Assouad dimensions of Moran sets and Cantor-like sets

Li, Wenwen; Li, Wenxia; Miao, Junjie; Xi, Lifeng

doi:10.1007/s11464-016-0539-6

Cited by 17 publications

(9 citation statements)

References 19 publications

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“…We begin, in § 2, by introducing our terminology and notation. There we also derive formulae for the (lower) Assouad dimensions of the associated sets C a , generalizing the formulae found in [19] and [24] for the special case of central Cantor sets. These formulae will be very useful for the proofs given later in the paper.…”

Section: Introductionmentioning

confidence: 69%

Assouad dimensions of complementary sets

García¹,

Hare²,

Mendivil³

2018

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Given a non-negative, decreasing sequence a with sum 1, we consider all the closed subsets of [0, 1] such that the lengths of their complementary open intervals are given by the terms of a, the so-called complementary sets. In this paper we determine the almost sure value of the Φ-dimensions of these sets given a natural model of randomness. The Φ-dimensions are intermediate Assouad-like dimensions which include the Assouad and quasi-Assouad dimensions as special cases. The answers depend on the size of Φ, with one size behaving like the Assouad dimension and the other, like the quasi-Assouad dimension.

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Section: Introductionmentioning

confidence: 69%

Assouad dimensions of complementary sets

García¹,

Hare²,

Mendivil³

2018

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…Assouad's original motivation was to study embedding problems, a subject where the Assouad dimension is still playing a fundamental rôle, see [Ol, OR, R]. The concept has also found a home in other areas of mathematics, including the theory of quasi-conformal mappings [H, L, MT], and more recently it is gaining substantial attention in the literature on fractal geometry [K,M,O,Fr3,LLMX,FHOR,ORS]. It is also worth noting that, due to its intimate relationship with tangents, it has always been present, although behind the scenes, in the pioneering work of Furstenberg on micro-sets and the related ergodic theory which goes back to the 1960s, see [Fu].…”

Section: The Assouad Dimensionmentioning

confidence: 99%

The Assouad dimension of randomly generated fractals

Fraser¹,

Miao²,

Troscheit³

2016

Ergod. Th. Dynam. Sys.

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We consider several different models for generating random fractals including random self-similar sets, random self-affine carpets, and fractal percolation. In each setting we compute either the almost sure or the Baire typical Assouad dimension and consider some illustrative examples. Our results reveal a common phenomenon in all of our models: the Assouad dimension of a randomly generated fractal is generically as big as possible and does not depend on the measure theoretic or topological structure of the sample space. This is in stark contrast to the other commonly studied notions of dimension like the Hausdorff or packing dimension.Mathematics Subject Classification 2010: primary: 28A80, 60J80; secondary: 37C45, 54E52, 82B43.

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“…see [6,17]. It is worth to point out that the Assouad dimension plays an important role in the theory of embeddings of metric spaces in Euclidean spaces and in the study of quasimmetric mappings, see [14,19,23].…”

Section: Resultsmentioning

confidence: 99%