Pomeau-Manneville Maps Research Articles

Generalised Lyapunov stretching in a weakly chaotic open dynamical system

Abstract Chaotic dynamical systems may be characterised by a positive Lyapunov exponent, which measures the exponential rate of separation of nearby trajectories. However in a wide range of so-called weakly chaotic systems, the separation of nearby trajectories is sub-exponential, for example stretched exponential, in time; and therefore in such cases the Lyapunov exponent vanishes. When a hole is introduced in chaotic systems, the Lyapunov exponent on the system’s fractal repeller can be related to the generation of entropy and the escape rate from the system via the escape rate formalism, but no suitable generalisation exists to weakly chaotic systems. In this work we show that in a paradigmatic one-dimensional weakly chaotic iterated map, the Pomeau-Manneville map, the generation of generalised Lyapunov stretching is completely suppressed in the presence of a hole. These results are shown based on numerical evidence, and explained with a fully analytic stochastic model.

Optimal Linear Response for Markov Hilbert–Schmidt Integral Operators and Stochastic Dynamical Systems

We consider optimal control problems for discrete-time random dynamical systems, finding unique perturbations that provoke maximal responses of statistical properties of the system. We treat systems whose transfer operator has an L^2 kernel, and we consider the problems of finding (i) the infinitesimal perturbation maximising the expectation of a given observable and (ii) the infinitesimal perturbation maximising the spectral gap, and hence the exponential mixing rate of the system. Our perturbations are either (a) perturbations of the kernel or (b) perturbations of a deterministic map subjected to additive noise. We develop a general setting in which these optimisation problems have a unique solution and construct explicit formulae for the unique optimal perturbations. We apply our results to a Pomeau–Manneville map and an interval exchange map, both subjected to additive noise, to explicitly compute the perturbations provoking maximal responses.

Critical Intermittency in Random Interval Maps

Critical intermittency stands for a type of intermittent dynamics in iterated function systems, caused by an interplay of a superstable fixed point and a repelling fixed point. We consider critical intermittency for iterated function systems of interval maps and demonstrate the existence of a phase transition when varying probabilities, where the absolutely continuous stationary measure changes between finite and infinite. We discuss further properties of this stationary measure and show that its density is not in L^q for any q>1. This provides a theory of critical intermittency alongside the theory for the well studied Manneville–Pomeau maps, where the intermittency is caused by a neutral fixed point.

Loss of Memory and Moment Bounds for Nonstationary Intermittent Dynamical Systems

We study nonstationary intermittent dynamical systems, such as compositions of a (deterministic) sequence of Pomeau-Manneville maps. We prove two main results: sharp bounds on memory loss, including the "unexpected" faster rate for a large class of measures, and sharp moment bounds for Birkhoff sums and, more generally, "separately H\"older" observables.

Pomeau-Manneville maps are global-local mixing

We prove that a large class of expanding maps of the unit interval with a $C^2$-regular indifferent point in 0 and full increasing branches are global-local mixing. This class includes the standard Pomeau-Manneville maps $T(x) = x + x^{p+1}$ mod 1 ($p \ge 1$), the Liverani-Saussol-Vaienti maps (with index $p \ge 1$) and many generalizations thereof.

Schmidt’s game and nonuniformly expanding interval maps

We study a class of nonuniformly expanding interval maps with a neutral fixed point at 0, a class that includes Manneville–Pomeau maps. We prove that the set of points whose forward orbits miss an interval with left endpoint 0 is strong winning for Schmidt’s game. Strong winning sets are dense, have full Hausdorff dimension, and satisfy a countable intersection property. Similar results were known for certain expanding maps, but these did not address the nonuniformly expanding case. Our analysis is complicated by the presence of infinite distortion and unbounded geometry.

Linear response for random dynamical systems

We study for the first time linear response for random compositions of maps, chosen independently according to a distribution P. We are interested in the following question: how does an absolutely continuous stationary measure (acsm) of a random system change when P changes smoothly to Pε? For a wide class of one dimensional random maps, we prove differentiability of acsm with respect to ε; moreover, we obtain a linear response formula. Our results cover random maps whose transfer operator does not necessarily admit a spectral gap. We apply our results to iid compositions, with respect to various distributions Pε, of uniformly expanding circle maps, Gauss-Rényi maps (random continued fractions) and Pomeau-Manneville maps. Our results yield an exact formula for the invariant density of random continued fractions; while for Pomeau-Manneville maps our results provide a precise relation between their linear response under certain random perturbations and their linear response under deterministic perturbations.

Shrinking targets and eventually always hitting points for interval maps

We study shrinking target problems and the set of eventually always hitting points. These are the points whose first n iterates will never have empty intersection with the nth target for sufficiently large n. We derive necessary and sufficient conditions on the shrinking rate of the targets for to be of full or zero measure especially for some interval maps including the doubling map, some quadratic maps and the Manneville–Pomeau map. We also obtain results for the Gauß map and correspondingly for the maximal digits in continued fraction expansions. In the case of -transformations we also compute the packing dimension of complementing already known results on the Hausdorff dimension of .

Weak chaos, Allee points, and intermittency emerging from niche construction in population models

The idea of niche construction is regarded as an interesting, if not essential, element of population dynamics. Classical results from population biology emerge from such a formulation, such as chaotic behavior, the self-organized emergence of Allee points, critical transitions, and hysteresis. Relaxing the assumption of a descending limb in classic 1-D map representations of chaos, the application of the Pomeau–Manneville map is suggested as more realistic in some situations. This alternative formulation does not allow for the emergence of a “self-organized” Allee point but introduces a new concept to the potential behavior of populations in discrete time, intermittency sensitively dependent on initial conditions, but with a distorted form referred to as weak chaos.

The pressure function for infinite equilibrium measures

Assume that (X, f) is a dynamical system and φ: X → [−∞, ∞) is a potential such that the f-invariant measure μφ equivalent to the φ-conformal measure is infinite, but that there is an inducing scheme F = fτ with a finite measure $$\mu_\phi^-$$ and polynomial tails $$\mu_\phi^-$$ (τ ≥ n) = O(n−β), β ∈ (0, 1). We give conditions under which the pressure of f for a perturbed potential φ + sψ relates to the pressure of the induced system as $$P(\phi + s\psi ) = (CP{(\overline {\phi + s\psi )} )^{1/\beta }}(1 + o(1)),$$ together with estimates for the o(1)-error term. This extends results from Sarig [S06] to the setting of infinite equilibrium states. We give several examples of such systems, thus improving on the results of Lopes [L93] for the Pomeau-Manneville map with potential φt = −tlogf′, as well as on the results by Bruin and Todd [BTo09, BTo12] on countably piecewise linear unimodal Fibonacci maps. In addition, limit properties of the family of measures μφ+sψ as s → 0 are studied and statistical properties (correlation coefficients and arcsine laws) under the limit measure are derived.

Targets, local weak -Gibbs measures and a generalized Bowen dimension formula

For a dynamical system, we study the set of points whose orbit approximates any chosen point at certain specified rates. Our basic setting is that of a left shift acting on topological Markov chains endowed with a local weak Gibbs measure. Our rates of recurrence are so fast that the corresponding set has measure zero, but we obtain a generalized Bowen formula for the Carathéodory dimension. For the case of Markov transformations with a countable partition and ‘big image’ property, a Bowen-type formula is obtained for the Hausdorff dimension of those exceptional sets. In particular, we apply our general results to Gauss and Luröth maps, a non-Bernoulli modification of a Gauss map and some inner functions. Since we only require the existence of weak Gibbs measures, we can deal with non Hölder potentials and we can also consider intermittent systems as the Manneville–Pomeau map.

Almost surely invariance principle for non-stationary and random intermittent dynamical systems

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.

Infinite measure renewal theorem and related results

We present abstract conditions under which a special flow over a probability preserving map with a non-integrable roof function is Krickeberg mixing. Our main condition is some version of the local central limit theorem for the underlying map. We check our assumptions for iid random variables (renewal theorem with infinite mean) and for suspensions over Pomeau-Manneville maps.

Anomalous diffusion and the Moses effect in an aging deterministic model

We decompose the anomalous diffusive behavior found in an aging system into its fundamental constitutive causes. The model process is a sum of increments that are iterates of a chaotic dynamical system, the Pomeau–Manneville map. The increments can have long-time correlations, fat-tailed distributions and be non-stationary. Each of these properties can cause anomalous diffusion through what is known as the Joseph, Noah and Moses effects, respectively. The model can have either sub- or super-diffusive behavior, which we find is generally due to a combination of the three effects. Scaling exponents quantifying each of the three constitutive effects are calculated using analytic methods and confirmed with numerical simulations. They are then related to the scaling of the distribution of the process through a scaling relation. Finally, the importance of the Moses effect in the anomalous diffusion of experimental systems is discussed.

Infinite mixing for one-dimensional maps with an indifferent fixed point

We study the properties of ‘infinite-volume mixing’ for two classes of intermittent maps: expanding maps with an indifferent fixed point at 0 preserving an infinite, absolutely continuous measure, and expanding maps with an indifferent fixed point at 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 systems. The first subclass includes, among others, the Farey map. The second class includes the standard Pomeau–Manneville map mod 1. Morevoer, we use global-local mixing to prove certain limit theorems for our intermittent maps.

On mixing and the local central limit theorem for hyperbolic flows

We formulate abstract conditions under which a suspension flow satisfies the local central limit theorem. We check the validity of these conditions for several systems including reward renewal processes, Axiom A flows, as well as the systems admitting Young’s tower, such as Sinai’s billiard with finite horizon, suspensions over Pomeau–Manneville maps, and geometric Lorenz attractors.

Intermittent quasistatic dynamical systems: weak convergence of fluctuations

Abstract 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 influences. 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, which concerned the case of quasistatic (uniformly) expanding systems.

Functional correlation decay and multivariate normal approximation for non-uniformly expanding maps

In the setting of intermittent Pomeau–Manneville maps with time dependent parameters, we show a functional correlation bound widely useful for the analysis of the statistical properties of the model. We give two applications of this result, by showing that in a suitable range of parameters the bound implies the conditions of the normal approximation methods of Stein and Rio. For a single Pomeau–Manneville map belonging to this parameter range, both methods then yield a multivariate central limit theorem with a rate of convergence.

Infinite invariant densities due to intermittency in a nonlinear oscillator.

Dynamical intermittency is known to generate anomalous statistical behavior of dynamical systems, a prominent example being the Pomeau-Manneville map. We present a nonlinear oscillator, i.e., a physical model in continuous time, whose properties in terms of weak ergodity breaking and aging have a one-to-one correspondence to the properties of the Pomeau-Manneville map. So for both systems in a wide range of parameters no physical invariant density exists. We show how this regime can be characterized quantitatively using the techniques of infinite invariant densities and the Thaler-Dynkin limit theorem. We see how expectation values exhibit aging in terms of scaling in time.

Linear response in the intermittent family: Differentiation in a weighted $C^0$-norm

We provide a general framework to study differentiability of SRB measures for one dimensional non-uniformly expanding maps. Our technique is based on inducing the non-uniformly expanding system to a uniformly expanding one, and on showing how the linear response formula of the non-uniformly expanding system is inherited from the linear response formula of the induced one. We apply this general technique to interval maps with a neutral fixed point (Pomeau-Manneville maps) to prove differentiability of the corresponding SRB measure. Our work covers systems that admit a finite SRB measure and it also covers systems that admit an infinite SRB measure. In particular, we obtain a linear response formula for both finite and infinite SRB measures. To the best of our knowledge, this is the first work that contains a linear response result for infinite measure preserving systems.