Weak Mixing Properties Research Articles

The weak mixing property on negatively curved manifolds

Given a complete manifold of negative curvature, we show that weak mixing is a generic property in the set of all probability measures invariant by the geodesic flow, as soon as the flow is topologically weakly mixing in restriction to its non-wandering set.

Transitivity and Shadowing Properties of Nonautonomous Discrete Dynamical Systems

This paper proves that some shadowing properties are sufficient conditions for being transitive or point-transitive for a nonautonomous discrete dynamical system. Moreover, considering weak mixing property and transitivity via Furstenberg family, this paper reveals the relationship for transitivity and mixing between [Formula: see text]-periodic systems and their induced autonomous discrete dynamical systems.

Eigenvalues of autocovariance matrix: A practical method to identify the Koopman eigenfrequencies.

To infer eigenvalues of the infinite-dimensional Koopman operator, we study the leading eigenvalues of the autocovariance matrix associated with a given observable of a dynamical system. For any observable f for which all the time-delayed autocovariance exist, we construct a Hilbert space H_{f} and a Koopman-like operator K that acts on H_{f}. We prove that the leading eigenvalues of the autocovariance matrix has one-to-one correspondence with the energy of f that is represented by the eigenvectors of K. The proof is associated to several representation theorems of isometric operators on a Hilbert space, and the weak-mixing property of the observables represented by the continuous spectrum. We also provide an alternative proof of the weakly mixing property. When f is an observable of an ergodic dynamical system which has a finite invariant measure μ,H_{f} coincides with closure in L^{2}(X,dμ) of Krylov subspace generated by f, and K coincides with the classical Koopman operator. The main theorem sheds light to the theoretical foundation of several semi-empirical methods, including singular spectrum analysis (SSA), data-adaptive harmonic analysis (DAHD), Hankel DMD, and Hankel alternative view of Koopman analysis (HAVOK). It shows that, when the system is ergodic and has finite invariant measure, the leading temporal empirical orthogonal functions indeed correspond to the Koopman eigenfrequencies. A theorem-based practical methodology is then proposed to identify the eigenfrequencies of K from a given time series. It builds on the fact that the convergence of the renormalized eigenvalues of the Gram matrix is a necessary and sufficient condition for the existence of K-eigenfrequencies. Numerical illustrating results on simple low dimensional systems and real interpolated ocean sea-surface height data are presented and discussed.

High protein and gliadin content improves tortilla quality of a weak gluten wheat

Flatbreads, like tortillas, are specialty breads that require extensible dough to reduce shrinking and sufficient protein content (10–12%, db) to maintain flexibility in storage, which are generally contradictory properties in wheat flour. This study compared wheat varieties bred for differing properties (high protein content, ruminant grazing quality, specialty breads) to identify potential desirable traits for tortillas and other flatbreads. Kernel characteristics were analyzed by SKCS, and the mixing properties were assessed with mixograph. Wheat flour proteins were profiled by SE-HPLC and RP-HPLC. Tortillas were produced and analyzed for quality throughout storage. Flour with a high protein content (14.0%, db), high concentration of gliadin relative to glutenin proteins (monomeric:polymeric proteins = 1.82), and relatively weak mixing properties (short mixograph time, <3 min) made excellent tortillas with large diameter and long shelf life. The relatively high gliadin concentration imparted dough extensibility and the high protein content allowed for flexibility throughout storage, though the same flour was previously found to produce poor bread quality. Assessment of hard wheat quality for applications beyond loaf bread (e.g., flatbreads) could allow for a wider range of “clean-label” flours.

Weak Mixing and Analyticity of the Pressure in the Ising Model

We prove that the pressure (or free energy) of the finite range ferromagnetic Ising model on $\mathbb{Z}^d$ is analytic as a function of both the inverse temperature $\beta$ and the magnetic field $h$ whenever the model has the exponential weak mixing property. We also prove the exponential weak mixing property whenever $h\neq 0$. Together with known results on the regime $h=0,\beta<\beta_c$, this implies both analyticity and weak mixing in all the domain of $(\beta,h)$ outside of the transition line $[\beta_c,\infty)\times \{0\}$. The proof of analyticity uses a graphical representation of the Glauber dynamic due to Schonmann and cluster expansion. The proof of weak mixing uses the random cluster representation.

Relative weak mixing is generic

A classical result of Halmos asserts that among measure preserving transformations the weak mixing property is generic. We extend Halmos’ result to the collection of ergodic extensions of a fixed, but arbitrary, aperiodic transformation T0. We then use a result of Ornstein and Weiss to extend this relative theorem to the general (countable) amenable group.

Weak mixing properties of interval exchange transformations and translation flows

Let $d >1$. In this paper we show that for an irreducible permutation $\pi$ which is not a rotation, the set of $[\lambda]\in \mathbb{P}_+^{d-1}$ such that the interval exchange transformation $f([\lambda],\pi)$ is not weakly mixing does not have full Hausdorff dimension. We also obtain an analogous statement for translation flows. In particular, it strengthens the result of almost sure weak mixing proved by G. Forni and the first author. We adapt here the probabilistic argument developed in their paper in order to get some large deviation results. We then show how the latter can be converted into estimates on the Hausdorff dimension of the set of bad parameters in the context of fast decaying cocycles, following a strategy developed by V. Delecroix and the first author.

Set-Valued Chaos in Linear Dynamics

We study several notions of chaos for hyperspace dynamics associated to continuous linear operators. More precisely, we consider a continuous linear operator $$T : X \rightarrow X$$ on a topological vector space X, and the natural hyperspace extensions $$\overline{T}$$ and $$\widetilde{T}$$ of T to the spaces $$\mathcal {K}(X)$$ of compact subsets of X and $$\mathcal {C}(X)$$ of convex compact subsets of X, respectively, endowed with the Vietoris topology. We show that, when X is a complete locally convex space (respectively, a locally convex space), then Devaney chaos (respectively, topological ergodicity) is equivalent for the maps T, $$\overline{T}$$ and $$\widetilde{T}$$ . Also, under very general conditions, we obtain analogous equivalences for Li-Yorke chaos. Finally, some remarks concerning the topological transitivity and weak mixing properties are included, extending results in Banks (Chaos Solitons Fractals 25(3):681–685, 2005) and Peris (Chaos Solitons Fractals 26(1):19–23, 2005).

Mixing properties of modified additive generators

We develop amatrix-graph approach to estimating themixing properties of bijective shift registers over a set of binary vectors. Such shift registers generalize, on the one hand, the class of ciphers based on the Feistel network and, on the other hand, the class of transformations of additive generators (the additive generators are the base for the Fish, Pike, andMush algorithms). It is worth noting that the original schemes of additive generators are found insecure due to their weak mixing properties. The article contains the results of investigations for the mixing properties of modified additive generators. For the mixing directed graph of a modified additive generator, we define the sets of arcs and cycles, obtain primitivity conditions, and give a bound for the exponent. We show that, the determination of parameters for the modified additive generator allows us to achieve a full mixing in a number of iterations that is substantially less than the number of vertices in the mixing digraph.

Weak mixing properties for non-singular actions

For a general group$G$we consider various weak mixing properties of non-singular actions. In the case where the action is actually measure preserving all these properties coincide, and our purpose here is to check which implications persist in the non-singular case.

Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards

This text is an expanded version of the lecture notes of a minicourse (with the same title of this text) delivered by the authors in the Bedlewo school ``Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory'' (from 4 to 16 July, 2011). &nbsp In the first part of this text, i.e., from Sections 1 to 5, we discuss the Teichmuller and moduli space of translation surfaces, the Teichmuller flow and the $SL(2,\mathbb{R})$-action on these moduli spaces and the Kontsevich--Zorich cocycle over the Teichmuller geodesic flow. We sketch two applications of the ergodic properties of the Teichmuller flow and Kontsevich--Zorich cocycle, with respect to Masur--Veech measures, to the unique ergodicity, deviation of ergodic averages and weak mixing properties of typical interval exchange transformations and translation flows. These applications are based on the fundamental fact that the Teichmuller flow and the Kontsevich--Zorich cocycle work as renormalization dynamics for interval exchange transformations and translation flows. &nbsp In the second part, i.e., from Sections 6 to 9, we start by pointing out that it is interesting to study the ergodic properties of the Kontsevich--Zorich cocycle with respect to invariant measures other than the Masur--Veech ones, in view of potential applications to the investigation of billiards in rational polygons (for instance). We then study some examples of measures for which the ergodic properties of the Kontsevich--Zorich cocycle are very different from the case of Masur--Veech measures. Finally, we end these notes by constructing some examples of closed $SL(2,\mathbb{R})$-orbits such that the restriction of the Teichmuller flow to them has arbitrary small rate of exponential mixing, or, equivalently, the naturally associated unitary $SL(2,\mathbb{R})$-representation has arbitrarily small spectral gap (and in particular it has complementary series).

Mixing type theorems for one-parameter semigroups of operators

In this paper, we present some results concerning strong and weak mixing properties of continuous one-parameter semigroups of operators on Banach spaces. We study also some stability problems for continuous one-parameter semigroups on Hilbert spaces. Some related problems are also discussed.

ON THE METRIC THEORY OF CONTINUED FRACTIONS IN POSITIVE CHARACTERISTIC

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{F}_q$ be the finite field of $q$ elements. An analogue of the regular continued fraction expansion for an element $\alpha $ in the field of formal Laurent series over $\mathbb{F}_q$ is given uniquely by $$\begin{equation*} \alpha = A_0(\alpha )+\cfrac {1}{A_1(\alpha )+\cfrac {1}{A_2(\alpha )+\ddots }}, \end{equation*}$$ where $(A_n(\alpha ))_{n=0}^\infty $ is a sequence of polynomials with coefficients in $\mathbb{F}_q$ such that $\deg (A_n(\alpha ))\ge 1$ for all $n\ge 1.$ We first prove the exactness of the continued fraction map in positive characteristic. This fact implies a number of strictly weaker properties. Particularly, we then use the weak-mixing property and ergodicity to establish various metrical results regarding the averages of partial quotients of continued fraction expansions. A sample result that we prove is that if $(p_n)_{n=1}^\infty $ denotes the sequence of prime numbers, we have $$\begin{equation*} \lim _{n\to \infty }\frac {1}{n}\sum _{j=1}^n \deg (A_{p_j}(\alpha )) = \frac {q}{q-1} \end{equation*}$$ for almost every $\alpha $ with respect to Haar measure. In the case where the sequence $(p_n)_{n=1}^\infty $ is replaced by $(n)_{n=1}^\infty ,$ this result is due to V. Houndonougbo, V. Berthe and H. Nakada. Our proofs rely on pointwise subsequence and moving average ergodic theorems.

Mixing properties for nonautonomous linear dynamics and invariant sets

We study mixing properties (topological mixing and weak mixing of arbitrary order) for nonautonomous linear dynamical systems that are induced by the corresponding dynamics on certain invariant sets. The kinds of nonautonomous systems considered here can be defined using a sequence ( T i ) i ∈ N of linear operators T i : X → X on a topological vector space X such that there is an invariant set Y for which the dynamics restricted to Y satisfies a certain mixing property. We then obtain the corresponding mixing property on the closed linear span of Y . We also prove that the class of nonautonomous linear dynamical systems that are weakly mixing of order n contains strictly the corresponding class with the weak mixing property of order n + 1 .

On weak mixing, minimality and weak disjointness of all iterates

AbstractThis article addresses some open questions about the relations between the topological weak mixing property and the transitivity of the map f×f2×⋯×fm, where f:X→X is a topological dynamical system on a compact metric space. The theorem stating that a weakly mixing and strongly transitive system is Δ-transitive is extended to a non-invertible case with a simple proof. Two examples are constructed, answering the questions posed by Moothathu [Diagonal points having dense orbit. Colloq. Math. 120(1) (2010), 127–138]. The first one is a multi-transitive non-weakly mixing system, and the second one is a weakly mixing non-multi-transitive system. The examples are special spacing shifts. The latter shows that the assumption of minimality in the multiple recurrence theorem cannot be replaced by weak mixing.

Weakly mixing operators on topological vector spaces

In this paper we review some known characterizations of the weak mixing property for operators on topological vector spaces, extend some of them, and obtain new ones.

Rank-one flows of transformations with infinite ergodic index

Abstract. A rank-one infinite measure preserving flow T = (T t ) t∈R is constructedsuch that for each t 6= 0, the Cartesian powers of the transformation T t are all ergodic. 0. IntroductionIn 1963 Kakutani and Parry discovered an interesting phenomenon in the theoryof infinite measure preserving maps. They showed that for each p > 0, there exists atransformation whose p-th Cartesian power is ergodic but (p+1)-th one is not [KP].Since then a number of other examples of transformations with exotic (from thepoint of view of the classical “probability preserving” ergodic theory) weak mixingproperties were constructed. See surveys [Da2] and [DaS3] for a detail discussion onthat. In [Da1], [DaS1] these examples were extended to infinite measure preservingactions of discrete countable Abelian groups. Weak mixing properties of infinitemeasure preserving actions of continuous Abelian groups such as R and R d wereunder consideration in [I–W]. In particular, a rank-one flow (i.e. R-action) whoseCartesian square is ergodic was constructed there. A rank-one infinite measurepreserving flow T = (T

Construction, ergodicity and rate of convergence of N-particle Langevin dynamics with singular potentials

We construct N-particle Langevin dynamics in \({\mathbb{R}^d}\) or in a cuboid region with periodic boundary for a wide class of N-particle potentials Φ and initial distributions which are absolutely continuous w.r.t. Lebesgue measure. The potentials are in particular allowed to have singularities and discontinuous gradients (forces). An important point is to prove an L p -uniqueness of the associated non-symmetric, non-sectorial degenerate elliptic generator. Analyzing the associated functional analytic objects, we also give results on the long-time behaviour of the dynamics, when the invariant measure is finite: Firstly, we prove the weak mixing property whenever it makes sense (i.e. whenever {Φ < ∞} is connected). Secondly, for a still quite large class of potentials we also give a rate of convergence of time averages to equilibrium when starting in the equilibrium distribution. In particular, all results apply to N-particle systems with pair interactions of Lennard–Jones type.

Extensions of dynamical systems without increasing the entropy

Let X be a compact metric space. For a large class of dynamical properties Λ (we call them hypertransitive properties) we prove that any continuous non-minimal selfmap f of X with property Λ can be extended to a triangular, i.e. skew product, map F(x, y) = (f(x), g(x, y)) in the space X × [0, 1] in such a way that F has the property Λ and all the fibre maps of F are monotone (hence the topological entropy of F is the same as that of f). For topological transitivity such an extension theorem was proved by Alsedà, Kolyada, Llibre and Snoha in 1999. Our result is much more general—in fact, the family of hypertransitive properties includes, except of transitivity, also total transitivity, weak mixing, χ-Banach transitivity and many other dynamical properties. Further we prove a similar extension theorem also for strong mixing. Moreover, both for hypertransitive Λ and for strong mixing, if we additionally assume that f has dense periodic points then we can guarantee the existence of an extension F which has all the required properties and also dense periodic points. Finally we show that for maps which are topologically exact (i.e. locally eventually onto) an analogous theorem does not work. However, we give an example showing that at least some of the topologically exact maps can be extended to triangular topologically exact maps with monotone fibres.

WEAK DEPENDENCE: MODELS AND APPLICATIONS TO ECONOMETRICS

In this paper we discuss weak dependence and mixing properties of some popular models. We also develop some of their econometric applications. Autoregressive models, autoregressive conditional heteroskedasticity (ARCH) models, and bilinear models are widely used in econometrics. More generally, stationary Markov modeling is often used. Bernoulli shifts also generate many useful stationary sequences, such as autoregressive moving average (ARMA) or ARCH(∞) processes. For Volterra processes, mixing properties obtain given additional regularity assumptions on the distribution of the innovations.We recall associated probability limit theorems and investigate the nonparametric estimation of those sequences.We first thank the editor for the huge amount of additional editorial work provided for this review paper. The efficiency of the numerous referees was especially useful. The error pointed out in Hall and Horowitz (1996) was the origin of the present paper, and we thank the referees for asking for a more detailed treatment of a correct proof for this paper in Section 2.3. Also we thank Marc Henry and Rafal Wojakowski for a very careful rereading of the paper. An anonymous referee has been particularly helpful in the process of revision of the paper. The authors thank him for his numerous suggestions of improvement, including important results on negatively associated sequences and a thorough update in standard English.