Weak separation condition, Assouad dimension, and Furstenberg homogeneity

Käenmäki, Antti; Rossi, Eino

doi:10.5186/aasfm.2016.4133

Cited by 36 publications

(38 citation statements)

References 27 publications

(46 reference statements)

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“…It generalizes the corresponding result of Farkas [4, Corollary 3.13] on self-similar sets. [11,Remark 3.7(2)]), the lack of exact overlaps implies the open set condition and we have s = P −1 (0).…”

Section: Dimension Drop Conjecturementioning

confidence: 99%

“…In fact, there exists unique

s ⩾ 0

for which

P (s) = 0

. It is a classical result that if

F

satisfies the open set condition, then

{prefixdim}_{normalH} false(F false) = P^{- 1} false(0 false)

; for the latest incarnation of this observation, see [, Proposition 3.5].…”

Section: Dimension Drop Conjecturementioning

confidence: 99%

“…Example We exhibit a conformal iterated function system

{φ_{1}, φ_{2}, φ_{3}}

double-struckR

for which the associated self‐conformal set

F \subset R

satisfies the weak separation condition but has

sup {# {normalΦ}^{*} false(x, r false) : x \in F and r > 0} = \infty,

where

{normalΦ}^{*} false(x, r false) = false{φ_{i} : prefixdiam (φ_{monospacei} false(F false)) ⩽ r < prefixdiam (φ_{i^{-}} false(F false)) and φ_{i} (F) \cap B (x, r) \neq \emptyset false}

for all

x \in R

and

r > 0

. Since, by [, Remark 3.7(1)], the condition

sup {# {normalΦ}^{*} false(x, r false) : x \in F and r > 0} < \infty

is equivalent to the original definition of Lau, Ngai and Wang , we see that our definition is strictly weaker in the non‐analytic case. Let

g false(x false) = \{\begin{matrix} left 0false \frac{1}{180} (9 x^{2} - 6 x + 1), & left i... \end{matrix}

…”

Section: Proof Of Theoremmentioning

confidence: 99%

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Self‐conformal sets with positive Hausdorff measure

Angelevska

Käenmäki

Troscheit

2020

Bull. London Math. Soc.

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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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Section: Dimension Drop Conjecturementioning

confidence: 99%

“…In fact, there exists unique

s ⩾ 0

for which

P (s) = 0

. It is a classical result that if

F

satisfies the open set condition, then

{prefixdim}_{normalH} false(F false) = P^{- 1} false(0 false)

; for the latest incarnation of this observation, see [, Proposition 3.5].…”

Section: Dimension Drop Conjecturementioning

confidence: 99%

“…Example We exhibit a conformal iterated function system

{φ_{1}, φ_{2}, φ_{3}}

double-struckR

for which the associated self‐conformal set

F \subset R

satisfies the weak separation condition but has

sup {# {normalΦ}^{*} false(x, r false) : x \in F and r > 0} = \infty,

where

{normalΦ}^{*} false(x, r false) = false{φ_{i} : prefixdiam (φ_{monospacei} false(F false)) ⩽ r < prefixdiam (φ_{i^{-}} false(F false)) and φ_{i} (F) \cap B (x, r) \neq \emptyset false}

for all

x \in R

and

r > 0

. Since, by [, Remark 3.7(1)], the condition

sup {# {normalΦ}^{*} false(x, r false) : x \in F and r > 0} < \infty

is equivalent to the original definition of Lau, Ngai and Wang , we see that our definition is strictly weaker in the non‐analytic case. Let

g false(x false) = \{\begin{matrix} left 0false \frac{1}{180} (9 x^{2} - 6 x + 1), & left i... \end{matrix}

…”

Section: Proof Of Theoremmentioning

confidence: 99%

See 1 more Smart Citation

Self‐conformal sets with positive Hausdorff measure

Angelevska

Käenmäki

Troscheit

2020

Bull. London Math. Soc.

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“…Over the last few years much progress has been made towards our understanding of this dimension and it is now a crucial part of fractal geometry, see e.g. [Che19, Fra14, FMT18, GHM16, KR16,Tro19] and references therein. Several other notions of dimension were derived from its definition and this family of Assouad-type dimensions has attracted much interest.…”

Section: Introductionmentioning

confidence: 99%

Assouad Spectrum Thresholds for Some Random Constructions

Troscheit¹

2019

Can. Math. Bull.

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The Assouad dimension of a metric space determines its extremal scaling properties. The derived notion of the Assouad spectrum fixes relative scales by a scaling function to obtain interpolation behaviour between the quasi-Assouad and box-counting dimensions. While the quasi-Assouad and Assouad dimensions often coincide, they generally differ in random constructions. In this paper we consider a generalised Assouad spectrum that interpolates between the quasi-Assouad to the Assouad dimension. For common models of random fractal sets we obtain a dichotomy of its behaviour by finding a threshold function where the quasi-Assouad behaviour transitions to the Assouad dimension. This threshold can be considered a phase transition and we compute the threshold for the Gromov boundary of Galton-Watson trees and one-variable random self-similar and self-affine constructions. We describe how the stochastically self-similar model can be derived from the Galton-Watson tree result.

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“…Here

{normalΓ}_{n} = false{monospacei |_{n} \in Σ_{n} : monospacei \in normalΓ false}

. Indeed, there exists

i \in {normalΣ}_{*}

which does not appear in any element of

normalΓ

(see [, § 2.1]). Since

Γ \subset {{monospacej}_{1} {monospacej}_{2} \dots \in Σ : | {monospacej}_{k} | = | i | and {monospacej}_{k} \neq i for all k \in N}

and, by iterating, we may assume that

| i | = 1

the claim follows from Proposition .…”

Section: Affinity Dimensionmentioning

confidence: 99%

Structure of equilibrium states on self-affine sets and strict monotonicity of affinity dimension

Käenmäki

Morris

2017

Proc. London Math. Soc.

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A fundamental problem in the dimension theory of self‐affine sets is the construction of high‐dimensional measures which yield sharp lower bounds for the Hausdorff dimension of the set. A natural strategy for the construction of such high‐dimensional measures is to investigate measures of maximal Lyapunov dimension; these measures can be alternatively interpreted as equilibrium states of the singular value function introduced by Falconer. While the existence of these equilibrium states has been well known for some years their structure has remained elusive, particularly in dimensions higher than two. In this article we give a complete description of the equilibrium states of the singular value function in the three‐dimensional case, showing in particular that all such equilibrium states must be fully supported. In higher dimensions we also give a new sufficient condition for the uniqueness of these equilibrium states. As a corollary, giving a solution to a folklore open question in dimension three, we prove that for a typical self‐affine set in R3, removing one of the affine maps which defines the set results in a strict reduction of the Hausdorff dimension.

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Weak separation condition, Assouad dimension, and Furstenberg homogeneity

Cited by 36 publications

References 27 publications

Self‐conformal sets with positive Hausdorff measure

Self‐conformal sets with positive Hausdorff measure

Assouad Spectrum Thresholds for Some Random Constructions

Structure of equilibrium states on self-affine sets and strict monotonicity of affinity dimension

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