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 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…
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 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.
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.
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 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.
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 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…
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…
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.
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.
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）
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…
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…
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 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 …
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 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 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…
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…
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.
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…
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…
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…
We consider a large class of piecewise expanding maps T of [0,1] with a neutral fixed point, and their associated Markov chain Y_i whose transition kernel is the Perron-Frobenius operator of T with respect to the absolutely continuous invariant probability measure. We give a large class of unbounded functions f for which the partial sums of f\circ T^i satisfy both a central limit theorem and a bounded law of the iterated logarithm. For the same class, we prove that the partial sums of f(Y_i) satisfy a strong invariance principle. When the class is larger, so that the partial sums of f\circ T^i…
This paper is about statistical properties of quasistatic dynamical systems. These are a class of non-stationary systems that model situations where the dynamics change very slowly over time due to external influence. We focus on the case where the time-evolution is described by intermittent interval maps (Pomeau-Manneville maps) with time-dependent parameters. In a suitable range of parameters, we obtain a description of the statistical properties as a stochastic diffusion, by solving a well-posed martingale problem. The results extend those of a related recent study due to Dobbs and Stenlund…
We consider the one parameter family $α\mapsto T_α$ ($α\in [0,1)$) of Pomeau-Manneville type interval maps $T_α(x)=x(1+2^αx^α)$ for $x \in [0,1/2)$ and $T_α(x)=2x-1$ for $x \in [1/2, 1]$, with the associated absolutely continuous invariant probability measure $μ_α$. For $α\in (0,1)$, Sarig and Gouëzel proved that the system mixes only polynomially with rate $n^{1-1/α}$ (in particular, there is no spectral gap). We show that for any $ψ\in L^q$, the map $α\to \int_0^1 ψ\, dμ_α$ is differentiable on $[0,1-1/q)$, and we give a (linear response) formula for the value of the derivative. This is the …
We consider the family of one-dimensional maps arising from the contracting Lorenz attractors studied by Rovella. Benedicks-Carleson techniques were used by Rovella to prove that there is a one-parameter family of maps whose derivatives along their critical orbits increase exponentially fast and the critical orbits have slow recurrent to the critical point. Metzger proved that these maps have a unique absolutely continuous ergodic invariant probability measure (SRB measure). Here we use the technique developed by Freitas and show that the tail set (the set of points which at a given time have …
