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Large dimensional sets not containing a given angle

Viktor Harangi (2011)

Open Mathematics

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...

Limsup random fractals.

Khoshnevisan, Davar, Peres, Yuval, Xiao, Yimin (2000)

Electronic Journal of Probability [electronic only]

Linear distortion of Hausdorff dimension and Cantor's function.

Oleksiy Dovgoshey, Vladimir Ryazanov, Olli Martio, Matti Vuorinen (2006)

Collectanea Mathematica

Let be a mapping from a metric space X to a metric space Y, and let α be a positive real number. Write dim (E) and Hs(E) for the Hausdorff dimension and the s-dimensional Hausdorff measure of a set E. We give sufficient conditions that the equality dim (f(E)) = αdim (E) holds for each E ⊆ X. The problem is studied also for the Cantor ternary function G. It is shown that there is a subset M of the Cantor ternary set such that Hs(M) = 1, with s = log2/log3 and dim(G(E)) = (log3/log2) dim (E), for...

Lipschitz equivalence of graph-directed fractals

Ying Xiong, Lifeng Xi (2009)

Studia Mathematica

This paper studies the geometric structure of graph-directed sets from the point of view of Lipschitz equivalence. It is proved that if E i i and F j j are dust-like graph-directed sets satisfying the transitivity condition, then E i ₁ and E i ₂ are Lipschitz equivalent, and E i and F j are quasi-Lipschitz equivalent when they have the same Hausdorff dimension.

Lower quantization coefficient and the F-conformal measure

Mrinal Kanti Roychowdhury (2011)

Colloquium Mathematicae

Let F = f ( i ) : 1 ≤ i ≤ N be a family of Hölder continuous functions and let φ i : 1 ≤ i ≤ N be a conformal iterated function system. Lindsay and Mauldin’s paper [Nonlinearity 15 (2002)] left an open question whether the lower quantization coefficient for the F-conformal measure on a conformal iterated funcion system satisfying the open set condition is positive. This question was positively answered by Zhu. The goal of this paper is to present a different proof of this result.

Markov operators on the space of vector measures; coloured fractals

Karol Baron, Andrzej Lasota (1998)

Annales Polonici Mathematici

We consider the family 𝓜 of measures with values in a reflexive Banach space. In 𝓜 we introduce the notion of a Markov operator and using an extension of the Fortet-Mourier norm we show some criteria of the asymptotic stability. Asymptotically stable Markov operators can be used to construct coloured fractals.

Measurable envelopes, Hausdorff measures and Sierpiński sets

Márton Elekes (2003)

Colloquium Mathematicae

We show that the existence of measurable envelopes of all subsets of ℝⁿ with respect to the d-dimensional Hausdorff measure (0 < d < n) is independent of ZFC. We also investigate the consistency of the existence of ℋ d -measurable Sierpiński sets.

Mesures invariantes pour les fractions rationnelles géométriquement finies

Guillaume Havard (1999)

Fundamenta Mathematicae

Let T be a geometrically finite rational map, p(T) its petal number and δ the Hausdorff dimension of its Julia set. We give a construction of the σ-finite and T-invariant measure equivalent to the δ-conformal measure. We prove that this measure is finite if and only if p ( T ) + 1 p ( T ) δ > 2 . Under this assumption and if T is parabolic, we prove that the only equilibrium states are convex combinations of the T-invariant probability and δ-masses at parabolic cycles.

Metric Diophantine approximation on the middle-third Cantor set

Yann Bugeaud, Arnaud Durand (2016)

Journal of the European Mathematical Society

Let μ ≥ 2 be a real number and let ℳ ( μ ) denote the set of real numbers approximable at order at least μ by rational numbers. More than eighty years ago, Jarník and, independently, Besicovitch established that the Hausdorff dimension of ℳ ( μ ) is equal to 2 / μ . We investigate the size of the intersection of ℳ ( μ ) with Ahlfors regular compact subsets of the interval [ 0 , 1 ] . In particular, we propose a conjecture for the exact value of the dimension of ℳ ( μ ) intersected with the middle-third Cantor set and give several results...

Minkowski sums of Cantor-type sets

Kazimierz Nikodem, Zsolt Páles (2010)

Colloquium Mathematicae

The classical Steinhaus theorem on the Minkowski sum of the Cantor set is generalized to a large class of fractals determined by Hutchinson-type operators. Numerous examples illustrating the results obtained and an application to t-convex functions are presented.

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