Quasi-Lower Dimension and Quasi-Lipschitz Mapping

Chen, Haipeng; Du, Yali; Wei, Chun

doi:10.1142/s0218348x17500347

Cited by 16 publications

(10 citation statements)

References 18 publications

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“…We denote by dim B E the lower box-counting dimension and refer the readers to [4,18] for the definition. For totally bounded sets E ⊂ X and any θ ∈ (0, 1), combining the results of Fraser [5], Fraser and Yu [8] and Chen et al [1], we have dim…”

Section: Introductionmentioning

confidence: 74%

“…Compared with the lower Assouad spectrum, the lower Assouad dimension is not a quasi-Lipschitz invariant. Motivated by this, Chen, Du and Wei [1] introduced the quasi lower spectrum to study how the lower Assouad dimension changes under the quasi-Lipschitz mappings. For any fixed θ ∈ (0, 1), they defined dim θ L E = sup s ≥ 0 | there exist constants ρ, c > 0, such that for any 0 < r ≤ R and then get a quasi-Lipschitz invariant.…”

Section: Introductionmentioning

confidence: 99%

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Lower Assouad-Type Dimensions of Uniformly Perfect Sets in Doubling Metric Spaces

Chen

Chang

2020

Fractals

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In this paper, we are concerned with the relationship among the lower Assouad type dimensions. For uniformly perfect sets in doubling metric spaces, we obtain a variational result between two different but closely related lower Assouad spectra. As an application, we show that the limit of the lower Assouad spectrum as θ tends to 1 equals to the quasi-lower Assouad dimension, which provides an equivalent definition to the latter. On the other hand, although the limit of the lower Assouad spectrum as θ tends to 0 exists, there exist uniformly perfect sets such that this limit is not equal to the lower box-counting dimension. Moreover, by the example of Cantor cut-out sets, we show that the new definition of quasi lower Assouad dimension is more accessible, and indicate that the lower Assouad dimension could be strictly smaller than the lower spectra and the quasi lower Assouad dimension.

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Section: Introductionmentioning

confidence: 74%

Section: Introductionmentioning

confidence: 99%

Lower Assouad-Type Dimensions of Uniformly Perfect Sets in Doubling Metric Spaces

Chen

Chang

2020

Fractals

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“…The analogous 'tale of two spectra' problem for the lower spectrum was considered in [41,42]. The quasi-lower dimension, dim qL F , was introduced in [41] and in [42] it was proved that…”

Section: Only Directly Implies That Lim Sup θ→1 Dim θmentioning

confidence: 99%

Assouad Dimension and Fractal Geometry

Fraser

2020

111

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Dimension theory of orthogonal projections vi Contents 10.2.2 The Assouad dimension of orthogonal projections 10.2.3 An application to the box dimensions of projections 10.2.4 The lower dimension and projections 10.3 Slices and intersections 11 Two famous problems in geometric measure theory 11.1 Distance sets 11.2 Kakeya sets 12 Conformal dimension 12.1 Lowering the Assouad dimension by quasi-symmetry PART THREE APPLICATIONS 13 Applications in embedding theory 13.1 Assouad's embedding theorem 13.1.1 Doubling and uniformly perfect metric spaces 13.1.2 Assouad's embedding theorem 13.2 The spiral winding problem 13.3 Almost bi-Lipschitz embeddings 14 Applications in number theory 14.1 Arithmetic progressions 14.1.1 Discrete Kakeya sets 14.2 Diophantine approximation 14.3 Definability of the integers 15 Applications in probability theory 15.1 Uniform dimension results 15.2 Dimensions of random graphs 16 Applications in functional analysis 16.1 Hardy inequalities 16.2 L p → L q bounds for spherical maximal operators 16.3 Connection with L p -norms 17 Future directions 17.1 Finite stability of modified lower dimension 17.2 Dimensions of measures 17.3 Weak tangents 17.4 Further questions of measurability 17.5 IFS attractors Contents vii 17.6 Random sets 17.7 General behaviour of the Assouad spectrum 17.8 Projections 17.9 Distance sets 17.10 The Hölder mapping problem and dimension 17.11 Dimensions of graphs References List of notation Index PART ONE THEORY 1 Fractal geometry and dimension theoryIn this introductory chapter we briefly discuss the history and development of fractal geometry and dimension theory. We introduce and motivate some important concepts such as Hausdorff and box dimension. As part of this discussion we encounter covers and packings, which are central notions in dimension theory, and introduce the dimension theory of measures. The emergence of fractal geometryA fractal can be described as an object which exhibits interesting features on a large range of scales, see Figure 1.1. In pure mathematics, the Sierpiński triangle, the middle third Cantor set, the boundary of the Mandelbrot set, and the von Koch snowflake are archetypal examples and, in 'real life', examples include the surface of a lung, the horizon of a forest, and the distribution of stars in the galaxy. The fractal story began in the nineteenth century with the appearance of a multitude of strange examples exhibiting what we now understand as fractal phenomena. These included Weierstrass' example of a continuous nowhere differentiable function, Cantor's construction of an uncountable set with zero length, and Brown's observations on the path taken by a piece of pollen suspended in water (Brownian motion). During the first half of the twentieth century the mathematical foundations for fractal geometry were laid down by, for example, Besicovitch, Bouligand, Hausdorff, Julia, Marstrand and Sierpiński, and the theory was unified and popularised by the extensive writings of Mandelbrot in the 1970s, for example [197]. It was Mandelbrot who co...

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“…We write dim qA F to denote the quasi-Assouad dimension of F and dim qL F to denote the quasilower dimension of F . These were introduced in [14] and [31] and, due to work in [13,23,27], it is known that…”

Section: Dimensions Of Sets and Measuresmentioning

confidence: 99%

A new perspective on the Sullivan dictionary via Assouad type dimensions and spectra

Fraser¹,

Liam²

2020

Preprint

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We conduct a detailed analysis of the Assouad type dimensions and spectra in the context of limit sets of geometrically finite Kleinian groups and Julia sets of parabolic rational maps. Our analysis includes the Patterson-Sullivan measure in the Kleinian case and the analogous conformal measure in the Julia set case. Our results constitute a new perspective on the Sullivan dictionary between Kleinian groups and rational maps. We show that there exist both strong correspondences and strong differences between the two settings. The differences we observe are particularly interesting since they come from dimension theory, a subject where the correspondence described by the Sullivan dictionary is especially strong.

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Quasi-Lower Dimension and Quasi-Lipschitz Mapping

Cited by 16 publications

References 18 publications

Lower Assouad-Type Dimensions of Uniformly Perfect Sets in Doubling Metric Spaces

Lower Assouad-Type Dimensions of Uniformly Perfect Sets in Doubling Metric Spaces

Assouad Dimension and Fractal Geometry

A new perspective on the Sullivan dictionary via Assouad type dimensions and spectra

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