Exponential Decay Of Correlations Research Articles

Stability of mixing rates for Axiom A attractors

There is a classical result that Axiom A attracting diffeomorphisms f:M↦M have exponential decay of correlations with respect to the Sinai–Ruelle–Bowen measure μ(f) on an attractor Λ. Following continuity results of Blank–Keller–Liverani (2002 Ruelle–Perron–Frobenius spectrum for Anosov maps Nonlinearity 15 1905–73) we shall describe settings where the rates of mixing are more regular (i.e. differentiable, smooth, etc).

Backward inducing and exponential decay of correlations for partially hyperbolic attractors

We study partially hyperbolic attractors ofC 2 diffeomorphisms on a compact manifold. For a robust (non-empty interior) class of such diffeomorphisms, we construct Sinai-Ruelle-Bowen measures, for which we prove exponential decay of correlations and the central limit theorem, in the space of Holder continuous functions. The techniques we develop (backward inducing, redundancy elimination algorithm) should be useful in the study of the stochastic properties of much more general non-uniformly hyperbolic systems.

Decay of correlations for piecewise invertible maps in higher dimensions

We study the mixing properties of equilibrium statesμ of non-Markov piecewise invertible mapsT:X→X, especially in the multidimensional case. Assuming mainly Holder continuity and that the topological pressure of the boundary is smaller than the total topological pressure, we establish exponential decay of correlations, i.e.,\(\left| {\int_x {\varphi \cdot \psi oT^n d\mu - \int_x {\varphi d\mu \cdot \int_x {\psi d\mu } } } } \right| \leqslant C \cdot e^{ - an} \) for all Holder functionsϕ,ψ :X→ℝ, alln≥0 and someC 0. We also obtain a Central Limit Theorem. Weakening the smoothness assumption, we get subexponential rates of decay.

CONVERGENCE IN DISTRIBUTION OF THE PERIODOGRAM OF CHAOTIC PROCESSES

In this work we analyze the convergence in distribution sense of the periodogram function (to the spectral density function) based on a time series of a stationary process Xt = (φ ◦ Tt)(X0) obtained from the iterations of a continuous transformation T invariant for an ergodic probability μ and a continuous function φ taking values in ℝ. We only assume a certain rate of convergence to zero for the autocovariance coefficient of the stochastic process, i.e. we assume there exist C > 0 and β > 2 such that |γX(h)| ≤ C|h|-β, for all h ∈ ℕ, where γX(h) = ∫(φ ◦ Th)(x) φ(x)dμ(x) - (∫ φ(x)dμ(x))2 is the h-autocovariance of the process. Our result applies to the case of exponential decay of correlation (or covariance), as it happens for a continuous expanding transformation T on the circle and a Holder potential φ. It can also be applied to the case when the transformation T has a fixed point with derivative equal to one.

Anosov diffeomorphisms and coupling

We prove the existence of a Sinai–Ruelle–Bowen (SRB) measure and the exponential decay of correlations for smooth observables for mixing Anosov C^{1+\alpha} diffeormorphisms on a d-dimensional (d \geq 2) Riemannian manifold. The novelty lies in the very simple method of proof. We construct explicitly a coupling between two initial densities so that under the action of the diffeomorphism, both components get closer and closer. The speed of this convergence can be explicitly estimated and is directly related to the speed of decay of the correlations. The existence of the SRB measure and its properties readily follow.

Gibbs states on the symbolic space over an infinite alphabet

We consider subshifts of finite type on the symbolic space generated by incidence matrices over a countably infinite alphabet. We extend the definition of topological pressure to this context and, as our main result, we construct a new class of Gibbs states of Holder continuous potentials on these symbol spaces. We establish some basic stochastic properties of these Gibbs states: exponential decay of correlations, central limit theorem and an a.s. invariance principle. This is accomplished via detailed studies of the associated Perron-Frobenius operator and its conjugate operator.

Spectrum of the Frobenius–Perron operator for systems with stochastic perturbation

We analyze the problem of evolution in a system with stochastic perturbation and point out that analytic properties of the noise present in the system might determine spectral properties of the evolution operator (Frobenius-Perron operator). We also propose a method for approximation of the spectrum and eigenvectors of the FP-operator by applying a suitable noise. Moreover we demonstrate that the eigenvalues of the FP-operator located outside the essential spectrum are robust not only against local perturbation but also against the non local perturbation and that these stable eigenvalues have a direct physical meaning: they determine the rate of the exponential decay of correlation in the system.

Statistical properties of hyperbolic systems with tangential singularities

We consider piecewise smooth, uniformly hyperbolic systems on aRiemannian manifold, where we allow the angle between the unstabledirection and the singularity manifolds to vanish. Under naturalassumptions we prove that such systems exhibit exponential decay ofcorrelations and satisfy a central limit theorem with respect to amixing Sinai-Ruelle-Bowen-measure. These results have been shown previously forsystems in which the angle between the singularity manifold and the unstabledirection is uniformly bounded away from zero.

Coupled map lattices with asynchronous updatings

We consider on M=(S 1) Z d a family of continuous local updatings, of finite range type or Lipschitz-continuous in all coordinates with summable Lipschitz-constants. We show that the infinite-dimensional dynamical system with independent identically Poisson-distributed times for the individual updatings is well-defined. We then consider the setting of analytically coupled uniformly expanding, analytic circle maps with weak, exponentially decaying interaction. We define transfer operators for the infinite-dimensional system, acting on Banach-spaces that include measures whose finite-dimensional marginals have analytic, exponentially bounded densities. We prove existence and uniqueness (in the considered Banach-space) of a probability measure and its exponential decay of correlations.

Spectrum of the Ruelle operator and exponential decay of correlations for open billiard flows

The paper deals with the billiard flow in the exterior of several strictly convex disjoint domains in the plane with smooth boundaries satisfying an additional (visibility) condition. Using a modification of the technique of Dolgopyat, we get spectral estimates for the Ruelle operator related to a Markov family for the nonwandering (trapping) set of the flow similar to those of Dolgopyat in the case of transitive Anosov flows on compact manifolds with smooth jointly non-integrable horocycle foliations. As a consequence, we get exponential decay of correlation for Hölder continuous potentials on the nonwandering set. Combining the spectral estimate for the Ruelle operator with an argument of Pollicott and Sharp, we also derive the existence of a meromorphic continuation of the dynamical zeta function of the billiard flow to a half-plane Re( s ) < h T - ε, where h T is the topological entropy of the billiard flow, and an asymptotic formula with an error term for the number π(λ) of closed orbits of least period λ > 0.

Strange Attractors with One Direction of Instability

We give simple conditions that guarantee, for strongly dissipative maps, the existence of strange attractors with a single direction of instability and certain controlled behaviors. Only the d= 2 case is treated in this paper, although our approach is by no means limited to two phase-dimensions. We develop a dynamical picture for the attractors in this class, proving they have many of the statistical properties associated with chaos: positive Lyapunov exponents, existence of SRB measures, and exponential decay of correlations. Other results include the geometry of fractal critical sets, nonuniform hyperbolic behavior, symbolic coding of orbits, and formulas for topological entropy.

Statistics of closest return for some non-uniformly hyperbolic systems

For non-uniformly hyperbolic maps of the interval with exponential decay of correlations we prove that the law of closest return to a given point when suitably normalized is almost surely asymptotically exponential. A similar result holds when the reference point is the initial point of the trajectory. We use the framework for non-uniformly hyperbolic dynamical systems developed by L. S. Young.

The existence of Burnett coefficients in the periodic Lorentz gas

The linear super-Burnett coefficient gives corrections to the diffusion equation in the form of higher derivatives of the density. Like the diffusion coefficient, it can be expressed in terms of integrals of correlation functions, but involving four different times. The power-law decay of correlations in real gases (with many moving particles) and the random Lorentz gas (with one moving particle and fixed scatterers) are expected to cause the super-Burnett coefficient to diverge in most cases. Here, we show that the expression for the super-Burnett coefficient of the periodic Lorentz gas converges as a result of exponential decay of correlations and a nontrivial cancellation of divergent contributions.

Transfer operators for coupled analytic maps

We consider analytically coupled circle maps (uniformly expanding and analytic) on the ${\mathbb Z}^d$-lattice with exponentially decaying interaction. We introduce Banach spaces for the infinite-dimensional system that include measures whose finite-dimensional marginals have analytic, exponentially bounded densities. Using residue calculus and ‘cluster expansion’-like techniques we define transfer operators on these Banach spaces. We get a unique (in the considered Banach spaces) probability measure that exhibits exponential decay of correlations.

Magnetic critical behavior of two-dimensional random-bond Potts ferromagnets in confined geometries.

We present a numerical study of two-dimensional random-bond Potts ferromagnets. The model is studied both below and above the critical value Qc=4, which discriminates between second- and first-order transitions in the pure system. Two geometries are considered, namely cylinders and square-shaped systems, and the critical behavior is investigated through conformal invariance techniques that were recently shown to be valid, even in the randomness-induced second-order phase transition regime Q>4. In the cylinder geometry, connectivity transfer matrix calculations provide a simple test to find the range of disorder amplitudes that is characteristic of the disordered fixed point. The scaling dimensions then follow from the exponential decay of correlations along the strip. Monte Carlo simulations of spin systems on the other hand are generally performed on systems of rectangular shape on the square lattice, but the data are then perturbed by strong surface effects. The conformal mapping of a semi-infinite system inside a square enables us to take into account boundary effects explicitly and leads to an accurate determination of the scaling dimensions. The techniques are applied to different values of Q in the range 3-64.

Rates of mixing for potentials of summable variation

It is well known that for subshifts of finite type and equilibrium measures associated to Hölder potentials we have exponential decay of correlations. In this article we derive explicit rates of mixing for equilibrium states associated to more general potentials.

Spatial variability of tidal gravity anomalies and its correlation with the effective elastic thickness of the lithosphere

Associations of the Earth tidal gravity response to physical properties of the lithosphere have been attempted at least for the last four decades. Although experimental data suggest this association, rigorous models have not yet been proposed. In this work, statistical tests are performed on the available World Gravity Earth Tides data set. Autocorrelation analysis shows that the M2 tidal gravity anomalies (TGAs) are significantly correlated up to a distance of about 500 km, with an approximately exponential correlation decay. The analysis of the latitudinal dependence of the anomalies shows that the anomaly variance, estimated inside of different latitude bands, follows a cos 4 ϕ curve within the ±45° latitude interval and defines the noise level for the M2 gravity anomaly data set. The regression analysis between M2 TGA and the lithosphere effective elastic thickness (EET) estimates shows that these quantities are significantly correlated, with a correlation coefficient of −0.82. The wide range of TGA and EET values, combined with a good global distribution of the data used in the regression analysis, makes the regression equation suitable to be used as a predictor for EET values in areas where M2 TGA data exist and meet the required quality criteria.

Decay of Correlations and Dispersing Billiards

We give a rigorous proof of exponential decay of correlations for all major classes of planar dispersing billiards: periodic Lorentz gases with and without horizon and dispersing billiard tables with corner points

Statistical properties of piecewise smooth hyperbolic systems in high dimensions

We study smooth hyperbolic systems with singularities and their SRBmeasures. Here we assume that the singularities are submanifolds, the hyperbolicityis uniform aside from the singularities, and one-sided derivatives exist on the singularities. We prove that the ergodic SRB measures exist, are finitely many, andmixing SRB measures enjoy exponential decay of correlations and a central limittheorem. These properties have been proved previously only for two-dimensionalsystems.

A note on the decay of correlations under δ-pinning

We prove that for a class of massless $\nabla\phi$ interface models on $\Ztwo$ an introduction of an arbitrary small pinning self-potential leads to exponential decay of correlation, or, in other words, to creation of mass.