On the Hausdorff dimension of some fractal sets

Przytycki, Feliks; Urbański, Mariusz

doi:10.4064/sm-93-2-155-186

Cited by 143 publications

(112 citation statements)

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“…The measure µ may be used to estimate the Hausdorff dimension of self-affine graphs studied by McMullen [26], Prsytycki and Urbański [33,40]. More details can be found in [37,39].…”

Section: It Is Easy To Show Thatmentioning

confidence: 99%

Phase transitions for the multifractal analysis of self-similar measures

Testud

2006

Nonlinearity

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Abstract. We are interested in the multifractal analysis of a class of self-similar measures with overlaps. This class, for which we obtain explicit formulae for the L qspectrum τ (q) as well as the singularity spectrum f (α), is sufficiently large to point out new phenomena in the multifractal structure of self-similar measures. We show that, unlike the classical quasi-Bernoulli case, the L q -spectrum τ (q) of the measures studied can have an arbitrarily large number of non-differentiability points (phase transitions). These singularities occur only for the negative values of q and yield to measures that do not satisfy the usual multifractal formalism. The weak quasi-Bernoulli property is the key point of most of the arguments.

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“…The measure µ may be used to estimate the Hausdorff dimension of self-affine graphs studied by McMullen [26], Prsytycki and Urbański [33,40]. More details can be found in [37,39].…”

Section: It Is Easy To Show Thatmentioning

confidence: 99%

Phase transitions for the multifractal analysis of self-similar measures

Testud

2006

Nonlinearity

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“…If µ is absolutely continuous with dµ/dx in L p for some p > 1, then the Hausdorff dimension of the graph of S is 2 + log ρ log 2 [HL1,PU] and the Hausdorff dimension of the level set is 1 + log ρ log 2 a.e. [HL2].…”

Section: Introductionmentioning

confidence: 99%

The local dimensions of the Bernoulli convolution associated with the golden number

1997

Trans. Amer. Math. Soc.

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Abstract. Let X 1 , X 2 , . . . be a sequence of i.i.d. random variables each taking values of 1 and −1 with equal probability. For 1/2 < ρ < 1 satisfying the equation 1 − ρ − · · · − ρ s = 0, let µ be the probability measure induced bybe the local dimension of µ at x whenever the limit exists. We prove that α * = − log 2 log ρ and α * = − log δ s log ρ − log 2 log ρ , where δ = ( √ 5 − 1)/2, are respectively the maximum and minimum values of the local dimensions. If s = 2, then ρ is the golden number, and the approximate numerical values are α * ≈ 1.4404 and α * ≈ 0.9404.

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“…Furthermore the result is true for all 0 < a < 1 if a small perturbation of 2~al in the sense of Kahane and Salem [9] is allowed [8]. The first result of the Rademacher series R(x) was also obtained by Przytycki and Urbanski [12] by using a dynamics argument; they also proved a more striking result: there exist some values of a ( 2a is a Pisot-Vijayaraghavan number) so that the graph of 7? (x) has Hausdorff dimension strictly less than 2-a.…”

Section: Introductionmentioning

confidence: 70%

Fractal dimensions and singularities of the Weierstrass type functions

Hu¹,

Lau²

1993

Trans. Amer. Math. Soc.

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Abstract. A new type of fractal measures Xs, 1 < s < 2, defined on the subsets of the graph of a continuous function is introduced. The ^-dimension defined by this measure is 'closer' to the Hausdorff dimension than the other fractal dimensions in recent literatures. For the Weierstrass type functions defined by W(x) = £S°'k-aig()Jx), where X > 1 , 0 < a < 1 , and g is an almost periodic Lipschitz function of order greater than a , it is shown that thê -dimension of the graph of W equals to 2 -a , this conclusion is also equivalent to certain rate of the local oscillation of the function. Some problems on the ' knot ' points and the nondifferentiability of W are also discussed.

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On the Hausdorff dimension of some fractal sets

Cited by 143 publications

References 0 publications

Phase transitions for the multifractal analysis of self-similar measures

Phase transitions for the multifractal analysis of self-similar measures

The local dimensions of the Bernoulli convolution associated with the golden number

Fractal dimensions and singularities of the Weierstrass type functions

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