Iterated invariance principle for slowly mixing dynamical systems

随机动力系统的迭代不变性原理

Iterated invariance principle for random dynamical systems

本文针对非一致扩张随机动力系统证明了弱迭代不变性原理，并利用鞅分解技术给出了快分量为一致扩张随机系统时快慢系统的淬火均匀化结果。

Law of iterated logarithm and invariance principle for one-parameter families of interval maps

We show that for almost every map in a transversal one-parameter family of piecewise expanding unimodal maps the Birkhoff sum of suitable observables along the forward orbit of the turning point satisfies the law of iterated logarithm. This result will follow from an almost sure invariance principle for the Birkhoff sum, as a function on the parameter space. Furthermore, we obtain a similar result for general one-parameter families of piecewise expanding maps on the interval.

Rates in almost sure invariance principle for slowly mixing dynamical systems

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…

Functional Correlation Bounds and Optimal Iterated Moment Bounds for Slowly-mixing Nonuniformly Hyperbolic Maps

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…

无限迭代函数系统的受限缓慢增长数字

Restricted slowly growing digits for infinite iterated function systems

本文研究无限迭代函数系统中数字序列受限缓慢增长时的Hausdorff维数，证明其不依赖于增长函数的缓慢程度，并推广至数字乘积情形。

Almost Sure Invariance Principle For Nonuniformly Hyperbolic Systems

We prove an almost sure invariance principle that is valid for general classes of nonuniformly expanding and nonuniformly hyperbolic dynamical systems. Discrete time systems and flows are covered by this result. In particular, the result applies to the planar periodic Lorentz flow with finite horizon. Statistical limit laws such as the central limit theorem, the law of the iterated logarithm, and their functional versions, are immediate consequences.