Weak separation properties for self-similar sets

We investigate the Hausdorff measure and content on a class of quasi self‐similar sets that include, for example, graph‐directed and sub self‐similar and self‐conformal sets. We show that any Hausdorff measurable subset of such a set has comparable Hausdorff measure and Hausdorff content. In particular, this proves that graph‐directed and sub self‐conformal sets with positive Hausdorff measure are Ahlfors regular, irrespective of separation conditions. When restricting to self‐conformal subsets of the real line with Hausdorff dimension strictly less than one, we additionally show that the weak separation condition is equivalent to Ahlfors regularity and its failure implies full Assouad dimension. In fact, we resolve a self‐conformal extension of the dimension drop conjecture for self‐conformal sets with positive Hausdorff measure by showing that their Hausdorff dimension fall below the expected value if and only if there are exact overlaps.

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“…In the self‐similar case, Zerner introduced the identity limit criterion,

{φ_{monospacei}^{- 1} \circ φ_{monospacej}}_{i, j} does not accumulate to the identity,

and showed that it is equivalent to the weak separation condition. The self‐conformal case is more complicated since we cannot use inverses.…”

Section: Introductionmentioning

confidence: 99%

Self‐conformal sets with positive Hausdorff measure

Angelevska

Käenmäki

Troscheit

2020

Bull. London Math. Soc.

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“…Then

# {scriptV}_{k} = {bolde}_{1} B^{k} 1^{t}

and

{trueprefixlim}_{k \to \infty} {({bolde}_{1} B^{k} 1^{t})}^{1 / k} = ρ

. Therefore, by [, Theorem 2], we have

{prefixdim}_{H} K = \underset{k \to \infty}{trueprefixlim} \frac{log # {scriptV}_{k}}{- k log r} = \frac{log ρ}{- log r} .

□

…”

Section: Lipschitz Equivalence Of Self‐similar Setsmentioning

confidence: 93%

“…In the absence of the OSC, it is much harder to compute the Hausdorff dimension. Nevertheless, the dimension formula is still attainable for a large class of self‐similar sets with overlaps (see ). Under our setting, we prefer to use the following formula.…”

Section: Lipschitz Equivalence Of Self‐similar Setsmentioning

confidence: 99%

Self‐similar sets, simple augmented trees and their Lipschitz equivalence

Luo

2018

J. London Math. Soc.

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Given an iterated function system (IFS) of contractive similitudes, the theory of Gromov hyperbolic graph on the IFS has been established recently. In the paper, we introduce a notion of simple augmented tree which is a Gromov hyperbolic graph. By generalizing a combinatorial device of rearrangeable matrix, we show that there exists a near‐isometry between the simple augmented tree and the symbolic space of the IFS, so that their hyperbolic boundaries are Lipschitz equivalent. We then apply this to consider the Lipschitz equivalence of self‐similar sets with or without assuming the open set condition. Moreover, we also provide a criterion for a self‐similar set to be a Cantor‐type set which completely answers an open question raised in [Luo and Lau, Adv. Math. 235 (2013) 555–579]. Our study extends the previous works.

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“…We remark that the above definition for the WSC is equivalent to that given by Lau and Ngai in [21], provided that K is not contained in a hyperplane of R d . For a proof, see Zerner [41,Theorem 1]. It is known that the open set condition implies the WSC [21].…”

Section: Preliminaries and The Wscmentioning

confidence: 99%

Multifractal formalism for self-similar measures with weak separation condition

Ding

Lau

2009

Journal de Mathématiques Pures et Appliquées

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For any self-similar measure μ on R d satisfying the weak separation condition, we show that there exists an open ball U 0 with μ(U 0) > 0 such that the distribution of μ, restricted on U 0 , is controlled by the products of a family of non-negative matrices, and hence μ| U 0 satisfies a kind of quasi-product property. Furthermore, the multifractal formalism for μ| U 0 is valid on the whole range of dimension spectrum, regardless of whether there are phase transitions. Moreover the dimension spectra of μ and μ| U 0 coincide for q 0. This result unifies and improves many of the recent works on the multifractal structure of self-similar measures with overlaps.

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Weak separation properties for self-similar sets

Abstract: Abstract. We develop a theory for self-similar sets in R s that fulfil the weak separation property of Lau and Ngai, which is weaker than the open set condition of Hutchinson.

Cited by 96 publications

References 9 publications

Self‐conformal sets with positive Hausdorff measure

Self‐conformal sets with positive Hausdorff measure

Self‐similar sets, simple augmented trees and their Lipschitz equivalence

Multifractal formalism for self-similar measures with weak separation condition

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