The Singularity Spectrum f(α) of Some Moran Fractals

We show that the multifractal decomposition behaves as expected for a family of sets E known as homogeneous Moran fractals associated with the Fibonacci sequence , using probability measures () associated with the Fibonacci sequence . For each value of a parameter (_min, _max), we define multifractal components E of E, and show that they are fractals in the sense of Taylor. We give the explicit formula for the dimension of E. Also our method can be used for the Moran fractals associated with some more general sequences.     

Measurable dynamics of maps on profinite groups

We study the measurable dynamics of transformations on profinite groups, in particular of those which factor through sufficiently many of the projection maps; these maps generalize the 1-Lipschitz maps on _p.     

Distributional limit theorems in infinite ergodic theory

We present a unified approach to the Darling-Kac theorem and the arcsine laws for occupation times and waiting times for ergodic transformations preserving an infinite measure. Our method is based on control of the transfer operator up to the first entrance to a suitable reference set rather than on the full asymptotics of the operator. We illustrate our abstract results by showing that they easily apply to a significant class of infinite measure preserving interval maps. We also show that some of the tools introduced here are useful in the setup of pointwise dual ergodic transformations.     

Imprecise Markov chains with absorption

We consider convergence of Markov chains with uncertain parameters, known as imprecise Markov chains, which contain an absorbing state. We prove that under conditioning on non-absorption the imprecise conditional probabilities converge independently of the initial imprecise probability distribution if some regularity conditions are assumed. This is a generalisation of a known result from the classical theory of Markov chains by Darroch and Seneta [6].     

Fractal Penrose tilings I. Construction and matching rules

Summary Quasiperiodic tilings of kite-and-dart type, widely used as models for quasicrystals with decagonal symmetry, are constructed by means of somewhat artificial matching rules for the tiles. The proof of aperiodicity uses a self-similarity property, or inflation procedure, which requires drawing auxiliary lines. We introduce a modification of the kite-and-dart tilings which comes very naturally with both properties: the tiles are strictly self-similar, and their fractal boundaries provide perfect matching rules.     

Limit sets of automatic sequences

We show that for every k-automatic sequence there exists a natural number p>0 such that the sequences of the form (k^pn+j)_n?0 with j=0,…,p−1 are scaling sequences for f. Moreover, we demonstrate that every limit set is the union of certain basic limit sets.     

Equidistribution quantitative des points de petite hauteur sur la droite projective

We introduce a new class of adelic heights on the projective line. We estimate their essential minimum and prove a result of equidistribution (at every place) for points of small height with estimates on the speed of convergence. To each rational function R in one variable and defined over a number field K, is associated a normalized height on the algebraic closure of K. We show that these dynamically defined heights are adelic in our sense, and deduce from this equidistribution results for preimages of points under R at every place of K. Our approach follows that of Bilu, and relies on potential theory in the complex plane, as well as in the Berkovich space associated to the projective line over , for each prime p. Le premier auteur tient à remercier chaleureusement le project MECESUP UCN0202, ainsi que l'ACI ``Systèmes Dynamiques Polynomiaux' qui ont permis son séjour à l'Université Catholique d'Antofagasta. Le deuxième auteur est partiellement soutenu par le projet FONDECYT N 1040683. Enfin, nous remercions le rapporteur pour sa lecture détaillée de l'article.     

The Lebesgue measure of the algebraic difference of two random Cantor sets

In this paper we consider a family of random Cantor sets on the line. We give some sufficient conditions when the Lebesgue measure of the arithmetic difference is positive. Combining this with the main result of a recent joint paper of the second author with M. Dekking we construct random Cantor sets F₁, F₂ such that the arithmetic difference set F₂ − F₁ does not contain any intervals but ?eb(F₂ − F₁)> 0 almost surely, conditioned on non-extinction.     

The Dimension of Subsets of Moran Sets Determined by the Success Run Behaviour of Their Codings

 Let be a Moran set associated with the set . Let Γ be a non-empty subset of with non-empty complement. Associated with the behaviour of success run of symbols from Γ in the coding space is a decomposition of F such that

Scattering Systems with Several Evolutions and Multidimensional Input/State/Output Systems

The one-to-one correspondence between one-dimensional linear (stationary, causal) input/state/output systems and scattering systems with one evolution operator, in which the scattering function of the scattering system coincides with the transfer function of the linear system, is well understood, and has significant applications in H^∞ control theory. Here we consider this correspondence in the d-dimensional setting in which the transfer and scattering functions are defined on the polydisk. Unlike in the onedimensional case, the multidimensional state space realizations and the corresponding multi-evolution scattering systems are not necessarily equivalent, and the cases d = 2 and d > 2 differ substantially. A new proof of Andô’s dilation theorem for a pair of commuting contraction operators and a new statespace realization theorem for a matrix-valued inner function on the bidisk are obtained as corollaries of the analysis.     

Sobolev spaces on an arbitrary metric space

We define Sobolev space W ^1,p for 1<p on an arbitrary metric space with finite diameter and equipped with finite, positive Borel measure. In the Euclidean case it coincides with standard Sobolev space. Several classical imbedding theorems are special cases of general results which hold in the metric case. We apply our results to weighted Sobolev space with Muckenhoupt weight.This work is supported by KBN grant no. 2 1057 91 01     

Morse-Sard定理在无限维时的一个反例

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅＭｏｒｓｅ Ｓａｒｄｔｈｅｏｒｅｍｉｓａｆｕｎｄａｍｅｎｔａｌｔｈｅｏｒｅｍｉｎａｎａｌｙｓｉｓ ,ｅｓｐｅｃｉａｌｌｙｉｎｔｈｅｂａｓｉｓｏｆｔｒａｎｓｖｅｒｓａｌｉｔｙｔｈｅｏｒｙａｎｄｄｉｆｆｅｒｅｎｔｉａｌｔｏｐｏｌｏｇｙ .ＴｈｅｃｌａｓｓｉｃａｌＭｏｒｓｅ Ｓａｒｄｔｈｅｏｒｅｍｓｔａｔｅｓｔｈａｔｔｈｅｉｍａｇｅｏｆｔｈｅｓｅｔｏｆｃｒｉｔｉｃａｌｐｏｉｎｔｓｏｆａｆｕｎｃｔｉｏｎ ｆ :Ｒｍ→ＲｌｏｆｃｌａｓｓＣｍ -ｌ+1ｈａｓｚｅｒｏＬｅｂｅｓｇｕｅｍ…     

Siegel’s theorem for Drinfeld modules

We prove a Siegel type statement for finitely generated -submodules of under the action of a Drinfeld module . This provides a positive answer to a question we asked in a previous paper. We also prove an analog for Drinfeld modules of a theorem of Silverman for nonconstant rational maps of over a number field.     

Chaotic unbounded differentiation operators

We construct dense sets of hypercyclic vectors for unbounded differention operators, including differentiation operators on the Hardy spaceH ₂, and the Laplacian operator onL ₂((), for any bounded open subset of ². Furthermore, we show that these operators are chaotic, in the sense of Devaney.     

An optimal extension of Marstrand’s plane-packing theorem

We prove that if F is a subset of the 2-dimensional unit sphere in $\mathbb{R}^3$, with Hausdorff dimension strictly greater than 1, and E is a subset of $\mathbb{R}^3$ such that for each $e \in F$, E contains a plane perpendicular to the vector e, then E must have positive 3-dimensional Lebesgue measure.Received: 16 April 2002     

On simultaneous non-vanishing of modular <Emphasis Type="Italic">L</Emphasis>-functions

For i = 1, , r, let f _i be newforms of weight 2k _i for Γ₀(N _i ) with trivial character. We consider the simultaneous non-vanishing problem for the central values of twisted L-functions of f _i . By using the Shimura correspondence, we give a certain relation between this problem and the kernel fields of 2-adic Galois representations associated to modular forms. Received: 28 January 2006     

On an extension of a lattice-continuous mapping with embeddability applications

It is well known that ifX andY are completely regularT ₂ spaces, then any continuous function,f, fromX toY, has a unique continuous extension,(f), fromX toY, whereX andY are the Stone—ech compactifications ofX andY, respectively. This function plays an important role in Stone—ech Theory, especially in questions pertaining to embeddability.In this paper, we first extend this construction to general Wallman spaces, and then apply the results to extend well-known embeddability theorems.