ℰ-complete andD-generic filters of partially ordered sets

In this paper we prove theorems which ensure the existence of -complete andD-generic filters of partially ordered sets.     

On the existence of ordered couplings of random sets — with applications

Letψ andϕ be two given random closed sets in a locally compact second countable topological spaceS. (They need not be based on the same probability space.) The main result gives necessary and sufficient conditions on the distributions ofψ andϕ, for the existence of two random closed sets and , based on the same probability space and such that their distributions coincide with those ofψ andϕ, resp., and a.s. This coupling result tells us in particular when a probability distribution onS is selectionable w.r.t. (the distribution of) a random closed set. An existence result for realizable thinnings of a simple point process is obtained by specializing it to supports of random measures. The coupling result is extended to random variables in a countably based continuous poset. As examples we mention various kinds of random capacities — in particular random measures — and random compact (saturated) sets. Moreover, the extended result tells us when a probability distribution onS is selectionable w.r.t. the distribution of a random compact (saturated) set.     

On monotone permutations of`ℓ-cyclically ordered sets

For an ℓ-cyclically ordered set M with the ℓ-cyclic order C let P(M) be the set of all monotone permutations on M. We define a ternary relation on the set P(M). Further, we define in a natural way a group operation (denoted by ·) on P(M). We prove that if the ℓ-cyclic order C is complete and ≠ 0, then (P(M),·, ) is a half cyclically ordered group.     

A proof of dilworth’s decomposition theorem for partially ordered sets

A short proof of the following theorem is given: LetP be a finite partially ordered set. If the maximal number of elements in an independent subset ofP isk, thenP is the union ofk chains.     

Sliding of self-similar sets

This paper deals with the Lipschitz equivalence of slidings of self-similar sets by graph-directed construction and martingale theory.     

Level sets of β-expansions

Let ${\{{\mathrm{\epsilon }}_n(x)\}}_{n?1}$ be the sequence of β-digits of a real number $x\in (0\text{,}1)$, with the golden number $\beta =(\sqrt{5}+1)/2$ as basis. For any $0?p?1/2$, any $0<\tau ?1$ and any real number a, we consider the level set consisting of numbers x such that ${\sum }_{n=1}^{\infty }({\mathrm{\epsilon }}_n(x)?p)/n^{\tau }=a$. We prove that the Hausdorff dimension of this set is independent of a and τ, and that it is equal to $\log f(p)/\log \beta$ where $f(p)={(1?p)}^{1?p}/({(1?2p)}^{1?2p}{p}^p)$. To cite this article: A. Fan, H. Zhu, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

��-Homogeneity of Borel sets

We give an affirmative answer to the following question: Is any Borel subset of a Cantor set C a sum of a countable number of pairwise disjoint h-homogeneous subspaces that are closed in X? It follows that every Borel set ${X \subset {\bf R}^n}$ can be partitioned into countably many h-homogeneous subspaces that are G _?? -sets in X.     

Dimensions of modified Besicovitch-Eggleston sets

We consider the variants of Besicovitch-Eggleston sets in symbolic spaces, and determine their fractal dimensions.     

On the structure of stationary sets

We isolate several classes of stationary sets of [k]ωand investigate implications among them. Under a large cardinal assumption, we prove a structure theorem for stationary sets.     

Asymptotically stable sets and the stability of ω-limit sets

Let C be the collection of continuous self-maps of the unit interval I=[0,1] to itself. For f∈C and x∈I, let ω(x,f) be the ω-limit set of f generated by x, and following Block and Coppel, we take Q(x,f) to be the intersection of all the asymptotically stable sets of f containing ω(x,f). We show that Q(x,f) tells us quite a bit about the stability of ω(x,f) subject to perturbations of either x or f, or both. For example, a chain recurrent point y is contained in Q(x,f) if and only if there are arbitrarily small perturbations of f to a new function g that give us y as a point of ω(x,g). We also study the structure of the map Q taking (x,f)∈I×C to Q(x,f). We prove that Q is upper semicontinuous and a Baire 1 function, hence continuous on a residual subset of I×C. We also consider the map given by x?Q(x,f), and find that this map is continuous if and only if it is a constant map; that is, only when the set is a singleton.     

Uniform distribution of prime powers and sets of recurrence and van der Corput sets in ℤ k

We establish new results on sets of recurrence and van der Corput sets in ? ^k which refine and unify some of the previous results obtained by Sárk?zy, Furstenberg, Kamae and Mèndes France, and Bergelson and Lesigne. The proofs utilize a general equidistribution result involving prime powers which is of independent interest.     

Dimension of divergence sets for dispersive equation

Consider the generalized dispersive equation defined by{iδtu+Ф(√-△)u=0,(x,t)∈R^n×R,u(x,0)=f(x),F∈F(R^n),(*)whereФ(√-△)is a pseudo-differential operator with symbolФ(|ζ|).In the present paper,assuming thatФsatisfies suitable growth conditions and the initial data in H^s(R^n),we bound the Hausdorff dimension of the sets on which the pointwise convergence of solutions to the dispersive equations(*)fails.These upper bounds of Hausdorff dimension shall be obtained via the Kolmogorov-Seliverstov-Plessner method.     

Differentiation of sets – The general case

In recent work by Khmaladze and Weil (2008) and by Einmahl and Khmaladze (2011), limit theorems were established for local empirical processes near the boundary of compact convex sets K   in R^d${\mathbb{R}}^d$. The limit processes were shown to live on the normal cylinder Σ of K, respectively on a class of set-valued derivatives in Σ. The latter result was based on the concept of differentiation of sets at the boundary ∂K of K, which was developed in Khmaladze (2007). Here, we extend the theory of set-valued derivatives to boundaries ∂F   of rather general closed sets F⊂R^d$F\subset {\mathbb{R}}^d$, making use of a local Steiner formula for closed sets, established in Hug, Last and Weil (2004).     

ω-Limit sets of smooth cylindrical cascades

Let f(x) be a smooth function on the circle S¹, x mod 1, \(\smallint _{S^1 } f(x)dx = 0\) , α be an irrational number, and q_n be the denominators of convergents of continued fractions. In this note a classification of ω-limit sets for the cylindrical cascade $$T:(x,y) \to (x + \alpha , y + f(x)),$$ x ε S¹, y ε R, is obtained. Criteria for the solvability of the equation g(x +α) — g(x)=f (x) are found. Estimates for the speed of decrease of the function $$h_{q_n } (x) = \sum _{i = 0}^{q_n - 1} f(x + i\alpha )$$ as n → ∞ are obtained.     

Products of non-σ-lower porous sets

In the present article we provide an example of two closed non-σ-lower porous sets A,B ? ? such that the product A × B is lower porous. On the other hand, we prove the following: Let X and Y be topologically complete metric spaces, let A ? X be a non-σ-lower porous Suslin set and let B ? Y be a non-σ-porous Suslin set. Then the product A × B is non-σ-lower porous. We also provide a brief summary of some basic properties of lower porosity, including a simple characterization of Suslin non-σ-lower porous sets in topologically complete metric spaces.     

Some topological properties of ω-covering sets

We prove the following theorems:1. There exists an -covering with the property s ₀.2. Under cov there exists X such that is not an -covering orX \ B is not an -covering].3. Also we characterize the property of being an -covering.     

Lipschitz equivalence of fractal sets in ?

Let T(q,D) be a self-similar (fractal) set generated by $ \left\{ {fi(x) = \frac{1} {q}(x + d_i )} \right\}_{i = 1}^N $ where integer q > 1 and D = {d ₁, d ₂, ??, d _N } ? ?. To show the Lipschitz equivalence of T(q,D) and a dust-like T(q,C), one general restriction is D ? ? by Peres et al. [Israel J Math, 2000, 117: 353?C379]. In this paper, we obtain several sufficient criterions for the Lipschitz equivalence of two self-similar sets by using dust-like graph-directed iterating function systems and combinatorial techniques. Several examples are given to illustrate our theory.     

Properties of sets admitting stable ɛ-selections

Sets in which some convex subsets admit local (global) continuous ?-selections are studied. In particular, it is shown that if, for any number ? > 0, some neighborhood O(x) of a point x in a Banach space X contains a dense (in O(x)) convex set K admitting an upper semicontinuous acyclic (in particular, continuous single-valued) ?-selection to an approximatively compact set M ? X, then x is a δ-sun point; if, in addition, X ∈ (R), then the set of all points nearest to x in M is a singleton.     

Alexander polynomials and Poincaré series of sets of ideals

Earlier the authors considered and, in some cases, computed Poincaré series for two sorts of multi-index filtrations on the ring of germs of functions on a complex (normal) surface singularity (in particular, on the complex plane). A filtration of the first class was defined by a curve (with several branches) on the surface singularity. A filtration of the second class (called divisorial) was defined by a set of components of the exceptional divisor of a modification of the surface singularity. Here we define and compute in some cases the Poincaré series corresponding to a set of ideals in the ring of germs of functions on a surface singularity. For the complex plane, this notion unites the two classes of filtrations described above.