Shrinking Target Problem Research Articles

Non-stationary Almost Sure Invariance Principle for Hyperbolic Systems with Singularities

We investigate a wide class of two-dimensional hyperbolic systems with singularities, and prove the almost sure invariance principle (ASIP) for the random process generated by sequences of dynamically H\"older observables. The observables could be unbounded, and the process may be non-stationary and need not have linearly growing variances. Our results apply to Anosov diffeomorphisms, Sinai dispersing billiards and their perturbations. The random processes under consideration are related to the fluctuation of Lyapunov exponents, the shrinking target problem, etc.

TWO-DIMENSIONAL SHRINKING TARGET PROBLEM IN BETA-DYNAMICAL SYSTEMS

In this paper, we investigate the two-dimensional shrinking target problem in beta-dynamical systems. Let $\unicode[STIX]{x1D6FD}>1$ be a real number and define the $\unicode[STIX]{x1D6FD}$-transformation on $[0,1]$ by $T_{\unicode[STIX]{x1D6FD}}:x\rightarrow \unicode[STIX]{x1D6FD}x\;\text{mod}\;1$. Let $\unicode[STIX]{x1D6F9}_{i}$ ($i=1,2$) be two positive functions on $\mathbb{N}$ such that $\unicode[STIX]{x1D6F9}_{i}\rightarrow 0$ when $n\rightarrow \infty$. We determine the Lebesgue measure and Hausdorff dimension for the $\limsup$ set $$\begin{eqnarray}W(T_{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6F9}_{1},\unicode[STIX]{x1D6F9}_{2})=\{(x,y)\in [0,1]^{2}:|T_{\unicode[STIX]{x1D6FD}}^{n}x-x_{0}|<\unicode[STIX]{x1D6F9}_{1}(n),|T_{\unicode[STIX]{x1D6FD}}^{n}y-y_{0}|<\unicode[STIX]{x1D6F9}_{2}(n)\text{ for infinitely many }n\in \mathbb{N}\}\end{eqnarray}$$ for any fixed $x_{0},y_{0}\in [0,1]$.

An application of lattice points counting to shrinking target problems

We apply lattice points counting results to solve a shrinking target problem in the setting of discrete time geodesic flows on hyperbolic manifolds of finite volume.

Dichotomy law for shrinking target problems in a nonautonomous dynamical system: Cantor series expansion

Dichotomy law for shrinking target problems in a nonautonomous dynamical system: Cantor series expansion

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 related to an effective version of the Oppenheim conjecture valid for almost all quadratic forms.

Almost sure invariance principle for sequential and non-stationary dynamical systems

We establish almost sure invariance principles, a strong form of approximation by Brownian motion, for non-stationary time-series arising as observations on dynamical systems. Our examples include observations on sequential expanding maps, perturbed dynamical systems, non-stationary sequences of functions on hyperbolic systems as well as applications to the shrinking target problem in expanding systems.

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 $(T,X,\mu)$. The question of whether $T^i x\in B_i$ infinitely often (i. o.) for $\mu$ a.e.\ $x$ is often called the shrinking target problem. In many dynamical settings it has been shown that if $E_n:=\sum_{i=1}^n \mu (B_i)$ 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 $\mu$ 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 $\lim_{n\to \infty} \frac{1}{a_n} \sum_{i=1}^{n} [1_{B_i} (T^i x)-\mu(B_i)] \to N(0,1)$ (in distribution) for a variety of hyperbolic and non-uniformly hyperbolic dynamical systems, the normalization constants are $a^2_n \sim E [\sum_{i=1}^n 1_{B_i} (T^i x)-\mu(B_i)]^2$. Dynamical systems to which our results apply include smooth expanding maps of the interval, Rychlik type maps, Gibbs-Markov maps, rational maps and, in higher dimensions, piecewise expanding maps. For such central limit theorems the main difficulty is to prove that the non-stationary variance has a limit in probability.

The shrinking target problem in the dynamical system of continued fractions

The shrinking target problem in the dynamical system of continued fractions

Shrinking target problems for beta-dynamical system

For any β > 1, let ([0, 1], Tβ) be the beta dynamical system. For a positive function ψ: ℕ → ℝ+ and a real number x0 ∈ [0, 1], we define \(\mathbb{D}\left( {T_\beta ,\psi ,x_0 } \right)\) the set of ψ-well approximable points by x0 as $$\left\{ {x \in \left[ {0,1} \right]:\left| {T_\beta ^n x - x_0 } \right| < \psi \left( n \right) for infinitely many n \in \mathbb{N}} \right\}.$$ In this note, by proving a structure lemma that any ball B(x, r) contains a regular cylinder of comparable length with r, we determine the Hausdorff dimension of the set \(\mathbb{D}\left( {T_\beta ,\psi ,x_0 } \right)\) completely for any β> 1 and any positive function ψ.

A note on Borel–Cantelli lemmas for non-uniformly hyperbolic dynamical systems

AbstractLet (Bi) be a sequence of measurable sets in a probability space (X,ℬ,μ) such that ∑ ∞n=1μ(Bi)=∞. The classical Borel–Cantelli lemma states that if the sets Bi are independent, then μ({x∈X:x∈Bi infinitely often})=1. Suppose (T,X,μ) is a dynamical system and (Bi) is a sequence of sets in X. We consider whether Tix∈Bi infinitely often for μ almost every x∈X and, if so, is there an asymptotic estimate on the rate of entry? If Tix∈Bi infinitely often for μ almost every x, we call the sequence (Bi) a Borel–Cantelli sequence. If the sets Bi :=B(p,ri) are nested balls of radius ri about a point p, then the question of whether Tix∈Bi infinitely often for μ almost every x is often called the shrinking target problem. We show, under certain assumptions on the measure μ, that for balls Bi if μ(Bi)≥i−γ, 0&lt;γ&lt;1, then a sufficiently high polynomial rate of decay of correlations for Lipschitz observables implies that the sequence is Borel–Cantelli. If μ(Bi)≥C1 /i, then exponential decay of correlations implies that the sequence is Borel–Cantelli. We give conditions in terms of return time statistics which quantify Borel–Cantelli results for sequences of balls such that μ(Bi)≥C/i. Corollaries of our results are that for planar dispersing billiards and Lozi maps, sequences of nested balls B(p,1/i) are Borel–Cantelli. We also give applications of these results to a variety of non-uniformly hyperbolic dynamical systems.

Lattice actions on the plane revisited

We study the action of a lattice Γ in the group G = SL(2, R) on the plane. We obtain a formula which simultaneously describes visits of an orbit Γu to either a fixed ball, or an expanding or contracting family of annuli. We also discuss the ‘shrinking target problem’. Our results are valid for an explicitly described set of initial points: all $${{\bf u} \in {\bf R}^2}$$ in the case of a cocompact lattice, and all u satisfying certain diophantine conditions in case $${\Gamma = {\rm SL}(2, \mathbb {Z})}$$ . The proofs combine the method of Ledrappier with effective equidistribution results for the horocycle flow on $${\Gamma {\backslash} G}$$ due to Burger, Strömbergsson, Forni and Flaminio.

The Shrinking Target Problem for Matrix Transformations of Tori

The Shrinking Target Problem for Matrix Transformations of Tori