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Last years, the search for a good theory of quantum dynamical entropy has been very much intensified. This is not only due to its usefulness in quantum probability but mainly because it is a very promising tool for the theory of quantum chaos. Nowadays, there are several constructions which try to fulfill this need, some of which are more mathematically inspired such as CNT (Connes, Narnhofer, Thirring), and the one proposed by Voiculescu, others are more inspired by physics such as ALF (Alicki,...
The aim of this paper is to compare various criteria leading to the central limit theorem and the weak invariance principle. These criteria are the martingale-coboundary decomposition developed by Gordin in Dokl. Akad. Nauk SSSR188 (1969), the projective criterion introduced by Dedecker in Probab. Theory Related Fields110 (1998), which was subsequently improved by Dedecker and Rio in Ann. Inst. H. Poincaré Probab. Statist.36 (2000) and the condition introduced by Maxwell and Woodroofe in Ann. Probab.28...
We compare the Hausdorff measures and dimensions with respect to the Euclidean and Heisenberg metrics on the first Heisenberg group. The result is a dimension jump described by two inequalities. The sharpness of our estimates is shown by examples. Moreover a comparison between Euclidean and H-rectifiability is given.
Let X be a Polish space, and let C₀ and C₁ be disjoint coanalytic subsets of X. The pair (C₀,C₁) is said to be complete if for every pair (D₀,D₁) of disjoint coanalytic subsets of there exists a continuous function such that and . We give several explicit examples of complete pairs of coanalytic sets.
The existence of non-Bernoullian actions with completely positive entropy is proved for a class of countable amenable groups which includes, in particular, a class of Abelian groups and groups with non-trivial finite subgroups. For this purpose, we apply a reverse version of the Rudolph-Weiss theorem.
Assume that no cardinal κ < 2ω is quasi-measurable (κ is quasi-measurable if there exists a κ-additive ideal of subsets of κ such that the Boolean algebra P(κ)/ satisfies c.c.c.). We show that for a metrizable separable space X and a proper c.c.c. σ-ideal II of subsets of X that has a Borel base, each point-finite cover ⊆
of X contains uncountably many pairwise disjoint subfamilies , with
-Bernstein unions ∪ (a subset A ⊆ X is
-Bernstein if A and X A meet each Borel
-positive subset...
Nous donnons, pour chaque niveau de complexité Γ, une caractérisation du type "test d'Hurewicz" des boréliens d'un produit de deux espaces polonais ayant toutes leurs coupes dénombrables ne pouvant pas être rendus Γ par changement des deux topologies polonaises.
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