Yeor Hafouta
We prove a local central limit theorem for "nonconventional" sums generated by some classes of sufficiently fast mixing sequences.
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Nonuniform expansion and diffusive noise imply random horseshoes and random Young towers
本文提出随机马蹄概念，证明非均匀扩张随机动力系统在扩散噪声下随机马蹄稠密且存在退火指数尾部的随机Young塔。
Martingale Methods for Maximal Large Deviations and Young Towers
本文利用鞅近似框架，为可逆动力系统推导定量最大大偏差估计，并应用于部分超bolic diffeomorphisms的杨结构及亚指数收缩扩张系统的大偏差估计。
In this article we will apply complex projective metrics to sequences of complex transfer operators generated by Young towers, countable shifts and other types of distance expanding maps (possibly time dependent) with countable degrees. We will derive a sequential Ruelle-Perron-Frobenius theorem for sequence of complex operators defined by such maps, and relying on it a variety of probabilistic limit theorems (finer than the CLT) for non-stationary processes follows.
We obtain quenched almost sure invariance principle (with convergence rate) for Random Young Tower. We apply our result to i.i.d perturbations of non-uniformly expanding maps. In particular, we answer one open question in \cite{BBM}.
We show that the direct product of maps with Young towers admits a Young tower whose return times decay at a rate which is bounded above by the slowest of the rates of decay of the return times of the component maps. An application of this result, together with other results in the literature, yields various statistical properties for the direct product of various classes of systems, including Lorenz-like maps, multimodal maps, piecewise $ C^2 $ interval maps with critical points and singularities, Hénon maps and partially hyperbolic systems.
We give geometric conditions that are necessary and sufficient for the existence of Sinai-Ruelle-Bowen (SRB) measures for $C^{1+α}$ surface diffeomorphisms, thus proving a version of the Viana conjecture. As part of our argument we give an original method for constructing first return Young towers, proving that every hyperbolic measure, and in particular every SRB measure, can be lifted to such a tower. This method relies on a new general result on hyperbolic branches and shadowing for pseudo-orbits in nonuniformly hyperbolic sets which is of independent interest.
The aim of this survey is twofold. First we show how the Markov tower construction is applicable for obtaining finer stochastic properties, like a local limit theorem of probability theory. Here the fundamental method is the study of the spectrum of the Fourier transform of the Perron--Frobenius operator. These ideas and results are applicable to all systems Young has been considering. Second, we survey the problem of recurrence of the planar Lorentz process. As an application of the results from the first part, we obtain a dynamical proof of recurrence for the finite horizon case. Here basica…
Random Young towers and quenched decay of correlations for predominantly expanding multimodal circle maps
本文研究主要扩张多模圆映射中随机动力系统的年轻塔结构和淬火相关性衰减特性。
Quenched invariance principle with a rate for random dynamical systems
该研究针对遍历系统驱动的随机Young塔，推导新的鞅-上边界分解以控制逼近鞅平方和，得到淬灭不变性原理的Wasserstein收敛速率，并将结果应用于三类随机动力系统。
In this paper we prove two results. First we show that dynamical systems with a $ϕ$-mixing measure have in the limit Poisson distributed return times almost everywhere. We use the Chen-Stein method to also obtain rates of convergence. Our theorem improves on previous results by allowing for infinite partitions and dropping the requirement that the invariant measure have finite entropy with respect to the given partition. As has been shown elsewhere, the limiting distribution at periodic points is not Poissonian (but compound Poissonian). Here we show that for all non-periodic points the return…
For Young systems, i.e. for hyperbolic systems without/with singularities satisfying Lai-Sang Young's axioms (which imply exponential decay of correlation and the CLT) a local CLT is proven. In fact, a unified version of the local CLT is found, covering among others the absolutely contionuous and the arithmetic cases. For the planar Lorentz process with a finite horizon this result implies a.) the local CLT and b.) the recurrence. For the latter case ($d=2$, finite horizon), combining the global CLT with abstract ergodic theoretic ideas, K. Schmidt, and J.-P. Conze, could already establish rec…
We show that dynamical systems with $ϕ$-mixing measures have local escape rates which are exponential with rate $1$ at non-periodic points and equal to the extremal index at periodic points. We apply this result to equilibrium states on subshifts of finite type, expanding interval maps, Gibbs states on conformal repellers and more generally to Young towers and by extension to all systems that can be modeled by a Young tower.
We prove a Berry-Esseen theorem, a local central limit theorem and (local) large and (global) moderate deviations principles for i.i.d. (uniformly) random non-uniformly expanding or hyperbolic maps with exponential first return times. Using existing results the problem is reduced to certain random (Young) tower extensions, which is the main focus of this paper. On the random towers we will obtain our results using contraction properties of random complex equivariant cones with respect to the complex Hilbert projective metric.
We consider invertible discrete-time dynamical systems having a hyperbolic product structure in some region of the phase space with infinitely many branches and variable recurrence time. We show that the decay of correlations of the SRB measure associated to that hyperbolic structure is related to the tail of the recurrence times. We also give sufficient conditions for the validity of the Central Limit Theorem. This generalizes previous results by Benedicks and Young.
We prove that, in Young towers with sufficiently small tails, the speed in the central limit theorem is O(1/\sqrt{n}), and the local limit theorem holds. This implies the same results for many non uniformly expanding dynamical systems, namely those for which a tower with sufficiently fast returns can be constructed.
We show that effective $2\ell$-multiple correlations imply quantitative $\ell$-multiple pointwise ergodic theorems. The result has a wide class of applications which include subgroup actions on homogeneous spaces, ergodic nilmanifold automorphisms, subshifts of finite type and Young towers.
We study higher order expansions both in the Berry-Esséen estimate (Edgeworth expansions) and in the local limit theorems for Birkhoff sums of chaotic probability preserving dynamical systems. We establish general results under technical assumptions, discuss the verification of these assumptions and illustrate our results by different examples (subshifts of finite type, Young towers, Sinai billiards, random matrix products), including situations of unbounded observables with integrability order arbitrarily close to the optimal moment condition required in the i.i.d. setting.
We formulate abstract conditions under which a suspension flow satisfies the local central limit theorem. We check the validity of these conditions for several systems including reward renewal processes, Axiom A flows, as well as the systems admitting Young tower, such as Sinai billiard with finite horizon, suspensions over Pomeau-Manneville maps, and geometric Lorenz attractors.
Erdos-Renyi limit laws give the length scale of a time-window over which time-averages in Birkhoff sums have a non-trivial almost-sure limit. We establish Erdos-Renyi type limit laws for Holder observables on dynamical systems modeled by Young Towers with exponential and polynomial tails. This extends earlier results on Erdos-Renyi limit laws to a broad class of dynamical systems with some degree of hyperbolicity.
We obtain central limit theorem, local limit theorems and renewal theorems for stationary processes generated by skew product maps $T(\om,x)=(\te\om,T_\om x)$ together with a $T$-invariant measure, whose base map $\te$ satisfies certain topological and mixing conditions and the maps $T_\om$ on the fibers are certain non-singular distance expanding maps. Our results hold true when $\te$ is either a sufficiently fast mixing Markov shift or a (non-uniform) Young tower with at least one periodic point and polynomial tails. %In fact, our conditions will be satisfied %when $\om$ is the whole orbit t…
We consider some nonuniformly hyperbolic invertible dynamical systems which are modeled by a Gibbs-Markov-Young tower. We assume a polynomial tail for the inducing time and a polynomial control of hyperbolicity, as introduced by Alves, Pinheiro and Azevedo. These systems admit a physical measure with polynomial rate of mixing. In this paper we prove that the distribution of the number of visits to a ball B(x, r) converges to a Poisson distribution as the radius r $\to$ 0 and after suitable normalization.
We prove that the Birkhoff sums for ``almost every'' relevant observable in the stadium billiard obey a non-standard limit law. More precisely, the usual central limit theorem holds for an observable if and only if its integral along a one-codimensional invariant set vanishes, otherwise a $\sqrt{n\log n}$ normalization is needed. As one of the two key steps in the argument, we obtain a limit theorem that holds in Young towers with exponential return time statistics in general, an abstract result that seems to be applicable to many other situations.
We consider perturbations of interval maps with indifferent fixed points, which we refer to as wobbly interval intermittent maps, for which stable laws for general Hölder observables fail. We obtain limit laws for such maps and Hölder observables. These limit laws are similar to the classical semistable laws previously established for random processes, but certain limitations imposed by the current dynamical set up are reflected in the main result. One of the considered examples is an interval map with a countable number of discontinuities, and to analyse it we need to construct a Markov/Young…
Hitting statistics for $ϕ$-mixing dynamical systems
本文研究 $\phi$-混合动力系统中击中率与逃逸率的一般极限，证明其在无短返回时为一，否则由极值指数公式刻画，并适用于双曲、扩张及具 Young tower 的系统。
We show that for systems that allow a Young tower construction with polynomially decaying correlations the return times to metric balls are in the limit Poisson distributed. We also provide error terms which are powers of logarithm of the radius. In order to get those uniform rates of convergence the balls centres have to avoid a set whose size is estimated to be of similar order. This result can be applied to non-uniformly hyperbolic maps and to any invariant measure that satisfies a weak regularity condition. In particular it shows that the return times to balls is Poissonian for SRB measure…
We study the number of visits to balls B_r(x), up to time t/mu(B_r(x)), for a class of non-uniformly hyperbolic dynamical systems, where mu is the SRB measure. Outside a set of `bad' centers x, we prove that this number is approximately Poissonnian with a controlled error term. In particular, when r-->0, we get convergence to the Poisson law for a set of centers of mu-measure one. Our theorem applies for instance to the Hénon attractor and, more generally, to systems modelled by a Young tower whose return-time function has a exponential tail and with one-dimensional unstable manifolds. Along t…
We prove the one-dimensional almost sure invariance principle with essentially optimal rates for slowly (polynomially) mixing deterministic dynamical systems, such as Pomeau-Manneville intermittent maps, with Hölder continuous observables. Our rates have form $o(n^γL(n))$, where $L(n)$ is a slowly varying function and $γ$ is determined by the speed of mixing. We strongly improve previous results where the best available rates did not exceed $O(n^{1/4})$. To break the $O(n^{1/4})$ barrier, we represent the dynamics as a Young-tower-like Markov chain and adapt the methods of Berkes-Liu-Wu and Cu…
Consider a nonuniformly hyperbolic map $ T $ modelled by a Young tower with tails of the form $ O(n^{-β}) $, $ β>2 $. We prove optimal moment bounds for Birkhoff sums $ \sum_{i=0}^{n-1}v\circ T^i $ and iterated sums $ \sum_{0\le i<j<n}v\circ T^i\, w\circ T^j $, where $ v,w:M\to \Bbb{R}$ are (dynamically) Hölder observables. Previously iterated moment bounds were only known for $ β>5$. Our method of proof is as follows; (i) prove that $ T $ satisfies an abstract functional correlation bound, (ii) use a weak dependence argument to show that the functional correlation bound implies moment estimat…
We consider a $ϕ$-mixing shift $T$ on a sequence space $Ω$ and study the number of returns $\{ T^kω\in U\}$ to a union $U$ of cylinders of length $n$ until the first return $\{ T^kω\in V\}$ to another union $V$ of cylinder sets of length $m$. It turns out that if probabilities of the sets $U$ and $V$ are small and of the same order then the above number of returns has approximately geometric distribution. Under appropriate conditions, we extend this result for some dynamical systems to geometric balls and Young towers with integrable tails. This work is motivated by a number of papers on asymp…
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