François Parreau

Ergodic properties of real cocycles and pseudo-homogeneous Banach spaces

Given an irrational rotation, in the space of real bounded variation functions it is proved that there are ergodic cocycles whose small perturbations remain ergodic; in fact, the set of ergodic cocycles has nonempty dense interior. Given a pseudo-homogeneous Banach space and an irrational rotation, we study the set of elements satisfying the mean ergodic theorem. Once such a space is not homogeneous, we prove it is not reflexive and not separable. In “natural” cases, up to L1-cohomology, the only elements satisfying the mean ergodic theorem are those from the closure of trigonometric polynomials. For pseudo-homogeneous spaces admitting a Koksma’s inequality ergodicity of the corresponding cylinder flows can be deduced from spectral properties of some circle extensions. In particular this is the case of Lebesgue spectrum (in the orthocomplement of the space of eigenfunctions) for the circle extension. Introduction By T we will mean the circle group {z ∈ C : |z| = 1} which most of the time will be treated as [0, 1) with addition mod 1; λ will denote Lebesgue measure. Each f : T −→ R (naturally identified with f : R −→ R periodic of period 1), f ∈ L(T) has its Fourier expansion

Random ergodic theorems and real cocycles

AbstractWe study mean convergence of ergodic averages

Joining primeness and disjointness from infinitely divisible systems

\frac{1}{N}\sum\nolimits_{n = 0}^{N - 1} {f^\circ \tau ^{k_n (\omega )} ( * )}

Prime Poisson suspensions

associated to a measure-preserving transformation or flow τ along the random sequence of times κn(ω) given by the Birkhoff sums of a measurable functionF for an ergodic measure-preserving transformationT.We prove that the sequence (kn(ω)) is almost surely universally good for the mean ergodic theorem, i.e., that, for almost every, ω, the averages (*) converge for every choice of τ, if and only if the “cocycle”F satisfies a cohomological condition, equivalent to saying that the eigenvalue group of the “associated flow” ofF is countable. We show that this condition holds in many natural situations.When no assumption is made onF, the random sequence (kn(ω)) is almost surely universally good for the mean ergodic theorem on the class of mildly mixing transformations τ. However, for any aperiodic transformationT, we are able to construct an integrable functionF for which the sequence (kn(ω)) is not almost surely universally good for the class of weakly mixing transformations.

The interior of the Šilov boundary of M(G) is trivial

If Tc1, T,c2 are two Gaussian automorphisms, where cl and a2 are concentrated on independent sets, then we have a dichotomy: either they are spectrally disjoint or they have a common factor. As an application, we construct non-rigid automorphisms which are spectrally determined.

Gaussian automorphisms whose ergodic self-joinings are Gaussian

We show that ergodic dynamical systems generated by infinitely divisible stationary processes are disjoint in the sense of Furstenberg from distally simple systems and systems whose maximal spectral type is singular with respect to the convolution of any two continuous measures.

Non singular transformations and spectral analysis of measures

We study the problem of lifting various mixing properties from a base automorphism T ∈ Aut(X,B, μ) to skew products of the form Tφ,S , where φ : X → G is a cocycle with values in a locally compact Abelian group G, S = (Sg)g∈G is a measurable representation of G in Aut(Y, C, ν) and Tφ,S acts on the product space (X × Y,B ⊗ C, μ⊗ ν) by Tφ,S(x, y) = (Tx, Sφ(x)(y)). It is also shown that whenever T is ergodic (mildly mixing, mixing) but Tφ,S is not ergodic (is not mildly mixing, not mixing), then on a non-trivial factor A ⊂ C of S the corresponding Rokhlin cocycle x 7→ Sφ(x)|A is a coboundary (a quasi-coboundary). Introduction Given an ergodic automorphism T of a standard Borel space (X,B, μ) we can study various extensions T of it. Among such extensions a special role is played by so called compact group extensions or, more generally, isometric extensions (see [8], [11] and [30]). In particular, one can ask which ergodic properties of T are lifted by isometric extensions. The two papers1 by Dan Rudolph [25] and [26] are beautiful examples of the mechanism that once the extension enjoys some “minimal” ergodic property then it shares some strong ergodic properties assumed to hold for its base. By iterating the procedure of taking isometric extensions we can hence lift ergodic properties of T to weakly mixing distal extensions of it. ∗Research partly supported by Polish MNiSzW grant N N201 384834 In [25] it is proved that Bernoullicity is lifted whenever the extension is weakly mixing, while in [26] it is shown that mixing (multiple mixing) lifts whenever the extension is weakly mixing.

Ergodicité et pureté des produits de Riesz

We establish a necessary and sufficient condition for a Poisson suspension to be prime. The proof is based on the Fock space structure of the