Carlangelo Liverani, Marco Martens
We investigate a class of area preserving non-uniformly hyperbolic maps of the two torus. First we establish some results on the regularity of the invariant foliations, then we use this knowledge to estimate the rate of mixing.
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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…
We study a random map $T$ which consists of intermittent maps $\{T_{k}\}_{k=1}^{K}$ and a position dependent probability distribution $\{p_{k,\varepsilon}(x)\}_{k=1}^{K}$. We prove existence of a unique absolutely continuous invariant measure (ACIM) for the random map $T$. Moreover, we show that, as $\varepsilon$ goes to zero, the invariant density of the random system $T$ converges in the $L^{1}$-norm to the invariant density of the deterministic intermittent map $T_{1}$. The outcome of this paper contains a first result on stochastic stability, in the strong sense, of intermittent maps.
Linear Response for Intermittent Circle Maps
本文通过锥技术研究间歇性圆映射的SRB测度弱微分性，证明线性响应并展示其统计稳定性。
We present a rigorous numerical scheme for the approximation of the linear response of the invariant density of a map with an indifferent fixed point, with explicit and computed estimates for the error and all the involved constants.
We study a class of random transformations built over finitely many intermittent maps sharing a common indifferent fixed point. Using a Young-tower technique, we show that the map with the fastest relaxation rate dominates the asymptotics. In particular, we prove that the rate of correlation decay for the annealed dynamics of the random map is the same as the sharp rate of correlation decay for the map with the fastest relaxation rate.
We consider intermittent maps T of the interval, with an absolutely continuous invariant probability measure μ. Kim showed that there exists a sequence of intervals A_n such that \sum μ(A_n)=\infty, but \{A_n\} does not satisfy the dynamical Borel-Cantelli lemma, i.e., for almost every x, the set \{n : T^n(x)\in A_n\} is finite. If \sum \Leb(A_n)=\infty, we prove that \{A_n\} satisfies the Borel-Cantelli lemma. Our results apply in particular to some maps T whose correlations are not summable.
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…
Doubly Intermittent Full Branch Maps with Critical Points and Singularities
研究一类具有两个中性不动点及临界点或奇异点的一维全分支映射，在温和假设下证明唯一混合绝对连续不变测度存在，并分析其相关衰减与极限定理。
We study the ergodic and statistical properties of a class of maps of the circle and of the interval of Lorenz type which present indifferent fixed points and points with unbounded derivative. These maps have been previously investigated in the physics literature. We prove in particular that correlations decay polynomially, and that suitable Limit Theorems (convergence to Stable Laws or Central Limit Theorem) hold for Hölder continuous observables. We moreover show that the return and hitting times are in the limit exponentially distributed.
We study an intermittent map which has exactly two ergodic invariant densities. The densities are supported on two subintervals with a common boundary point. Due to certain perturbations, leakage of mass through subsets, called holes, of the initially invariant subintervals occurs and forces the subsystems to merge into one system that has exactly one invariant density. We prove that the invariant density of the perturbed system converges in the $L^1$-norm to a particular convex combination of the invariant densities of the intermittent map. In particular, we show that the ratio of the weights…
We consider a family of Pomeau-Manneville type interval maps $T_α$, parametrized by $α\in (0,1)$, with the unique absolutely continuous invariant probability measures $ν_α$, and rate of correlations decay $n^{1-1/α}$. We show that despite the absence of a spectral gap for all $α\in (0,1)$ and despite nonsummable correlations for $α\geq 1/2$, the map $α\mapsto \int φ\, dν_α$ is continuously differentiable for $φ\in L^{q}[0,1]$ for $q$ sufficiently large.
Doubly Intermittent Maps with Critical Points, Unbounded Derivatives and Regularly Varying Tail
该研究针对带临界点、无界导数与正则变化尾的双重间歇映射，证明了温和假设下唯一混合绝对连续不变测度的存在性并给出有限性条件，推广了已有结果，还分析了慢变函数决定测度有限性的边界情形。
In the setting of abstract Markov maps, we prove results concerning the convergence of renormalized Birkhoff sums to normal laws or stable laws. They apply to one-dimensional maps with a neutral fixed point at 0 of the form $x+x^{1+α}$, for $α\in (0,1)$. In particular, for $α>1/2$, we show that the Birkhoff sums of a Hölder observable $f$ converge to a normal law or a stable law, depending on whether $f(0)=0$ or $f(0)\not=0$. The proof uses spectral techniques introduced by Sarig, and Wiener's Lemma in noncommutative Banach algebras.
We present results on the broadband nature of the power spectrum $S(ω)$, $ω\in(0,2π)$, for a large class of nonuniformly expanding maps with summable and nonsummable decay of correlations. In particular, we consider a class of intermittent maps $f:[0,1]\to[0,1]$ with $f(x)\approx x^{1+γ}$ for $x\approx 0$, where $γ\in(0,1)$. Such maps have summable decay of correlations when $γ\in(0,\frac12)$, and $S(ω)$ extends to a continuous function on $[0,2π]$ by the classical Wiener-Khintchine Theorem. We show that $S(ω)$ is typically bounded away from zero for Hölder observables. Moreover, in the nonsum…
Improved polynomial rates of memory loss for nonstationary intermittent dynamical systems
本文研究非平稳间歇动力系统，在首返回时间尾部多项式衰减非一致的条件下，证明了尖锐的多项式记忆损失速率，并应用于Pomeau–Manneville型间歇映射及无界导数间歇映射的随机遍历复合。
We establish some statistical properties of the hyperbolic times for a class of nonuniformly expanding dynamical systems. The maps arise as factors of area preserving maps of the unit square via a geometric Baker's map type construction, exhibit intermittent dynamics, and have unbounded derivatives. The geometric approach captures various examples from the literature over the last thirty years. The statistics of these maps are controlled by the order of tangency that a certain "cut function" makes with the boundary of the square. Using a large deviations result of Melbourne and Nicol we obtain…
Conditional limit laws for open intermittent dynamics
该研究针对开放间歇LSV映射推导条件极限定理，发现其极限分布不同于封闭系统的高斯或稳定定律，为两类可观测量均得到新分布。
We study intermittent maps from the point of view of metastability. Small neighbourhoods of an intermittent fixed point and their complements form pairs of almost-invariant sets. Treating the small neighbourhood as a hole, we first show that the absolutely continuous conditional invariant measures (ACCIMs) converge to the ACIM as the length of the small neighbourhood shrinks to zero. We then quantify how the escape dynamics from these almost-invariant sets are connected with the second eigenfunctions of Perron-Frobenius (transfer) operators when a small perturbation is applied near the intermi…
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 a…
We establish self-norming central limit theorems for non-stationary time series arising as observations on sequential maps possessing an indifferent fixed point. These transformations are obtained by perturbing the slope in the Pomeau-Manneville map. We also obtain quenched central limit theorems for random compositions of these maps.
It is well known that the Liverani-Saussol-Vaienti map satisfies a central limit theorem for Hölder observables in the parameter regime where the correlations are summable. We show that when $C^2$ observables are considered, the variance of the limiting normal distribution is a $C^1$ function of the parameter. We first show this for the first return map to the base of the second branch by studying the Green-Kubo formula, then conclude the result for the original map using Kac's lemma and relying on linear response.
Global-local mixing for infinite measure dynamical systems
本文研究了无限测度动力系统中全局-局部混合性质，涵盖了间歇映射和复平面上的抛物线有理映射，提出了相关数学证明。
We study the escape dynamics in the presence of a hole of a standard family of intermittent maps of the unit interval with neutral fixed point at the origin (and finite absolutely continuous invariant measure). Provided that the hole (is a cylinder that) does not contain any neighborhood of the origin, the surviving volume is shown to decay at polynomial speed with time. The associated polynomial escape rate depends on the density of the initial distribution, more precisely, on its behavior in the vicinity of the origin. Moreover, the associated normalized push forward measures are proved to c…
We establish almost sure invariance principles (ASIP), a strong form of approximation by Brownian motion, for non-stationary time series arising as observations on sequential maps possessing an indifferent fixed point. These transformations are obtained by perturbing the slope in the Pomeau-Manneville map. Quenched ASIP for random compositions of these maps is also obtained.
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.
Decay of correlations for critically intermittent systems
本文研究具有超吸引不动点的随机间歇动力系统，证明其测度性质随随机参数和映射阶数发生相变，并揭示系统在具有绝对连续不变概率测度时呈混合性且相关性仅多项式衰减，与确定性映射的指数衰减形成对比。
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…
Linear response for intermittent maps with critical point
本文研究具有中性不动点和非平坦临界点的双参数映射族，利用 Baladi-Todd 锥技巧证明 SRB 测度积分对参数的可微性，并给出方向导数公式。
Absolute Winning Exceptional Sets for Intermittent Interval Maps
研究曼内维尔 - 波梅奥型区间映射中轨道闭包错过规定可数集的点集，通过直接证明、利用有限分支获胜定理及拟对称拉回论证三部分，证明该点集在麦克马伦意义下是绝对获胜的。
Quenched Mixing Rates for Doubly Intermittent Maps
研究两类含特定奇点区间映射随机组合的淬火混合率，对皮科夫斯基族给出等变样本测度，为格罗斯曼 - 霍纳族构造等变测度族，利用随机杨塔证明有界与赫尔德可观测量纤维相关性的淬火多项式衰减，速率由淬火返回时间尾部估计确定。
Using quantitative perturbation theory for linear operators, we prove spectral gap for transfer operators of various families of intermittent maps with almost constant potentials ("high-temperature" regime). Hölder and bounded p-variation potentials are treated, in each case under a suitable assumption on the map, but the method should apply more generally. It is notably proved that for any Pommeau-Manneville map, any potential with Lispchitz constant less than 0.0014 has a transfer operator acting on Lip([0, 1]) with a spectral gap; and that for any 2-to-1 unimodal map, any potential with tot…
We consider a family $\{ T_{r} \colon [0, 1] \circlearrowleft \}_{r \in [0, 1]}$ of Markov interval maps interpolating between the Tent map $T_{0}$ and the Farey map $T_{1}$. Letting $\mathcal{P}_{r}$ denote the Perron-Frobenius operator of $T_{r}$, we show, for $β\in [0, 1]$ and $α\in (0, 1)$, that the asymptotic behaviour of the iterates of $\mathcal{P}_{r}$ applied to observables with a singularity at $β$ of order $α$ is dependent on the structure of the $ω$-limit set of $β$ with respect to $T_{r}$. Having a singularity it seems that such observables do not fall into any of the function cla…
In this paper, we prove a criterion for the local ergodicity of non-uniformly hyperbolic symplectic maps with singularities. Our result is an extension of a theorem of Liverani and Wojtkowski.
We prove that symplectic twist maps defined on the cotangent bundle of the d-dimensional torus that have no conjugate points are $C^0$ integrable, i.e. the cotangent bundle is continuously foliated by a family of invariant Lagrangian graphs.
We study dynamical systems arising as time-dependent compositions of Pomeau-Manneville-type intermittent maps. We establish central limit theorems for appropriately scaled and centered Birkhoff-like partial sums, with estimates on the rate of convergence. For maps chosen from a certain parameter range, but without additional assumptions on how the maps vary with time, we obtain a self-normalized CLT provided that the variances of the partial sums grow sufficiently fast. When the maps are chosen randomly according to a shift-invariant probability measure, we identify conditions under which the …
Linear response asymmetry between SRB and physical measures for families of intermittent maps with a transition point
研究间歇映射族在临界参数值处SRB测度从有限到无限总质量过渡时的线性响应，揭示了过渡导致的不对称性，包括SRB测度的光滑参数依赖性和物理测度在临界点的连续性与不可微性，以及大规模统计性质退化量化描述。
We establish sharp bounds on the mixing rates of a class of two dimensional non-uniformly hyperbolic symplectic maps. This provides a primer on how to investigate such questions in a concrete example and, at the same time, it solves a controversy between previous rigorous results and numerical experiments.
Statistical aspects of mean field coupled intermittent maps
本文研究 Pomeau-Manneville 情形下平均场弱耦合间歇映射的无限系统，证明其存在唯一物理稳态，并刻画了绝对连续及适当正则状态的收敛性。
Regularity of the variance in quenched CLT for random intermittent dynamical systems
研究随机间歇动力系统中淬火CLT中方差连续可微的条件及统计稳定性方法
We study the behavior of the empirical distribution function of iterates of intermittent maps in the Hilbert space of square inegrable functions with respect to Lebesgue measure. In the long-range dependent case, we prove that the empirical distribution function, suitably normalized, converges to a degenerate stable process, and we give the corresponding almost sure result. We apply the results to the convergence of the Wasserstein distance between the empirical measure and the invariant measure. We also apply it to obtain the asymptotic distribution of the corresponding Cramér-von-Mises stati…
We study the dynamics of a symplectic twist map without conjugate points. We show that in a neighborhood of a totally periodic Lagrangian manifold, there exists a large family of invariant Lagrangian tori on which the twist map is conjugated to a translation of non-resonant vector.
If a system mixes too slowly, putting a hole in it can completely destroy the richness of the dynamics. Here we study this instability for a class of intermittent maps with a family of slowly mixing measures. We show that there are three regimes: 1) standard hyperbolic-like behavior where the rate of mixing is faster than the rate of escape through the hole, there is a unique limiting absolutely continuous conditionally invariant measure (accim) and there is a complete thermodynamic description of the dynamics on the survivor set; 2) an intermediate regime, where the rate of mixing and escape …
We establish a functional limit theorem for the joint-law of occupations near and away from indifferent fixed points of interval maps, and of waits for the occupations away from these points, in the sense of strong distributional convergence. It is a functional and joint-distributional extension of Darling--Kac type limit theorem, of Lamperti type generalized arcsine laws for occupation times, and of Dynkin and Lamperti type generalized arcsine laws for waiting times, at the same time.
We construct a $C^2$ symplectic twist map f of the annulus that has an essential invariant curve C such that: - C is not differentiable; - the dynamics of f restricted to C is conjugated to the one of a Denjoy counter-example; - C is at the boundary of an instability zone for f.
We study bifurcations of cubic homoclinic tangencies in two-dimensional symplectic maps. We distinguish two types of cubic homoclinic tangencies, and each type gives different first return maps derived to diverse conservative cubic Hénon maps with quite different bifurcation diagrams. In this way, we establish the structure of bifurcations of periodic orbits in two parameter general unfoldings generalizing to the conservative case the results previously obtained for the dissipative case. We also consider the problem of 1:4 resonance for the conservative cubic Hénon maps.
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…
Phase transitions in temperature for intermittent maps
本文针对特定空间内的\textsc{Hölder}连续势，刻画间歇映射的温度相变及鲁棒相变，揭示温度相变与遍历优化的关联。
We compute some dependence coefficients for the stationary Markov chain whose transition kernel is the Perron-Frobenius operator of an expanding map $T$ of $[0, 1]$ with a neutral fixed point. We use these coefficients to prove a central limit theorem for the partial sums of $f\circ T^i$, when $f$ belongs to a large class of unbounded functions from $[0, 1]$ to ${\mathbb R}$. We also prove other limit theorems and moment inequalities.
We study non-stationary stochastic processes arising from sequential dynamical systems built on maps with a neutral fixed points and prove the existence of Extreme Value Laws for such processes. We use an approach developed in \cite{FFV16}, where we generalised the theory of extreme values for non-stationary stochastic processes, mostly by weakening the uniform mixing condition that was previously used in this setting. The present work is an extension of our previous results for concatenations of uniformly expanding maps obtained in \cite{FFV16}.
We study random transformations built from intermittent maps on the unit interval that share a common neutral fixed point. We focus mainly on random selections of Pomeu-Manneville-type maps $T_α$ using the full parameter range $0< α< \infty$, in general. We derive a number of results around a common theme that illustrates in detail how the constituent map that is fastest mixing (i.e.\ smallest $α$) combined with details of the randomizing process, determines the asymptotic properties of the random transformation. Our key result (Theorem 1.1) establishes sharp estimates on the position of retur…
Breakdown of rotational tori in 2D and 4D conservative and dissipative standard maps
通过实现Lindstedt级数与Padé逼近、Newton方法及Greene方法扩展，研究2D和4D保守与耗散标准映射中旋转不变环面的破裂阈值，发现Padé与Newton方法结果可靠，而Greene方法扩展因计算复杂在4D非辛映射中尚无定论。
In infinite ergodic theory, two distributional limit theorems are well-known. One is characterized by the Mittag-Leffler distribution for time averages of $L^1(m)$ functions, i.e., integrable functions with respect to an infinite invariant measure. The other is characterized by the generalized arc-sine distribution for time averages of non-$L^1(m)$ functions. Here, we provide another distributional behavior of time averages of non-$L^1(m)$ functions in one-dimensional intermittent maps where each has an indifferent fixed point and an infinite invariant measure. Observation functions considered…
Intermittent dynamics is characterized by long periods of different types of dynamical characteristics, for instance almost periodic dynamics alternated by chaotic dynamics. Critical intermittency is intermittent dynamics that can occur in iterated function systems, and involves a superattracting periodic orbit. This paper will provide and study examples of iterated function systems by two rational maps on the Riemann sphere that give rise to critical intermittency. The main ingredient for this is a superattracting fixed point for one map that is mapped onto a common repelling fixed point by t…
Intermittent--synchronization in non-weakly coupled piecewise linear expanding map lattice: a geometric-combinatorics method
本文针对非弱耦合双节点分段线性扩张映射格，提出几何-组合方法，给出绝对连续不变测度唯一性与间歇同步的充分必要条件。
Phase diagram for intermittent maps
本文刻画了间歇映射在赫尔德势函数空间中的相图，证明存在间歇相与平稳相，并揭示相变边界与压力函数失去实解析性及基态之间的联系。
We study the properties of `infinite-volume mixing' for two classes of intermittent maps: expanding maps $[0,1] \longrightarrow [0,1]$ with an indifferent fixed point at 0 preserving an infinite, absolutely continuous measure, and expanding maps $\mathbb{R}^+ \longrightarrow \mathbb{R}^+$ with an indifferent fixed point at $+\infty$ preserving the Lebesgue measure. All maps have full branches. While certain properties are easily adjudicated, the so-called global-local mixing, namely the decorrelation of a global and a local observable, is harder to prove. We do this for two subclasses of syste…
Thermodynamic formalism for intermittent maps with multiple neutral fixed points and phase transitions
本文针对具有多个中性不动点的间歇映射与不连续势函数发展热力学形式主义，证明满足双曲性假设时存在具正熵且关联指数衰减的唯一平衡态，还刻画了单参数族的相变并证明临界参数处多平衡态可共存，方法基于共形测度分析。
Survivor-conditioned renewal laws and observable bounds for open intermittent maps
针对开放Pomeau–Manneville映射，通过确定性的更新定理解释了幸存者条件李雅普诺夫拉伸被抑制的现象，并推广到可观测量的有界性和收敛性。
We prove that solutions $u$ of the equation $f=u-u\circ T$ are automatically Holder continuous when $f$ is Holder continuous, and $T$ is non uniformly expanding and Markov. This result applies in particular to Young towers and to intermittent maps.
We consider a sequences of symplectic twist maps corresponding to Frenkel Kontorova variational functional. For such a sequence without conjugate points we construct an invariant Lagrangian subbundle. Using the ideas by E.Hopf from Riemannian geometry we show that absence of conjugate points is a rare property for this class of maps.
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