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Characterization of local dimension functions of subsets of ℝ d

L. Olsen (2005)

Colloquium Mathematicae

For a subset E ⊆ ℝ d and x ∈ ℝ d , the local Hausdorff dimension function of E at x is defined by d i m H , l o c ( x , E ) = l i m r ↘ 0 d i m H ( E ∩ B ( x , r ) ) where d i m H denotes the Hausdorff dimension. We give a complete characterization of the set of functions that are local Hausdorff dimension functions. In fact, we prove a significantly more general result, namely, we give a complete characterization of those functions that are local dimension functions of an arbitrary regular dimension index.

Concerning Sets of the First Baire Category with Respect to Different Metrics

Maria Moszyńska, Grzegorz Sójka (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

We prove that if ϱ H and δ are the Hausdorff metric and the radial metric on the space ⁿ of star bodies in ℝ, with 0 in the kernel and with radial function positive and continuous, then a family ⊂ ⁿ that is meager with respect to ϱ H need not be meager with respect to δ. Further, we show that both the family of fractal star bodies and its complement are dense in ⁿ with respect to δ.

Conformal measures for rational functions revisited

Manfred Denker, R. Mauldin, Z. Nitecki, Mariusz Urbański (1998)

Fundamenta Mathematicae

We show that the set of conical points of a rational function of the Riemann sphere supports at most one conformal measure. We then study the problem of existence of such measures and their ergodic properties by constructing Markov partitions on increasing subsets of sets of conical points and by applying ideas of the thermodynamic formalism.

Construction of functions with prescribed Hölder and chirp exponents.

Stéphane Jaffard (2000)

Revista Matemática Iberoamericana

We show that the Hölder exponent and the chirp exponent of a function can be prescribed simultaneously on a set of full measure, if they are both lower limits of continuous functions. We also show that this result is optimal: In general, Hölder and chirp exponents cannot be prescribed outside a set of Hausdorff dimension less than one. The direct part of the proof consists in an explicit construction of a function determined by its orthonormal wavelet coefficients; the optimality is the direct consequence...

Continuous dependence on parameters of certain self-affine measures, and their singularity

Daoxin Ding (2011)

Czechoslovak Mathematical Journal

In this paper, we first prove that the self-affine sets depend continuously on the expanding matrix and the digit set, and the corresponding self-affine measures with respect to the probability weight behave in much the same way. Moreover, we obtain some sufficient conditions for certain self-affine measures to be singular.

Convergence and uniqueness problems for Dirichlet forms on fractals

Roberto Peirone (2000)

Bollettino dell'Unione Matematica Italiana

M 1 è un particolare operatore di minimizzazione per forme di Dirichlet definite su un sottoinsieme finito di un frattale K che è, in un certo senso, una sorta di frontiera di K . Viene talvolta chiamato mappa di rinormalizzazione ed è stato usato per definire su K un analogo del funzionale u ↦ ∫ grad ⁢ u 2 e un moto Browniano. In questo lavoro si provano alcuni risultati sull'unicità dell'autoforma (rispetto a M 1 ), e sulla convergenza dell'iterata di M 1 rinormalizzata. Questi risultati sono collegati con l'unicità...

Correlation dimension for self-similar Cantor sets with overlaps

Károly Simon, Boris Solomyak (1998)

Fundamenta Mathematicae

We prove a classification theorem of the “Glimm-Effros” type for Borel order relations: a Borel partial order on the reals either is Borel linearizable or includes a copy of a certain Borel partial order ≤ 0 which is not Borel linearizable.

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