Rank-one weak mixing for nonsingular transformations

We construct rank-one infinite measure preserving transformations satisfying each of the following dynamical properties: (1) ContinuousL ^∞ spectrum, conservativek-fold cartesian products but nonergodic cartesian square; (2) ergodick-fold cartesian products; (3) nonconservative cartesian square. We show how to modify the construction of (1) to obtain type III^λ transformations with similar properties. Supported in part by National Science Foundation grant DMS-9214077.     

Topological weak mixing and quasi-Bohr systems

A minimal dynamical system (X, T) is called quasi-Bohr if it is a non-trivial equicontinuous extension of a proximal system. We show that if (X, T) is a minimal dynamical system which is not weakly mixing then some minimal proximal extension of (X, T) admits a nontrivial quasi-Bohr factor. (In terms of Ellis groups the corresponding statement is:AG′=G implies weak mixing.) The converse does not hold. In fact there are nontrivial quasi-Bohr systems which are weakly mixing of all orders. Our main tool in the proof is a theorem, of independent interest, which enhances the general structure theorem for minimal systems.     

On weak mixing in lattice models

Summary. For lattice models on ℤ ^d , weak mixing is the property that the influence of the boundary condition on a finite decays exponentially with distance from that region. For a wide class of models on ℤ², including all finite range models, we show that weak mixing is a consequence of Gibbs uniqueness, exponential decay of an appropriate form of connectivity, and a natural coupling property. In particular, on ℤ², the Fortuin-Kasteleyn random cluster model is weak mixing whenever uniqueness holds and the connectivity decays exponentially, and the q-state Potts model above the critical temperature is weak mixing whenever correlations decay exponentially, a hypothesis satisfied if q is sufficiently large. Ratio weak mixing is the property that uniformly over events A and B occurring on subsets Λ and Γ, respectively, of the lattice, |P(A∩B)/P(A)P(B)−1| decreases exponentially in the distance between Λ and Γ. We show that under mild hypotheses, for example finite range, weak mixing implies ratio weak mixing. Received: 27 August 1996 / In revised form: 15 August 1997     

Exponential weak Bernoulli mixing for collet-eckmann maps

We prove exponential weak Bernoulli mixing for invariant measures of certain piecewise monotone interval maps studied in [BK] and [KN]. In particular we prove this for unimodal maps with negative Schwarzian derivative satisfying lim , wherec is the unique critical point ofT.     

Characterization of the generic unfolding of a weak focus

In this paper we prove that the orbital class of a generic real analytic family unfolding a weak focus is determined by the conjugacy class of its Poincaré monodromy and vice versa. We solve the embedding problem by means of quasiconformal surgery on the formal normal form. The surgery yields an integrable abstract almost complex 2-manifold equipped with an elliptic foliation. The monodromy of the latter coincides with the second iterate of a germ of prescribed family of real analytic diffeomorphisms undergoing a flip bifurcation.     

Characterizing mixing, weak mixing and transitivity of induced hyperspace dynamical systems

Hyperspace dynamical system (E²,f²) induced by a given dynamical system (E,f) has been recently investigated regarding topological mixing, weak mixing and transitivity that characterize orbit structure. However, the Vietoris topology on E² employed in these studies is non-metrizable when E is not compact metrizable, e.g., E=Rn. Consequently, metric related dynamical concepts of (E²,f²) such as sensitivity on initial conditions and metric-based entropy, could not even be defined. Moreover, a condition on (E²,f²) equivalent to the transitivity of (E,f) has not been established in the literature. On the other hand, Hausdorff locally compact second countable spaces (HLCSC) appear naturally in dynamics. When E is HLCSC, the hit-or-miss topology on E² is again HLCSC, thus metrizable. In this paper, the concepts of co-compact mixing, co-compact weak mixing and co-compact transitivity are introduced for dynamical systems. For any HLCSC system (E,f), these three conditions on (E,f) are respectively equivalent to mixing, weak mixing and transitivity on (E²,f²) (hit-or-miss topology equipped). Other noticeable properties of co-compact mixing, co-compact weak mixing and co-compact transitivity such as invariants for topological conjugacy, as well as their relations to mixing, weak mixing and transitivity, are also explored.     

Relative weak injectivity of operator system pairs

The concept of a relatively weakly injective pair of operator systems is introduced and studied in this paper, motivated by relative weak injectivity in the C*-algebra category. E. Kirchberg [11] proved that the C^?${\mathrm{C}}^{?}$-algebra C^?(_F∞)${\mathrm{C}}^{?}({\mathbb{F}}_{\infty })$ of the free group _F∞${\mathbb{F}}_{\infty }$ on countably many generators characterises relative weak injectivity for pairs of C^?${\mathrm{C}}^{?}$-algebras by means of the maximal tensor product. One of the main results of this paper shows that C^?(_F∞)${\mathrm{C}}^{?}({\mathbb{F}}_{\infty })$ also characterises relative weak injectivity in the operator system category. A key tool is the theory of operator system tensor products  and .     

Solvability of linear equations within weak mixing sets

We introduce a new class of “random“ subsets of natural numbers, WM sets. This class contains normal sets (sets whose characteristic function is a normal binary sequence). We establish necessary and sufficient conditions for solvability of systems of linear equations within every WM set and within every normal set. We also show that any partition-regular system of linear equations with integer coefficients is solvable in any WM set.     

On the strong mixing and weak Bernoulli conditions

Summary A strictly stationary random sequence is constructed which has exactly two states and satisfies the strong mixing condition but not the weak Bernoulli condition.This research was partially supported by the National Science Foundation and the Office of Naval Research     

Funny rank-one weak mixing for nonsingular Abelian actions

We construct funny rank-one infinite measure preserving free actionsT of a countable Abelian groupG satisfying each of the following properties: (1)T _g1×…×T_gk is ergodic for each finite sequenceg ₁,…,g _k ofG-elements of infinite order, (2)T×T is nonconservative, (3)T×T is nonergodic but allk-fold Cartesian products are conservative, and theL ^∞-spectrum ofT is trivial, (4) for eachg of infinite order, allk-fold Cartesian products ofT _g are ergodic, butT _2g×T_g is nonconservative. A topological version of this theorem holds. Moreover, given an AT-flowW, we construct nonsingularG-actionsT with similar properties and such that the associated flow ofT isW. Orbit theory is used in an essential way here. The work was supported in part by INTAS 97-1843 and CRDF grant UM1-2092.     

Spectrally determined growth is generic

Let be the infinitesimal generator of a -semigroup of operators in a Hilbert space. We consider the class of operators , where is bounded. It is proved that the spectrum of determines the growth of the associated semigroup for ``most" operators (in the sense of Baire category).

     

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 δ-scrambled set for some δ > 0 if and only if it has a fixed point and is not uniformly rigid. We also provide two methods for the construction of completely scrambled systems which are weakly mixing, proximal and uniformly rigid.     

Relative entropy and mixing properties of infinite dimensional diffusions

Summary. Let η be a diffusion process taking values on the infinite dimensional space T ^Z , where T is the circle, and with components satisfying the equations dη _i =σ _i (η) dW _i +b _i (η) dt for some coefficients σ _i and b _i , i∈Z. Suppose we have an initial distribution μ and a sequence of times t _n →∞ such that lim _{n →∞}μS _tn =ν exists, where S _t is the semi-group of the process. We prove that if σ _i and b _i are bounded, of finite range, have uniformly bounded second order partial derivatives, and inf _{i ,η}σ _i (η)>0, then ν is invariant. Received: 12 September 1996 / In revised form: 10 November 1997     

A class of generalized Ornstein transformations with the weak mixing property

It is shown that there is a class of Ornstein transformations, distinct from the Ornstein mixing class, which are almost surely weak mixing     

Saari's conjecture is true for generic vector fields

The simplest non-collision solutions of the -body problem are the ``relative equilibria', in which each body follows a circular orbit around the centre of mass and the shape formed by the bodies is constant. It is easy to see that the moment of inertia of such a solution is constant. In 1970, D. Saari conjectured that the converse is also true for the planar Newtonian -body problem: relative equilibria are the only constant-inertia solutions. A computer-assisted proof for the 3-body case was recently given by R. Moeckel, Trans. Amer. Math. Soc. (2005). We present a different kind of answer: proofs that several generalisations of Saari's conjecture are generically true. Our main tool is jet transversality, including a new version suitable for the study of generic potential functions.

     

A weak invariance principle for self-normalized products of sums of mixing sequences

Let variables in the {X, Xn, n ≥ 1} be a sequence of strictly stationary φ-mixing positive random domain of attraction of the normal law. Under some suitable conditions the principle for self-normalized products of partial sums is obtained.     

The toda flow on a generic orbit is integrable

We construct action-angle variables for the Toda flow on generic orbits of the coadjoint action of the lower triangular group on its dual Lie algebra.