Weakly Mixing 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.

Tied-down occupation times of infinite ergodic transformations

We prove distributional limit theorems (conditional and integrated) for the occupation times of certain weakly mixing, pointwise dual ergodic transformations at “tied-down” times immediately after “excursions”. The limiting random variables include the local times of q-stable Lévy-bridges (1 < q ≤ 2) and the transformations involved exhibit “tied-down renewal mixing” properties which refine rational weak mixing. Periodic local limit theorems for Gibbs—Markov and AFU maps are also established.

On mean sensitive tuples

In this paper we introduce and study several mean forms of sensitive tuples. It is shown that the topological or measure-theoretical entropy tuples are correspondingly mean sensitive tuples under certain conditions (minimal in the topological setting or ergodic in the measure-theoretical setting). Characterizations of the question when every non-diagonal tuple is mean sensitive are presented. Among other results we show that under minimality assumption a topological dynamical system is weakly mixing if and only if every non-diagonal tuple is mean sensitive and so as a consequence every minimal weakly mixing topological dynamical system is mean n -sensitive for any integer n ≥ 2 . Moreover, the notion of weakly sensitive in the mean tuple is introduced and it turns out that this property has some special lift property. As an application we obtain that the maximal mean equicontinuous factor for any topological dynamical system can be induced by the smallest closed invariant equivalence relation containing all weakly sensitive in the mean pairs.

On transitive and chaotic dynamics of linear semiflows

In this paper, we study the underlying relationship between transitivity and chaos for linear semiflows, which provides a better understanding of chaotic dynamics in topological spaces. We show that syndetically transitive linear semiflows with abelian acting semigroups are thickly syndetically-transitive and weakly mixing. As a result, Devaney chaos implies the weak mixing for linear semiflows with abelian acting semigroups. Besides, we give the characterizations for multi-dimensional Li-Yorke chaotic linear semiflows and show that if a linear semiflow (S,X) on a Polish space satisfies one of the following conditions: (i) S is discrete; (ii) dimX=∞; (iii) S is abelian, then Devaney chaos implies multi-dimensional Li-Yorke chaos. This answers a latest question posed in [12] (Question 2.19) to some extent. We also show that any topologically transitive linear semiflow (S,X) on a Polish space X with a translation-invariant metric appears to be densely Li-Yorke chaotic if IntX(Bx)=∅ for any compact subset B of S and any transitive point x∈X. Moreover, if the acting semigroup S is abelian, we present a weaker condition for the densely multi-dimensional Li-Yorke chaos.

Further discussion about transitivity and mixing of continuous maps on compact metric spaces

Let (X, d) be a compact metric space and f : X → X be a continuous map. This work investigates the relationships among various forms of transitivity, such as syndetically transitive, totally transitive, and strongly transitive. Some necessary and sufficient conditions, or sufficient conditions for f to be totally (respectively, strongly and syndetically) transitive, weakly mixing, mixing, and minimal, are obtained. Then, let (K(X),f̄) be the natural extension of (X, f), where K(X) is the family of all nonempty compact subsets of X. The above properties of f̄ are further studied. It is proved that syndetical transitivity, total transitivity, and weak mixing of f̄ are equivalent, and if f̄ is strongly transitive, then f̄ is mixing.

Dynamics of Weakly Mixing Nonautonomous Systems

For a commutative nonautonomous dynamical system we show that topological transitivity of the nonautonomous system induced on probability measures (hyperspaces) is equivalent to the weak mixing of the induced systems. Several counter examples are given for the results which are true in autonomous but need not be true in nonautonomous systems. Wherever possible sufficient conditions are obtained for the results to hold true. For a commutative periodic nonautonomous system on intervals, it is proved that weak mixing implies Devaney chaos. Given a periodic nonautonomous system, it is shown that sensitivity is equivalent to some stronger forms of sensitivity on a closed unit interval.

Weak mixing for nonsingular Bernoulli actions of countable amenable groups

Let G G be an amenable discrete countable infinite group, let A A be a finite set, and let ( μ g ) g ∈ G (\mu _g)_{g\in G} be a family of probability measures on A A such that inf g ∈ G min a ∈ A μ g ( a ) &gt; 0 \inf _{g\in G}\min _{a\in A}\mu _g(a)&gt;0 . It is shown (among other results) that if the Bernoulli shiftwise action of G G on the infinite product space ⨂ g ∈ G ( A , μ g ) \bigotimes _{g\in G}(A,\mu _g) is nonsingular and conservative, then it is weakly mixing. This answers in the positive a question by Z. Kosloff, who proved recently that the conservative Bernoulli Z d \mathbb {Z}^d -actions are ergodic. As a byproduct, we prove a weak version of the pointwise ratio ergodic theorem for nonsingular actions of G G .

Ergodic properties of the ideal gas model for infinite billiards

In this paper we study ergodic properties of the Poisson suspension (the ideal gas model) of the billiard flow (bt)t∈R on the plane with a Λ-periodic pattern (Λ⊂R2 is a lattice) of polygonal scatterers. We prove that if the billiard table is additionally rational then for a.e. direction θ∈S1 the Poisson suspension of the directional billiard flow (btθ)t∈R is weakly mixing. This gives the weak mixing of the Poisson suspension of (bt)t∈R. We also show that for a certain class of such rational billiards (including the periodic version of the classical wind-tree model) the Poisson suspension of (btθ)t∈R is not mixing for a.e. θ∈S1.

Weak mixing in switched systems

Given a switched system, we introduce weakly mixing sets of type 1, 2 and Xiong chaotic sets of type 1, 2 with respect to a given set and show that they are equivalent, respectively.

Quenched invariance principles for the discrete Fourier transforms of a stationary process

In this paper, we study the asymptotic behavior of the normalized cadlag functions generated by the discrete Fourier transforms of a stationary centered square-integrable process, started at a point. We prove that the quenched invariance principle holds for averaged frequencies under no assumption other than ergodicity, and that this result holds also for almost every fixed frequency under a certain generalization of the Hannan condition and a certain rotated form of the Maxwell and Woodroofe condition which, under a condition of weak dependence that we specify, is guaranteed for a.e. frequency. If the process is in particular weakly mixing, our results describe the asymptotic distributions of the normalized discrete Fourier transforms at every frequency other than $0$ and $\pi$ under the generalized Hannan condition. We prove also that under a certain regularity hypothesis the conditional centering is irrelevant for averaged frequencies, and that the same holds for a given fixed frequency under the rotated Maxwell and Woodroofe condition but not necessarily under the generalized Hannan condition. In particular, this implies that the hypothesis of regularity is not sufficient for functional convergence without random centering at a.e. fixed frequency. The proofs are based on martingale approximations and combine results from Ergodic theory of recent and classical origin with approximation results by contemporary authors and with some facts from Harmonic Analysis and Functional Analysis.

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.

Chaos, Mixing, Weakly Mixing and Exactness

In this paper, new definitions of chaos, exact chaos, mixing chaos, and weak mixing chaos called θ-chaos, θ exact chaos, θ-mixing chaos are introduced and extended to topological spaces. Our purpose is to investigate other types of transitivity, chaos and mixing, because when we confirm not existence of θ-transitivity we confirm not existence of other types of transitive functions and its absence cannot find other types of transitivity. So the author must confirm the existence of this type of transitivity. We have proved that these chaotic definitions are all preserved under θ r-conjugation.

Quantum Groups, Property (T), and Weak Mixing

For second countable discrete quantum groups, and more generally second countable locally compact quantum groups with trivial scaling group, we show that property (T) is equivalent to every weakly mixing unitary representation not having almost invariant vectors. This is a generalization of a theorem of Bekka and Valette from the group setting and was previously established in the case of low dual by Daws, Skalsi, and Viselter. Our approach uses spectral techniques and is completely different from those of Bekka--Valette and Daws--Skalski--Viselter. By a separate argument we furthermore extend the result to second countable nonunimodular locally compact quantum groups, which are shown in particular not to have property (T), generalizing a theorem of Fima from the discrete setting. We also obtain quantum group versions of characterizations of property (T) of Kerr and Pichot in terms of the Baire category theory of weak mixing representations and of Connes and Weiss in term of the prevalence of strongly ergodic actions.

Almost sure convergence of the multiple ergodic average for certain weakly mixing systems

The family of pairwise independently determined (PID) systems, i.e. those for which the independent joining is the only self joining with independent 2-marginals, is a class of systems for which the long standing open question by Rokhlin, of whether mixing implies mixing of all orders, has a positive answer. We show that in the class of weakly mixing PID one finds a positive answer for another long-standing open problem, whether the multiple ergodic averages \begin{equation*} \frac 1 N\sum_{n=0}^{N-1}f_1(T^nx)\cdots f_d(T^{dn}x), \quad N\to \infty, \end{equation*} almost surely converge.

Conditions for rational weak mixing

We exhibit rationally ergodic, spectrally weakly mixing measure preserving transformations which are not subsequence rationally weakly mixing and give a condition for smoothness of renewal sequences.

Weak mixing for locally compact quantum groups

We generalize the notion of weakly mixing unitary representations to locally compact quantum groups, introducing suitable extensions of all standard characterizations of weak mixing to this setting. These results are used to complement the non-commutative Jacobs–de Leeuw–Glicksberg splitting theorem of Runde and the author [Ergodic theory for quantum semigroups. J. Lond. Math. Soc. (2) 89(3) (2014), 941–959]. Furthermore, a relation between mixing and weak mixing of state-preserving actions of discrete quantum groups and the properties of certain inclusions of von Neumann algebras, which is known for discrete groups, is demonstrated.

Invariant scrambled sets, uniform rigidity and weak mixing

We show that for a non-trivial transitive dynamical system, it has a dense Mycielski invariant strongly scrambled set if and only if it has a fixed point, and it has a dense Mycielski invariant $\delta$-scrambled set for some $\delta>0$ if and only if it has a fixed point and not uniformly rigid. We also provide two methods for the construction of completely scrambled systems which are weakly mixing, proximal and uniformly rigid.

On recurrence over subsets and weak mixing

We study properties of weakly mixing sets (of order n) in relation to proximality, sensitivity, scrambled tuples, Xiong chaotic sets and independent sets. Our main emphasis is on the structure of the set of transfer times N.U A; V/ between open sets U and V , both intersecting a weakly mixing set A. We find several conditions on properties of the set A that are equivalent to weak mixing. We also prove that on topological graphs weakly mixing sets of order 2 can be approximated arbitrarily closely by a weakly mixing set of all orders. This property is known to hold on the unit interval but is not true in general (there are systems with weakly mixing sets of order n but not nC 1).

Ratner’s property and mild mixing for smooth flows on surfaces

Let${\mathcal{T}}=(T_{t}^{f})_{t\in \mathbb{R}}$be a special flow built over an IET$T:\mathbb{T}\rightarrow \mathbb{T}$of bounded type, under a roof function$f$with symmetric logarithmic singularities at a subset of discontinuities of$T$. We show that${\mathcal{T}}$satisfies the so-called switchable Ratner’s property which was introduced in Fayad and Kanigowski [On multiple mixing for a class of conservative surface flows.Invent. Math.to appear]. A consequence of this fact is that such flows are mildly mixing (before, they were only known to be weakly mixing [Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations.J. Mod. Dynam.3(2009), 35–49] and not mixing [Ulcigrai. Absence of mixing in area-preserving flows on surfaces.Ann. of Math.(2)173(2011), 1743–1778]). Thus, on each compact, connected, orientable surface of genus greater than one there exist flows that are mildly mixing and not mixing.

Tower multiplexing and slow weak mixing

A technique is presented for multiplexing two ergodic measure preserving transformations together to derive a third limiting transformation. This technique is used to settle a question regarding rigidity of weak mixing transformations. Namely, given any rigidity sequence for an ergodic measure preserving transformation, there exists a weak mixing transformation which is rigid along the same sequence. This establishes a wide range of rigidity sequences for weakly mixing dynamical systems.