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1
Content available remote

A multifractal analysis of an interesting class of measures

100%
Bisbas A.
Colloquium Mathematicum
|
1996
|
tom 69
|
nr 1
37-42
2
Content available remote

A Fourier analytical characterization of the Hausdorff dimension of a closed set and of related Lebesgue spaces

100%
Triebel H., Winkelvoss H.
Studia Mathematica
|
1996
|
tom 121
|
nr 2
149-166
EN
Let Γ be a closed set in $ℝ^n$ with Lebesgue measure |Γ| = 0. The first aim of the paper is to give a Fourier analytical characterization of Hausdorff dimension of Γ. Let 0 < d < n. If there exist a Borel measure µ with supp µ ⊂ Γ and constants $c_{1} > 0$ and $c_{2} > 0$ such that $c_{1}r^{d} ≤ µ (B(x,r)) ≤ c_{2}r^{d}$ for all 0 < r < 1 and all x ∈ Γ, where B(x,r) is a ball with centre x and radius r, then Γ is called a d-set. The second aim of the paper is to provide a link between the related Lebesgue spaces $L_{p}(Γ)$, 0 < p ≤ ∞, with respect to that measure µ on the hand and the Fourier analytically defined Besov spaces $B^s_{p,q}(ℝ^n)$ (s ∈ ℝ, 0 < p ≤ ∞, 0 < q ≤ ∞) on the other hand.
3
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A family of singular functions and its relation to harmonic fractal analysis and fuzzy logic

100%
de Amo E., Carrillo M. D., Fernández-Sánchez J.
Open Mathematics
|
2016
|
tom 14
|
nr 1
1039-1052
EN
We study a parameterized family of singular functions which appears in a paper by H. Okamoto and M. Wunsch (2007). Various properties are revisited from the viewpoint of fractal geometry and probabilistic techniques. Hausdorff dimensions are calculated for several sets related to these functions, and new properties close to fractal analysis and strong negations are explored.
4
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Dynamical boundary of a self-similar set

100%
Morán M.
Fundamenta Mathematicae
|
1999
|
tom 160
|
nr 1
1-14
EN
Given a self-similar set E generated by a finite system Ψ of contracting similitudes of a complete metric space X we analyze a separation condition for Ψ, which is obtained if, in the open set condition, the open subset of X is replaced with an open set in the topology of E as a metric subspace of X. We prove that such a condition, which we call the restricted open set condition, is equivalent to the strong open set condition. Using the dynamical properties of the forward shift, we find a canonical construction for the largest open set V satisfying the restricted open set condition. We show that the boundary of V in E, which we call the dynamical boundary of E, is made up of exceptional points from a topological and measure-theoretic point of view, and it exhibits some other boundary-like properties. Using properties of subself-similar sets, we find a method which allows us to obtain the Hausdorff and packing dimensions of the dynamical boundary and the overlapping set in the case when X is the n-dimensional Euclidean space and Ψ satisfies the open set condition. We show that, in this case, the dimension of these sets is strictly less than the dimension of the set E.
5
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Heisenberg Hausdorff Dimension of Besicovitch Sets

100%
Venieri L.
Analysis and Geometry in Metric Spaces
|
2014
|
tom 2
|
nr 1
EN
We consider (bounded) Besicovitch sets in the Heisenberg group and prove that Lp estimates for the Kakeya maximal function imply lower bounds for their Heisenberg Hausdorff dimension.
6
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Generalized dimension compression under mappings of exponentially integrable distortion

100%
Zapadinskaya A.
Open Mathematics
|
2011
|
tom 9
|
nr 2
356-363
EN
We prove a dimension compression estimate for homeomorphic mappings of exponentially integrable distortion via a modulus of continuity result by D. Herron and P. Koskela [Mappings of finite distortion: gauge dimension of generalized quasicircles, Illinois J. Math., 2003, 47(4), 1243–1259]. The essential sharpness of our estimate is demonstrated by an example.
7
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Large dimensional sets not containing a given angle

100%
Harangi V.
Open Mathematics
|
2011
|
tom 9
|
nr 4
757-764
EN
We say that a set in a Euclidean space does not contain an angle α if the angle determined by any three points of the set is not equal to α. The goal of this paper is to construct compact sets of large Hausdorff dimension that do not contain a given angle α ∈ (0,π). We will construct such sets in ℝn of Hausdorff dimension c(α)n with a positive c(α) depending only on α provided that α is different from π/3, π/2 and 2π/3. This improves on an earlier construction (due to several authors) that has dimension c(α) log n. The main result of the paper concerns the case of the angles π/3 and 2π/3. We present self-similar sets in ℝn of Hausdorff dimension $c{{\sqrt[3]{n}} \mathord{\left/ {\vphantom {{\sqrt[3]{n}} {\log n}}} \right. \kern-\nulldelimiterspace} {\log n}}$ with the property that they do not contain the angles π/3 and 2π/3. The constructed sets avoid not only the given angle α but also a small neighbourhood of α.
8
Content available remote

On the Hausdorff Dimension of CAT(κ) Surfaces

100%
Constantine D., Lafont J.-F.
Analysis and Geometry in Metric Spaces
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2016
|
tom 4
|
nr 1
EN
We prove that a closed surface with a CAT(κ) metric has Hausdorff dimension = 2, and that there are uniform upper and lower bounds on the two-dimensional Hausdorff measure of small metric balls. We also discuss a connection between this uniformity condition and some results on the dynamics of the geodesic flow for such surfaces. Finally,we give a short proof of topological entropy rigidity for geodesic flow on certain CAT(−1) manifolds.
9
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Infinite Graph-Directed Systems and Hausdorff Dimension

88%
Priyadarshi A.
Waves, Wavelets and Fractals
|
2017
|
tom 3
|
nr 1
84-95
EN
In this paper we study infinite graph-directed iterated function systems on compact metric spaces given by contractive ‘infinitesimal similitudes’. We derive formula for the Hausdorff dimension of the ‘invariant set’ for such a system in terms of the spectral radii of the naturally associated family of the ‘Perron- Frobenius operators’. The results in this paper generalizes the results obtained in [20], where finite graphdirected systems and infinite iterated function systems are considered
10
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The Beta(p,1) extensions of the random (uniform) Cantor sets

88%
Pestana D., Aleixo S., Leonel Rocha J.
Discussiones Mathematicae Probability and Statistics
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2009
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tom 29
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nr 2
199-221
EN
Starting from the random extension of the Cantor middle set in [0,1], by iteratively removing the central uniform spacing from the intervals remaining in the previous step, we define random Beta(p,1)-Cantor sets, and compute their Hausdorff dimension. Next we define a deterministic counterpart, by iteratively removing the expected value of the spacing defined by the appropriate Beta(p,1) order statistics. We investigate the reasons why the Hausdorff dimension of this deterministic fractal is greater than the Hausdorff dimension of the corresponding random fractals.
11
Content available remote

Multifractal properties of the sets of zeroes of Brownian paths

88%
Dolgopyat D., Sidorov V.
Fundamenta Mathematicae
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1995
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tom 147
|
nr 2
157-171
EN
We study Brownian zeroes in the neighborhood of which one can observe a non-typical growth rate of Brownian excursions. We interpret the multifractal curve for the Brownian zeroes calculated in [6] as the Hausdorff dimension of certain sets. This provides an example of the multifractal analysis of a statistically self-similar random fractal when both the spacing and the size of the corresponding nested sets are random.
12
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Content available

Stochastic diffrential equations on Banach spaces and their optimal feedback control

75%
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2012
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tom 32
|
nr 1
87-109
EN
In this paper we consider stochastic differential equations on Banach spaces (not Hilbert). The system is semilinear and the principal operator generating a C₀-semigroup is perturbed by a class of bounded linear operators considered as feedback operators from an admissible set. We consider the corresponding family of measure valued functions and present sufficient conditions for weak compactness. Then we consider applications of this result to several interesting optimal feedback control problems. We present results on existence of optimal feedback operators.
13
Content available remote

Coverings and dimensions in infinite profinite groups

75%
Maga P.
Open Mathematics
|
2013
|
tom 11
|
nr 2
246-253
EN
Answering a question of Miklós Abért, we prove that an infinite profinite group cannot be the union of less than continuum many translates of a compact subset of box dimension less than 1. Furthermore, we show that it is consistent with the axioms of set theory that in any infinite profinite group there exists a compact subset of Hausdorff dimension 0 such that one can cover the group by less than continuum many translates of it.
14
Content available remote

$L^q$-spectrum of the Bernoulli convolution associated with the golden ratio

63%
Lau K.-S., Ngai S.-M.
Studia Mathematica
|
1998
|
tom 131
|
nr 3
225-251
EN
Based on a set of higher order self-similar identities for the Bernoulli convolution measure for (√5-1)/2 given by Strichartz et al., we derive a formula for the $L^q$-spectrum, q >0, of the measure. This formula is the first obtained in the case where the open set condition does not hold.
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