establish a dichotomy for the rate of the decay of the Cesàro averages of correlations of sufficiently regular functions for typical interval exchange transformations (IET) which are not rigid rotations (for which weak mixing had been previously established in the works of Katok-Stepin, Veech, and Avila-Forni). We show that the rate of decay is either logarithmic or polynomial, according to whether the IET is of rotation class (i.e., it can be obtained as the induced map of a rigid rotation) or not. In the latter case, we also establish that the spectral measures of Lipschitz functions have…
Sharp large deviations for some hyperbolic systems
对于某些模型，我们证明了一个关于区间以亚指数速度缩小的急剧大偏差原理，该模型涉及与公理A流的马尔可夫族相关的庞加莱映射，该公理A流限制在满足一些附加正则性假设的基本集合$Λ$上。
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。
We derive new variants of the quantitative Borel--Cantelli lemma and apply them to analysis of statistical properties for some dynamical systems. We consider intermittent maps of $(0,1]$ which have absolutely continuous invariant probability measures. In particular, we prove that every sequence of intervals with left endpoints uniformly separated from zero is the strong Borel--Cantelli sequence with respect to such map and invariant measure.
Diophantine properties of IETs and general systems: Quantitative proximality and connectivity
我们提出了一般系统的收缩目标的结果，重点是应用程序的间隔交换变换（$(J,T)$），$J=[0,1)$。特别地，我们证明了，如果IET $(J,T)$是遍历（相对于勒贝格测度$\lam$），那么等式\[ \liminf_{n\to\infty}\limits n |T^n(x)-y|=0 \tag{A1} \]对$\lam\ttimes\lam$ -a成立。a. $(x,y)\in J^2$。遍历性假设是必要的：结果并不延伸到所有的最小迭代。（A1）中的因子$n$是最佳的（e.例如，在一个实施例中，它不能被$n \ln(\ln(\ln n))$取代。另一方面，对于勒贝格几乎所有的3-INUS $(J,T)$，我们证明，对于所有的$\eps>0$ \[ \liminf_{n\to\infty}\limits n^\eps |T^n(x)-T^n(y)|= \infty,\quad \text{for Lebesgue a. a.} (x,y)\in J^2. \tag{A2} \]这应该是对比的平等$ \liminf_{n\to\infty}\limits |T^n(x)-T^n(y)|=0, $的a。a. $(x,y)\in J^2$，它成立，因为$(J^2, T\times T)$是遍历的（因为一般的3-整数$(J,T)$是弱混合的）。我们还证明了没有3-IET是强拓扑混合的。
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