Coalition convex preference orders are almost surely convex

AbstractLet E be a separable linear topological space, which admits a complete metric compatible with the topology, and (Ω, A, μ) a complete probability space. Let ⩾∈A⊗B(E)⊗B(E). Then ⩾ is coalition convex if and only if for almost all ω, ⩾ ω is convex

On series of translates of positive functions II

AbstractIn this paper we continue our investigation of series of the form ∑λ ∈ Λ ƒ(x + λ). Given a sequence of natural numbers n1 < n2 < … we are interested in sets Λ of the form where 0 < α < 1. In case α = 1q, where q > 1 is an integer, there is a zero-one law showing that for every measurable the above sum either converges almost everywhere or diverges almost everywhere. However, for any other value of α ∈ (0, 1) there is no such zero-one law

Generic Properties of Open Billiards

AbstractThe purpose of this paper is to show that for a denseGδset of three smooth convex bodies with nowhere vanishing curvature (in theCktopology, 2≤k≤∞), the open billiard obtained form these convex bodies determines a potential (the one that defines the natural escape measure of this billiard) which is non-lattice. This result generalizes one of the results obtained in a previous work of A. Lopes and R. Markarian [1]

Positively regular homeomorphisms of Euclidean spaces

AbstractA homeomorphism of Rn onto itself is called positively regular (or EC+) iff its family of non-negative iterates is pointwise equicontinuous. For EC+ homeomorphism of Rn such that some point of Rn has bounded positive semi-orbit, the nucleus M is defined, and the following theorems are proved.Theorem 1. If such a homeomorphism h:Rn→Rn has compact nucleus M, then M is a fully invariant compact AR. Further, for n≠4,5,h:Rn/M→Rn/M is conjugate to a contraction on Rn.Theorem 2. In Rn,n≠4,5,M compact iff there existsa disk D such that h(D)⊆IntD.Theorem 3. In R2, either M is a disk and h|M is a rotation, or h|M is periodic. The relationship between M and the irregular set of ? is also studied

Dimension and Measures for a Curvilinear Sierpinski Gasket or Apollonian Packing

In this paper we apply some results about general conformal iterated function systems toA, the residual set of a standard Apollonian packing or a curvilinear Sierpinski gasket. Within this context, it is straightforward to show thath, the Hausdorff dimension ofA, is greater than 1 and the packing dimension and the upper and lower box counting dimensions are all the same as the Hausdorff dimension. Among other things, we verify Sullivan's result that 0<Hh(A)<∞ and Ph(A)=∞