Correspondence Between Semi-Measures and Small Systems
Correspondence between subadditive integrals and small systems
Corrigenda to: "Optimal domains for the kernel operator associated with Sobolev's inequality" (Studia Math. 158 (2003), 131-152)
Corrigendum of Mean Quadratic Variations and Fourier Asymptotics of Self-similar Measures.
Corrigendum on "Semigroup-theoretical proofs of the central limit theorem and other theorems of analysis.
Corrigendum to (n,2)-sets have full Hausdorff dimension.
In the paper (n,2)-sets have full Hausdorff dimension, appeared in Rev. Mat. Iberoamericana 20 (2004), 381-393, the author claimed that an (n,2)-set must have full Hausdorff dimension. However, as pointed out by Terence Tao and John Bueti, the proof contains an error.
Corrigendum to the paper: “A remark on the weighted averages for superadditive processes”.
Countable contraction mappings in metric spaces: invariant sets and measure
We consider a complete metric space (X, d) and a countable number of contraction mappings on X, F = {F i: i ∈ ℕ}. We show the existence of a smallest invariant set (with respect to inclusion) for F. If the maps F i are of the form F i(x) = r i x + b i on X = ℝd, we prove a converse of the classic result on contraction mappings, more precisely, there exists a unique bounded invariant set if and only if r = supi r i is strictly smaller than 1. Further, if ρ = {ρ k}k∈ℕ is a probability sequence, we...
Countable tightness in the spaces of regular probability measures
We prove that if K is a compact space and the space P(K × K) of regular probability measures on K × K has countable tightness in its weak* topology, then L₁(μ) is separable for every μ ∈ P(K). It has been known that such a result is a consequence of Martin's axiom MA(ω₁). Our theorem has several consequences; in particular, it generalizes a theorem due to Bourgain and Todorčević on measures on Rosenthal compacta.
Countably piecewise expanding transformations without absolutely continuous invariant measure
Counting and convergence in ergodic theory
Courte démonstration du théorème ergodique sur-additif
Coverable Radon measures in topological spaces with covering properties
Coverable standard measures with the chain condition and the Lebesgue decomposition
Covering locally compact groups by less than many translates of a compact nullset
Gruenhage asked if it was possible to cover the real line by less than continuum many translates of a compact nullset. Under the Continuum Hypothesis the answer is obviously negative. Elekes and Stepr mans gave an affirmative answer by showing that if is the well known compact nullset considered first by Erdős and Kakutani then ℝ can be covered by cof() many translates of . As this set has no analogue in more general groups, it was asked by Elekes and Stepr mans whether such a result holds for...
Covering spaces and irreducible partitions
An irreducible partition of a space is a partition of that space into solid sets with a certain minimality property. Previously, these partitions were studied using the cup product in cohomology. This paper obtains similar results using the fundamental group instead. This allows the use of covering spaces to obtain information about irreducible partitions. This is then used to generalize Knudsen's construction of topological measures on the torus. We give examples of such measures that are invariant...
Covering the real line with translates of a zero-dimensional compact set
We construct a compact set C of Hausdorff dimension zero such that cof(𝒩) many translates of C cover the real line. Hence it is consistent with ZFC that less than continuum many translates of a zero-dimensional compact set can cover the real line. This answers a question of Dan Mauldin.
Covering theorems for uniqueness and extended uniqueness sets
Criterios de convergencia de medidas de conjuntos Riemann integrables.
Curvature bounds for neighborhoods of self-similar sets
In some recent work, fractal curvatures and fractal curvature measures , , have been determined for all self-similar sets in , for which the parallel neighborhoods satisfy a certain regularity condition and a certain rather technical curvature bound. The regularity condition is conjectured to be always satisfied, while the curvature bound has recently been shown to fail in some concrete examples. As a step towards a better understanding of its meaning, we discuss several equivalent formulations...