We consider a family $\mathcal F$ of maps with two branches and a common neutral fixed point $0$ such that the order of tangency at $0$ belongs to some interval $[α_0, α_1]\subset (0, 1)$. Maps in $\mathcal F$ do not necessarily share a common Markov partition. At each step a member of $\mathcal F$ is chosen independently with respect to the uniform distribution on $[α_0, α_1]$. We show that the construction of the random tower in Bahsoun-Bose-Ruziboev \cite{BBR} with \emph{general return time} can be carried out for random compositions of such maps. Thus their general results are applicable and gives upper bounds for the quenched decay of correlations of form $n^{1-1/α_0+δ}$ for any $δ>0$.
We introduce random towers to study almost sure rates of correlation decay for random partially hyperbolic attractors. Using this framework, we obtain abstract results on almost sure exponential, stretched exponential and polynomial correlation decay rates. We then apply our results to small random perturbations of Axiom A attractors, small perturbations of derived from Anosov partially hyperbolic systems and to solenoidal attractors with random intermittency.
Quenched Mixing Rates for Doubly Intermittent Maps
School of Mathematical Sciences, Hebei Normal University（河北师范大学数学科学学院）；V.I.Romanovskiy Institute of Mathematics, Academy of Sciences of the Republic of Uzbekistan（乌兹别克斯坦共和国科学院V.I.罗曼诺夫斯基数学研究所）
We obtain large deviations estimates for both sequential and random compositions of intermittent maps. We also address the question of whether or not centering is necessary for the quenched central limit theorems (CLT) obtained by Nicol, Török and Vaienti for random dynamical systems comprised of intermittent maps. Using recent work of Abdelkader and Aimino, Hella and Stenlund we extend the results of Nicol, Török and Vaienti on quenched central limit theorems (CLT) for centered observables over random compositions of intermittent maps: first by enlarging the parameter range over which the que…
Intermittent maps of Pomeau-Manneville type are well-studied in one-dimension, and also in higher dimensions if the map happens to be Markov. In general, the nonconformality of multidimensional intermittent maps represents a challenge that up to now is only partially addressed. We show how to prove sharp polynomial bounds on decay of correlations for a class of multidimensional intermittent maps. In addition we show that the optimal results on statistical limit laws for one-dimensional intermittent maps hold also for the maps considered here. This includes the (functional) central limit theore…
In this paper we show how to apply classical probabilistic tools for partial sums $\sum_{j=0}^{n-1}φ\circτ^j$ generated by a skew product $τ$, built over a sufficiently well mixing base map and a random expanding dynamical system. Under certain regularity assumptions on the observable $φ$, we obtain a central limit theorem (CLT) with rates, a functional CLT, an almost sure invariance principle (ASIP), a moderate deviations principle, several exponential concentration inequalities and Rosenthal type moment estimates for skew products with $α, ϕ$ or $ψ$ mixing base maps and expanding on the aver…
Correcting the proof of Theorem 3.2 and Corollary 5.2 in Almost sure rates of mixing for i.i.d. unimodal maps by V. Baladi, M. Benedicks, V. Maume-Deschamps, Ann. E.N.S. (2002)
Equidistribution for nonuniformly expanding dynamical systems, and application to the almost sure invariance principle
设$T \colon M \to M$是一个非一致扩张的动力系统，如Logistic映射或间歇映射。令$v \colon M \to \mathbb{R}^d$是一个可观测的和$v_n = \sum_{k=0}^{n-1} v \circ T^k$表示伯克霍夫和。给定$M$上的一个概率测度$μ$，我们把$v_n$看作概率空间$(M, μ)$上的一个离散时间随机过程.在光滑遍历理论中有各种各样的自然选择的$μ$，如勒贝格措施，或绝对连续的$T$不变的措施。它们产生不同的随机过程。我们调查这些过程之间的关系。我们表明，在一个大类的措施，它是可能的耦合（重新定义一个新的概率空间）每两个过程，使他们几乎肯定是彼此接近，与明确的估计“接近”。这项工作的目的是关闭一个缺口，证明几乎必然的不变原理的非均匀双曲变换墨尔本和尼科尔。
We consider an independently identically distributed random dynamical system generated by finitely many, non-uniformly expanding Markov interval maps with a finite number of branches. Assuming a topologically mixing condition and the uniqueness of equilibrium state for the associated skew product map, we establish a samplewise (quenched) almost-sure level-2 weighted equidistribution of "random cycles", with respect to a natural stationary measure as the periods of the cycles tend to infinity. This result implies an analogue of Bowen's theorem on periodic orbits of topologically mixing Axiom A …
Natural measures and statistical properties of non-statistical maps with multiple neutral fixed points
Universidade Federal do Rio Janeiro（里约热内卢联邦大学）；University of Warwick（华威大学）；Sharif University of Technology（谢里夫理工大学）；Institute for Research in Fundamental Sciences (IPM)（伊朗基础科学研究院）
We consider a conservative ergodic measure-preserving transformation $T$ of a $σ$-finite measure space $(X,\mathcal{B},μ)$ with $μ(X)=\infty$. Given an observable $f:X\to \mathbb{R}$ we study the almost sure asymptotic behaviour of the Birkhoff sums $S_Nf(x) := \sum_{j=1}^N\, (f\circ T^{j-1})(x)$. In infinite ergodic theory it is well known that the asymptotic behaviour of $S_Nf(x)$ strongly depends on the point $x\in X$, and if $f\in L^1(X,μ)$, then there exists no real valued sequence $(b(N))$ such that $\lim_{N\to\infty} S_Nf(x)/b(N)=1$ almost surely. In this paper we show that for dynamica…
