Pomeau-Manneville Maps Research Articles

Central limit theorems for sequential and random intermittent dynamical systems

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.

Rare events for the Manneville–Pomeau map

We prove a dichotomy for Manneville–Pomeau maps f:[0,1]→[0,1]: given any point ζ∈[0,1], either the Rare Events Point Processes (REPP), counting the number of exceedances, which correspond to entrances in balls around ζ, converge in distribution to a Poisson process; or the point ζ is periodic and the REPP converge in distribution to a compound Poisson process. Our method is to use inducing techniques for all points except 0 and its preimages, extending a recent result Haydn (2014), and then to deal with the remaining points separately. The preimages of 0 are dealt with applying recent results in Aytaç (2015). The point ζ=0 is studied separately because the tangency with the identity map at this point creates too much dependence, which causes severe clustering of exceedances. The Extremal Index, which measures the intensity of clustering, is equal to 0 at ζ=0, which ultimately leads to a degenerate limit distribution for the partial maxima of stochastic processes arising from the dynamics and for the usual normalising sequences. We prove that using adapted normalising sequences we can still obtain non-degenerate limit distributions at ζ=0.

Clustering of extreme events created by multiple correlated maxima

We consider stochastic processes arising from dynamical systems by evaluating an observable function along the orbits of the system. The novelty is that we will consider observables achieving a global maximum value (possible infinite) at multiple points with special emphasis for the case where these maximal points are correlated or bound by belonging to the same orbit of a certain chosen point. These multiple correlated maxima can be seen as a new mechanism creating clustering of extreme observations, i.e., the occurrence of several extreme observations concentrated in the time frame. We recall that clustering was intimately connected with periodicity when the maximum was achieved at a single point. We will study this mechanism for creating clustering and will address the existence of limiting Extreme Value Laws, the repercussions on the value of the Extremal Index, the impact on the limit of Rare Events Points Processes, the influence on clustering patterns and the competition of domains of attraction. We also consider briefly and for comparison purposes multiple uncorrelated maxima. The systems considered include expanding maps of the interval such as Rychlik maps but also maps with an indifferent fixed point such as Manneville–Pomeau maps.

Extensive numerical investigations on the ergodic properties of two coupled Pomeau–Manneville maps

We present extensive numerical investigations on the ergodic properties of two identical Pomeau–Manneville maps interacting on the unit square through a diffusive linear coupling. The system exhibits anomalous statistics, as expected, but with strong deviations from the single intermittent map: Such differences are characterized by numerical experiments with densities which do not have singularities in the marginal fixed point, escape and Poincaré recurrence time statistics that share a power-law decay exponent modified by a clear dimensional scaling, while the rate of phase-space filling and the convergence of ensembles of Lyapunov exponents show a stretched instead of pure exponential behavior. In spite of the lack of rigorous results about this system, the dependence on both the intermittency and the coupling parameters appears to be smooth, paving the way for further analytical development. We remark that dynamical exponents appear to be independent of the (nonzero) coupling strength.

A New Approach for Image Encryption Based on Cyclic Rotations and Multiple Blockwise Diffusions Using Pomeau-Manneville and Sin Maps

In this paper an efficient image encryption scheme based on cyclic rotations and multiple blockwise diffusions with two chaotic maps is proposed. A Sin map is used to generate round keys for the encryption/decryption process. A Pomeau-Manneville map is used to generate chaotic values for permutation, pixel value rotation and diffusion operations. The encryption scheme is composed of three stages: permutation, pixel value rotation and diffusion. The permutation stage performs four operations on the image: row shuffling, column shuffling, cyclic rotation of all the rows and cyclic rotation of all the columns. This stage reduces the correlation significantly among neighboring pixels. The second stage performs circular rotation of pixel values twice by scanning the image horizontally and vertically. The amount of rotation is based on M × N chaotic values. The last stage performs the diffusion four times by scanning the image in four different ways: block of 8 × 8 pixels, block of 16 × 16 pixels, principal diagonally, and secondary diagonally. Each of the above four diffusions performs the diffusion in two directions (forwards and backwards) with two previously diffused pixels and two chaotic values. This stage makes the scheme resistant to differential attacks. The security and performance of the proposed method is analyzed systematically by using the key space, entropy, statistical, differential and performance analysis. The experimental results confirm that the proposed method is computationally efficient with high security.

Multifractal Analysis for Quotients of Birkhoff Sums for Countable Markov Maps

This paper is devoted to study multifractal analysis of quotients of Birkhoff averages for countable Markov maps. We prove a variational principle for the Hausdorff dimension of the level sets. Under certain assumptions we are able to show that the spectrum varies analytically in parts of its domain. We apply our results to show that the Birkhoff spectrum for the Manneville-Pomeau map can be discontinuous, showing the remarkable differences with the uniformly hyperbolic setting. We also obtain results describing the Birkhoff spectrum of suspension flows. Examples involving continued fractions are also given.

On stochastic stability of expanding circle maps with neutral fixed points

It is well known that the Manneville–Pomeau map with a parabolic fixed point of the form is stochastically stable for α ≥ 1 and the limiting measure is the Dirac measure at the fixed point. In this paper, we show that if α ∈ (0, 1), then it is also stochastically stable. Indeed, the stationary measure of the random map converges strongly to the absolutely continuous invariant measure for the deterministic system as the noise tends to zero.

The Compound Poisson Limit Ruling Periodic Extreme Behaviour of Non-Uniformly Hyperbolic Dynamics

We prove that the distributional limit of the normalised number of returns to small neighbourhoods of periodic points of non-uniformly hyperbolic dynamical systems is compound Poisson. The returns to small balls around a fixed point in the phase space correspond to the occurrence of rare events, or exceedances of high thresholds, so that there is a connection between the laws of Return Times Statistics and Extreme Value Laws. The fact that the fixed point in the phase space is a repelling periodic point implies that there is a tendency for the exceedances to appear in clusters whose average sizes is given by the Extremal Index, which depends on the expansion of the system at the periodic point. We recall that for generic points, the exceedances, in the limit, are singular and occur at Poisson times. However, around periodic points, the picture is different: the respective point processes of exceedances converge to a compound Poisson process, so instead of single exceedances, we have entire clusters of exceedances occurring at Poisson times with a geometric distribution ruling its multiplicity. The systems to which our results apply include: general piecewise expanding maps of the interval (Rychlik maps), maps with indifferent fixed points (Manneville-Pomeau maps) and Benedicks-Carleson quadratic maps.

Renormalization, Thermodynamic Formalism and Quasi-Crystals in Subshifts

We examine the thermodynamic formalism for a class of renormalizable dynamical systems which in the symbolic space is generated by the Thue-Morse substitution, and in complex dynamics by the Feigenbaum-Coullet-Tresser map. The basic question addressed is whether fixed points V of a renormalization operator $${\mathcal{R}}$$ acting on the space of potentials are such that the pressure function $${\beta \mapsto \mathcal{P}(-\beta V)}$$ exhibits phase transitions. This extends the work by Baraviera, Leplaideur and Lopes on the Manneville-Pomeau map, where such phase transitions were indeed detected. In this paper, however, the attractor of renormalization is a Cantor set (rather than a single fixed point), which admits various classes of fixed points of $${\mathcal{R}}$$ , some of which do and some of which do not exhibit phase transitions. In particular, we show it is possible to reach, as a ground state, a quasi-crystal before temperature zero by freezing a dynamical system.

A generalized central limit theorem in infinite ergodic theory

We prove a generalized central limit theorem for dynamical systems with an infinite ergodic measure which induce a Gibbs-Markov map on some subset, provided the return time to this subset has regularly varying tails. We adapt a method designed by Csaki and Foldes for observables of random walks to show that the partial sums of some functions of the system-the return time and the observable-are asymptotically independent. Some applications to random walks and Pomeau-Manneville maps are discussed.

THE POTENTIAL POINT OF VIEW FOR RENORMALIZATION

For the shift σ in Σ = {0, 1}ℕ, we define the renormalization for potentials by [Formula: see text] We show that for a good H, there is a unique fixed point for [Formula: see text]. It is the Hofbauer potential V*.We show that the stable set of the Hofbauer potential, i. e. the set of potentials V such that [Formula: see text] converges to V* is characterized by the germ of these potentials close to 0∞= 000….Then, we make connections with the Manneville–Pomeau map f : [0, 1]↺. In particular we show that the lift in Σ of log f′ is in the stable set of V*.In the second part, we characterize "good" H, such that σ2◦ H = H ◦ σ.In the last part, we study the thermodynamic formalism for some special potentials in the stable set of V*. They are called virtual Manneville–Pomeau maps.

Multifractal analysis and phase transitions for hyperbolic and parabolic horseshoes

We effect a complete study of the thermodynamic formalism, the entropy spectrum of Birkhoff averages, and the ergodic optimization problem for a family of parabolic horseshoes. We consider a large class of potentials that are not necessarily regular, and we describe both the uniqueness of equilibrium measures and the occurrence of phase transitions for nonregular potentials in this class. Our approach consists in reducing the problems to the study of renewal shifts. We also describe applications of this approach to hyperbolic horseshoes as well as to noninvertible maps, both parabolic (with the Manneville-Pomeau map) and uniformly expanding. This allows us to recover in a unified manner several results scattered in the literature. For the family of hyperbolic horseshoes, we also describe the dimension spectrum of equilibrium measures of a class of potentials that are not necessarily regular. In particular, the dimension spectra need not be strictly convex.

Separation of trajectories and its relation to entropy for intermittent systems with a zero Lyapunov exponent

One-dimensional intermittent maps with stretched exponential δx(t)∼δx(0)e(λ(α)t(α)) separation of nearby trajectories are considered. When t→∞ the standard Lyapunov exponent λ=∑(i=0)(t-1)ln|M'(x(i))|/t is zero (M' is a Jacobian of the map). We investigate the distribution of λ(α)=∑(i=0)(t-1)ln|M'(x(i))|/t(α), where α is determined by the nonlinearity of the map in the vicinity of marginally unstable fixed points. The mean of λ(α) is determined by the infinite invariant density. Using semianalytical arguments we calculate the infinite invariant density for the Pomeau-Manneville map, and with it we obtain excellent agreement between numerical simulation and theory. We show that α(λ(α)) is equal to Krengel's entropy and to the complexity calculated by the Lempel-Ziv compression algorithm. This generalized Pesin's identity shows that (λ(α)) and Krengel's entropy are the natural generalizations of usual Lyapunov exponent and entropy for these systems.

Multifractal analysis of non-uniformly hyperbolic systems

We prove a multifractal formalismfor Birkhoff averages of continuous functions in the case of some non-uniformly hyperbolic maps, which includes interval examples such as the Manneville-Pomeau map.

Large deviations for intermittent maps

In this paper we study large deviation results for the Manneville–Pomeau map and related transformations to indifferent fixed points. In particular, we consider conditions under which the associated error term is polynomial or even exponential. For typical observables, polynomial estimates are optimal. However, under suitable conditions, the exponential error term arises from the compactness of the space of measures, despite the indifference of the fixed point.

Pesin-Type Identity for Intermittent Dynamics with a Zero Lyaponov Exponent

Pesin's identity provides a profound connection between the Kolmogorov-Sinai entropy h_{KS} and the Lyapunov exponent lambda. It is well known that many systems exhibit subexponential separation of nearby trajectories and then lambda=0. In many cases such systems are nonergodic and do not obey usual statistical mechanics. Here we investigate the nonergodic phase of the Pomeau-Manneville map where separation of nearby trajectories follows deltax_{t}=deltax_{0}e;{lambda_{alpha}t;{alpha}} with 0<alpha<1. The limit distribution of lambda_{alpha} is the inverse Lévy function. The average lambda_{alpha} is related to the infinite invariant density, and most importantly to entropy. Our work gives a generalized Pesin's identity valid for systems with an infinite invariant density.

Decay of correlations for some partially hyperbolic diffeomorphisms

In this paper we study a $C^{1+\alpha}$-partially hyperbolic diffeomorphism f of which restriction on one dimensional center unstable direction behaves as Manneville-Pomeau map. We show that f admits a unique ergodic SRB measure with polynomial upper bounds on correlations for Hölder continuous functions.

Sub-actions for young towers

Let $T:X\to X$ be a dynamical system, and $\phi: X\to \mathbb{R}$ a function on $X$. A function $\theta:X\to \mathbb{R}$ is called a sub-action if $\theta$ satisfies the equation $\phi \leq \theta \circ T - \theta + m(\phi, T)$ where $m(\phi, T)=$sup{$\int \phi d\mu:\mu$ is an invariant probability measure for $ T$}. The existence and regularity of sub-actions are important for the study of optimizing measures. We prove the existence of Holder sub-actions for Lipschitz functions on certain classes of Manneville-Pomeau type maps. We also construct locally Holder sub-actions for Lipschitz functions on Young Towers. In some settings (uniform hyperbolicity and Manneville-Pomeau maps) this implies Holder sub-actions for the underlying system modeled by the Tower.

Decay of correlations for non-Hölder observables

We consider the general question of estimating decay of correlationsfor non-uniformly expanding maps, for classes of observables whichare much larger than the usual class of Hölder continuous functions.Our results give new estimates for many non-uniformly expandingsystems, including Manneville-Pomeau maps, many one-dimensionalsystems with critical points, and Viana maps . In manysituations, we also obtain a Central Limit Theorem for a much largerclass of observables than usual. Our main tool is an extension of the coupling method introduced byL.-S. Young for estimating rates of mixing on certain non-uniformlyexpanding tower maps.

On the variational principle for the topological entropy of certain non-compact sets

For a continuous transformation f of a compact metric space (X,d) and any continuous function \phi on X we consider sets of the formK_{\alpha} =\bigg\{x\in X:\lim_{n\to\infty} \frac 1n \sum_{i=0}^{n-1} \phi( f^i(x))=\alpha \bigg\},\quad\alpha\in\R.For transformations satisfying the specification property we prove the following Variational Principleh_{\rm top}(f,K_{\alpha}) = \sup\bigg( h_\mu(f): \mu\text{ is invariant and } \int\phi \,d\mu=\alpha \bigg),where h_{\rm top}(f,\cdot) is the topological entropy of non-compact sets. Using this result we are able to obtain a complete description of the multifractal spectrum for Lyapunov exponents of the so-called Manneville–Pomeau map, which is an interval map with an indifferent fixed point.We also consider multi-dimensional multifractal spectra and establish a contraction principle.