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Semicontinuity of dimension and measure for locally scaling fractals

L. B. Jonker, J. J. P. Veerman (2002)

Fundamenta Mathematicae

The basic question of this paper is: If you consider two iterated function systems close to each other in an appropriate topology, are the dimensions of their respective invariant sets close to each other? It is well known that the Hausdorff dimension (and Lebesgue measure) of the invariant set does not depend continuously on the iterated function system. Our main result is that (with a restriction on the "non-conformality" of the transformations) the Hausdorff dimension is a lower semicontinuous...

Separation properties for self-conformal sets

Yuan-Ling Ye (2002)

Studia Mathematica

For a one-to-one self-conformal contractive system w j j = 1 m on d with attractor K and conformality dimension α, Peres et al. showed that the open set condition and strong open set condition are both equivalent to 0 < α ( K ) < . We give a simple proof of this result as well as discuss some further properties related to the separation condition.

Sets of β -expansions and the Hausdorff measure of slices through fractals

Tom Kempton (2016)

Journal of the European Mathematical Society

We study natural measures on sets of β -expansions and on slices through self similar sets. In the setting of β -expansions, these allow us to better understand the measure of maximal entropy for the random β -transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing, leading...

Singular distributions, dimension of support, and symmetry of Fourier transform

Gady Kozma, Alexander Olevskiĭ (2013)

Annales de l’institut Fourier

We study the “Fourier symmetry” of measures and distributions on the circle, in relation with the size of their supports. The main results of this paper are:(i) A one-side extension of Frostman’s theorem, which connects the rate of decay of Fourier transform of a distribution with the Hausdorff dimension of its support;(ii) A construction of compacts of “critical” size, which support distributions (even pseudo-functions) with anti-analytic part belonging to l 2 .We also give examples of non-symmetry...

Size minimizing surfaces

Thierry De Pauw (2009)

Annales scientifiques de l'École Normale Supérieure

We prove a new existence theorem pertaining to the Plateau problem in 3 -dimensional Euclidean space. We compare the approach of E.R. Reifenberg with that of H. Federer and W.H. Fleming. A relevant technical step consists in showing that compact rectifiable surfaces are approximatable in Hausdorff measure and in Hausdorff distance by locally acyclic surfaces having the same boundary.

Some generic properties of concentration dimension of measure

Józef Myjak, Tomasz Szarek (2003)

Bollettino dell'Unione Matematica Italiana

Let K be a compact quasi self-similar set in a complete metric space X and let M 1 K denote the space of all probability measures on K , endowed with the Fortet-Mourier metric. We will show that for a typical (in the sense of Baire category) measure in M 1 K the lower concentration dimension is equal to 0 , while the upper concentration dimension is equal to the Hausdorff dimension of K .

Some properties of packing measure with doubling gauge

Sheng-You Wen, Zhi-Ying Wen (2004)

Studia Mathematica

Let g be a doubling gauge. We consider the packing measure g and the packing premeasure g in a metric space X. We first show that if g ( X ) is finite, then as a function of X, g has a kind of “outer regularity”. Then we prove that if X is complete separable, then λ s u p g ( F ) g ( B ) s u p g ( F ) for every Borel subset B of X, where the supremum is taken over all compact subsets of B having finite g -premeasure, and λ is a positive number depending only on the doubling gauge g. As an application, we show that for every doubling gauge...

Some remarks on restriction of the Fourier tranform for general measures.

Per Sjölin, Fernando Soria (1999)

Publicacions Matemàtiques

In this paper we establish a formal connection between the average decay of the Fourier transform of functions with respect to a given measure and the Hausdorff behavior of that measure. We also present a generalization of the classical restriction theorem of Stein and Tomas replacing the sphere with sets of prefixed Hausdorff dimension n - 1 + α, with 0 &lt; α &lt; 1.

Sous-ensembles de courbes Ahlfors-régulières et nombres de Jones.

Hervé Pajot (1996)

Publicacions Matemàtiques

We prove that an Ahlfors-regular set (with dimension one) E ⊂ Rn which verifies a βq-version of P. W. Jones' geometric lemma is included in an Ahlfors-regular curve Γ.This theorem is due to G. David and S. Semmes, we give a more direct proof.

Spaces of σ-finite linear measure

Ihor Stasyuk, Edward D. Tymchatyn (2013)

Colloquium Mathematicae

Spaces of finite n-dimensional Hausdorff measure are an important generalization of n-dimensional polyhedra. Continua of finite linear measure (also called continua of finite length) were first characterized by Eilenberg in 1938. It is well-known that the property of having finite linear measure is not preserved under finite unions of closed sets. Mauldin proved that if X is a compact metric space which is the union of finitely many closed sets each of which admits a σ-finite linear measure then...

Strong bifurcation loci of full Hausdorff dimension

Thomas Gauthier (2012)

Annales scientifiques de l'École Normale Supérieure

In the moduli space d of degree  d rational maps, the bifurcation locus is the support of a closed ( 1 , 1 ) positive current T bif which is called the bifurcation current. This current gives rise to a measure μ bif : = ( T bif ) 2 d - 2 whose support is the seat of strong bifurcations. Our main result says that supp ( μ bif ) has maximal Hausdorff dimension 2 ( 2 d - 2 ) . As a consequence, the set of degree  d rational maps having ( 2 d - 2 ) distinct neutral cycles is dense in a set of full Hausdorff dimension.

Strong measure zero and meager-additive sets through the prism of fractal measures

Ondřej Zindulka (2019)

Commentationes Mathematicae Universitatis Carolinae

We develop a theory of sharp measure zero sets that parallels Borel’s strong measure zero, and prove a theorem analogous to Galvin–Mycielski–Solovay theorem, namely that a set of reals has sharp measure zero if and only if it is meager-additive. Some consequences: A subset of 2 ω is meager-additive if and only if it is -additive; if f : 2 ω 2 ω is continuous and X is meager-additive, then so is f ( X ) .

Szpilrajn type theorem for concentration dimension

Jozef Myjak, Tomasz Szarek (2002)

Fundamenta Mathematicae

Let X be a locally compact, separable metric space. We prove that d i m T X = i n f d i m L X ' : X ' i s h o m e o m o r p h i c t o X , where d i m L X and d i m T X stand for the concentration dimension and the topological dimension of X, respectively.

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