Hausdorff Dimension Spectrum of Self-Affine Carpets Indexed by Nonlinear Fibre-Coding

2009 ◽  
Author(s):  
Yongxin Gui
Keyword(s):  
Hausdorff Dimension ◽  
Dimension Spectrum ◽  
Hausdorff Dimension Spectrum

scholarly journals Hereditarily just infinite profinite groups with complete Hausdorff dimension spectrum

2019 ◽  
Vol 18 (11) ◽  
pp. 1950216
Author(s):  
Yiftach Barnea ◽  
Matteo Vannacci
Keyword(s):  
Hausdorff Dimension ◽  
Wreath Products ◽  
Inverse Limits ◽  
Normal Subgroups ◽  
Profinite Groups ◽  
Dimension Spectrum ◽  
Product Action ◽  
Hausdorff Dimension Spectrum

We prove that the inverse limits of certain iterated wreath products in product action have complete Hausdorff dimension spectrum with respect to their unique maximal filtration of open normal subgroups. Moreover we can produce explicitly subgroups with a specified Hausdorff dimension.

Dimension spectra of lines1

Computability ◽  
2021 ◽  
pp. 1-28
Author(s):  
Neil Lutz ◽  
D.M. Stull
Keyword(s):  
Hausdorff Dimension ◽  
Kolmogorov Complexity ◽  
Euclidean Plane ◽  
Unit Interval ◽  
Packing Dimension ◽  
Hausdorff Dimensions ◽  
Dimension Spectrum ◽  
Algorithmic Dimension

This paper investigates the algorithmic dimension spectra of lines in the Euclidean plane. Given any line L with slope a and vertical intercept b, the dimension spectrum sp ( L ) is the set of all effective Hausdorff dimensions of individual points on L. We draw on Kolmogorov complexity and geometrical arguments to show that if the effective Hausdorff dimension dim ( a , b ) is equal to the effective packing dimension Dim ( a , b ), then sp ( L ) contains a unit interval. We also show that, if the dimension dim ( a , b ) is at least one, then sp ( L ) is infinite. Together with previous work, this implies that the dimension spectrum of any line is infinite.

Representation of families of sets by measures, dimension spectra and Diophantine approximation

2000 ◽  
Vol 128 (1) ◽  
pp. 111-121 ◽  
Author(s):  
K. J. FALCONER
Keyword(s):  
Dynamical Systems ◽  
Hausdorff Dimension ◽  
Diophantine Approximation ◽  
Multifractal Spectrum ◽  
Kleinian Groups ◽  
Local Dimension ◽  
Dimension Spectrum ◽  
Hyperbolic Dynamical Systems

A family of sets {Fd}d is said to be ‘represented by the measure μ’ if, for each d, the set Fd comprises those points at which the local dimension of μ takes some specific value (depending on d). Finding the Hausdorff dimension of these sets may then be thought of as finding the dimension spectrum, or multifractal spectrum, of μ. This situation pertains surprisingly often, with many familiar families of sets representable by measures which have simple dimension spectra. Examples are given from Diophantine approximation, Kleinian groups and hyperbolic dynamical systems.

-SPECTRUM OF SELF-SIMILAR MEASURES WITH OVERLAPS IN THE ABSENCE OF SECOND-ORDER IDENTITIES

2018 ◽  
Vol 106 (1) ◽  
pp. 56-103 ◽  
Author(s):  
SZE-MAN NGAI ◽  
YUANYUAN XIE
Keyword(s):  
Hausdorff Dimension ◽  
Finite Type ◽  
Iterated Function Systems ◽  
Second Order ◽  
Higher Dimension ◽  
Closed Formula ◽  
Dimension Spectrum ◽  
Iterated Function ◽  
Self Similar ◽  
Set Up

For the class of self-similar measures in $\mathbb{R}^{d}$ with overlaps that are essentially of finite type, we set up a framework for deriving a closed formula for the $L^{q}$-spectrum of the measure for $q\geq 0$. This framework allows us to include iterated function systems that have different contraction ratios and those in higher dimension. For self-similar measures with overlaps, closed formulas for the $L^{q}$-spectrum have only been obtained earlier for measures satisfying Strichartz’s second-order identities. We illustrate how to use our results to prove the differentiability of the $L^{q}$-spectrum, obtain the multifractal dimension spectrum, and compute the Hausdorff dimension of the measure.

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