Gibbs properties of self-conformal measures and the multifractal formalism

Ding, Feng

doi:10.1017/s0143385706000952

Cited by 37 publications

(44 citation statements)

References 52 publications

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“…Although false in general, the multifractal formalism has been verified for many natural measures, including self-similar and graph self-similar measures satisfying the open set condition (see [3,6,7,26] and the references therein), self-similar measures with the weak separation property (see [4,5,8,9,13,14,22,23,24,25]), and self-conformal measures (see [10,28,29,31]). However, little is known for selfaffine measures, except some special cases (see [1,2,20,27]), and this motivated our work.…”

Section: Introduction Letmentioning

confidence: 99%

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Multifractal formalism for self-affine measures with overlaps

Deng

Ngai

2009

Arch. Math.

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Section: Introduction Letmentioning

confidence: 99%

“…Even though the IFSs do not have the weak separation property, they satisfy the asymptotic weak separation condition in [10].…”

Section: Introduction Letmentioning

confidence: 99%

Multifractal formalism for self-affine measures with overlaps

Deng

Ngai

2009

Arch. Math.

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“…For example, Feng et al [4,5] have recently proved that all self-similar measures (and, in particular, self-similar measures not satisfying the OSC) satisfies some version of the multifractal formalism, and Lau et al [10][11][12] and Testud [18] have investigated multifractal properties of various special classes of self-similar measures not satisfying the OSC.…”

Section: Main Results -Part 1: Bounds For the L Q -Spectramentioning

confidence: 99%

Bounds for the Lq-spectra of a self-similar multifractal not satisfying the Open Set Condition

Olsen

2009

Journal of Mathematical Analysis and Applications

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By now the L q -spectra of self-similar measures satisfying the so-called Open Set Condition is well understood. However, if the Open Set Condition is not satisfied, then almost nothing is known. In this paper we provide non-trivial bounds for the L q -spectra of an arbitrary self-similar measure. We emphasize that we are considering arbitrary self-similar measures (and sets) which are not assumed to satisfy the Open Set Condition or similar separation conditions.By now the L q -spectra of self-similar measures satisfying the so-called Open Set Condition is well understood. However, if the Open Set Condition is not satisfied, then almost nothing is known. In this paper we provide non-trivial bounds for the L q -spectra of an arbitrary self-similar measure, see Theorem 1.1 and its corollaries, and non-trivial bounds for the mixed L q -spectra of finite families of arbitrary self-similar measures, see Theorem 2.1. We emphasize that we are considering arbitrary self-similar measures and sets which are not assumed to satisfy any separation conditions; in particular, we are not assuming that the Open Set Condition is satisfied. As an application of our results we obtain bounds for the L q -spectra of the (2, 3)-Bernoulli convolution, see Section 3.

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“…Thus, we will focus on the problem of providing non-trivial estimates for the L spectra and Rényi dimension of inhomogeneous and homogeneous self-similar measures not satisfying separability conditions like the IOSC and OSC. If the OSC is not satisfied then we can find only sporadic studies of various special classes of measures (see [10,11,16,17,24,25]) for which something is known about the L spectra or other multifractal properties. For the inhomogeneous case, to the best of our knowledge, no investigation was performed so far.…”

Section: Introductionmentioning

confidence: 99%

The L q spectra and Rényi dimension of generalized inhomogeneous self-similar measures

Liszka

2014

Open Mathematics

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Very recently bounds for the L spectra of inhomogeneous self-similar measures satisfying the InhomogeneousOpen Set Condition (IOSC), being the appropriate version of the standard Open Set Condition (OSC), were obtained. However, if the IOSC is not satisfied, then almost nothing is known for such measures. In the paper we study the L spectra and Rényi dimension of generalized inhomogeneous self-similar measures, for which we allow an infinite number of contracting similarities and probabilities depending on positions. As an application of the results, we provide a systematic approach to obtaining non-trivial bounds for the L spectra and Rényi dimension of inhomogeneous self-similar measures not satisfying the IOSC and of homogeneous ones not satisfying the OSC. We also provide some non-trivial bounds without any separation conditions. MSC:28A80

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Gibbs properties of self-conformal measures and the multifractal formalism

Abstract: We prove that for any self-conformal measures, without any separation conditions, the multifractal formalism partially holds. The result follows by establishing certain Gibbs properties for self-conformal measures.

Cited by 37 publications

References 52 publications

Multifractal formalism for self-affine measures with overlaps

Multifractal formalism for self-affine measures with overlaps

Bounds for the Lq-spectra of a self-similar multifractal not satisfying the Open Set Condition

The L q spectra and Rényi dimension of generalized inhomogeneous self-similar measures

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