Separation Conditions for Iterated Function Systems with Overlaps

Deng, Qi-Rong; Lau, Ka–Sing; Ngai, Sze-Man

doi:10.1090/conm/600/11928

Cited by 11 publications

(26 citation statements)

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“…Not all points with a symbolic representation in a loop class external to the essential set need be non-essential. To see this we observe that (1,4,14,22,30,37,30,37,30,37, . .…”

Section: Minimal Polynomial Of Approx Value Of Size Of Reducementioning

confidence: 85%

Local dimensions of measures of finite type

Hare¹,

Matthews²

2016

J. Fractal Geom.

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Abstract. We study the multifractal analysis of a class of equicontractive, self-similar measures of finite type, whose support is an interval. Finite type is a property weaker than the open set condition, but stronger than the weak open set condition. Examples include Bernoulli convolutions with contraction factor the inverse of a Pisot number and self-similar measures associated with m-fold sums of Cantor sets with ratio of dissection 1/R for integer R ≤ m.We introduce a combinatorial notion called a loop class and prove that the set of attainable local dimensions of the measure at points in a positive loop class is a closed interval. We prove that the local dimensions at the periodic points in the loop class are dense and give a simple formula for those local dimensions. These self-similar measures have a distinguished positive loop class called the essential class. The set of points in the essential class has full Lebesgue measure in the support of the measure and is often all but the two endpoints of the support. Thus many, but not all, measures of finite type have at most one isolated point in their set of local dimensions.We give examples of Bernoulli convolutions whose sets of attainable local dimensions consist of an interval together with an isolated point. As well, we give an example of a measure of finite type that has exactly two distinct local dimensions.

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“…Not all points with a symbolic representation in a loop class external to the essential set need be non-essential. To see this we observe that (1,4,14,22,30,37,30,37,30,37, . .…”

Section: Minimal Polynomial Of Approx Value Of Size Of Reducementioning

confidence: 85%

Local dimensions of measures of finite type

Hare¹,

Matthews²

2016

J. Fractal Geom.

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“…It is well known that the WSC is strictly weaker than the OSC and the finite type condition . We refer to a survey paper for the detailed discussion on various separation conditions in fractal geometry.…”

Section: Simple Augmented Treesmentioning

confidence: 99%

“…It is easy to see that

S_{14} = S_{21}

(see Figure ), hence the OSC does not hold for any

λ \in [\frac{7}{4}, 2]

. However, the IFS always satisfies the WSC . If

λ \in (\frac{7}{4}, 2)

, all

K_{λ}

are Lipschitz equivalent to each other as their augmented trees have the same quotient space (see Figure (a)).…”

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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“…IFSs that have the finite type property admit only finitely many neighbour sets, [3]. (Strictly speaking, neighbour sets were only defined for the generations r k min for k ∈ N, but, as shown in [1,Prop. 5.4], this makes no difference.)…”

Section: Finite Neighbour Conditionmentioning

confidence: 99%

“…We call this the convex (generalized) finite type condition and write (G)FTC co . The following relationships are well known, (c.f., [1,12,13,15]): [12]. They showed that the answer, in general, is negative for IFS on R d for d ≥ 2, but for all known examples, the interior of the convex hull of the self-similar set is empty.…”

Section: Introductionmentioning

confidence: 99%