Dimensions of measures under orthogonal projections

1996 ◽  
Vol 28 (2) ◽  
pp. 344-345
Author(s):  
Martina Zähle
Keyword(s):  
Hausdorff Dimension ◽  
Packing Dimension ◽  
Packing Measure ◽  
Orthogonal Projections

Let dimH, E be the Hausdorff dimension and dimP, E the packing dimension of the subset E of ℝn given by the unique exponent where the corresponding Hausdorff or packing measure of E jumps from infinity to zero.

Dimensions of measures under orthogonal projections

1996 ◽  
Vol 28 (02) ◽  
pp. 344-345
Author(s):  
Martina Zähle
Keyword(s):  
Hausdorff Dimension ◽  
Packing Dimension ◽  
Packing Measure ◽  
Orthogonal Projections

Let dimH, E be the Hausdorff dimension and dimP, E the packing dimension of the subset E of ℝ n given by the unique exponent where the corresponding Hausdorff or packing measure of E jumps from infinity to zero.

On Hausdorff and packing dimension of product spaces

1996 ◽  
Vol 119 (4) ◽  
pp. 715-727 ◽  
Author(s):  
J. D. Howroyd
Keyword(s):  
Hausdorff Dimension ◽  
Metric Spaces ◽  
Main Idea ◽  
Packing Dimension ◽  
Packing Measure ◽  
Product Spaces ◽  
Image Position

AbstractWe show that for arbitrary metric spaces X and Y the following dimension inequalities hold:where ‘dim’ denotes Hausdorff dimension and ‘Dim’ denotes packing dimension. The main idea of the proof is to use modified constructions of the Hausdorff and packing measure to deduce appropriate inequalities for the measure of X × Y.

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.

scholarly journals Packing dimension and measure of homogeneous Cantor sets

2006 ◽  
Vol 74 (3) ◽  
pp. 443-448 ◽  
Author(s):  
H.K. Baek
Keyword(s):  
Lower Bound ◽  
Explicit Formula ◽  
Cantor Sets ◽  
Packing Dimension ◽  
Packing Measure ◽  
Hausdorff Measures ◽  
Packing Dimensions

For a class of homogeneous Cantor sets, we find an explicit formula for their packing dimensions. We then turn our attention to the value of packing measures. The exact value of packing measure for homogeneous Cantor sets has not yet been calculated even though that of Hausdorff measures was evaluated by Qu, Rao and Su in (2001). We give a reasonable lower bound for the packing measures of homogeneous Cantor sets. Our results indicate that duality does not hold between Hausdorff and packing measures.

scholarly journals Hausdorff dimension and uniform exponents in dimension two

2018 ◽  
Vol 167 (02) ◽  
pp. 249-284 ◽  
Author(s):  
YANN BUGEAUD ◽  
YITWAH CHEUNG ◽  
NICOLAS CHEVALLIER
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Packing Dimension ◽  
Two Dimensional ◽  
Joint Work ◽  
Unpublished Work

AbstractIn this paper we prove that the Hausdorff dimension of the set of (nondegenerate) singular two-dimensional vectors with uniform exponentμin (1/2, 1) is equal to 2(1 −μ) forμ⩾$\sqrt2/2$, whereas forμ<$\sqrt2/2$it is greater than 2(1 −μ) and at most equal to (3 − 2μ)(1 − μ)/(1 −μ+μ2). We also establish that this dimension tends to 4/3 (which is the dimension of the set of singular two-dimensional vectors) whenμtends to 1/2. These results improve upon previous estimates of R. Baker, joint work of the first author with M. Laurent, and unpublished work of M. Laurent. Moreover, we prove a lower bound for the packing dimension, which appears to be strictly greater than the Hausdorff dimension for μ ⩾ 0.565. . . .

The packing measure of self-affine carpets

1994 ◽  
Vol 115 (3) ◽  
pp. 437-450 ◽  
Author(s):  
Yuval Peres
Keyword(s):  
Packing Dimension ◽  
Packing Measure ◽  
Initial Pattern

AbstractWe show that the seif-affine sets considered by McMullen [15] and Bedford [2] have infinite packing measure in their packing dimension θ except when all non-empty rows of the initial pattern have the same number of rectangles. More precisely, the packing measure is infinite in the gauge tθ|logt|−1 and zero in the gauge tθ|logt|−1−δ for any δ > 0.

scholarly journals Some typical properties of dimensions of sets and measures

2005 ◽  
Vol 2005 (3) ◽  
pp. 239-254
Author(s):  
Józef Myjak
Keyword(s):  
Hausdorff Dimension ◽  
Correlation Dimension ◽  
Packing Dimension ◽  
Box Dimension ◽  
Local Dimension

This paper contains a review of recent results concerning typical properties of dimensions of sets and dimensions of measures. In particular, we are interested in the Hausdorff dimension, box dimension, and packing dimension of sets and in the Hausdorff dimension, box dimension, correlation dimension, concentration dimension, and local dimension of measures.

DIMENSION ANALYSIS OF CONTINUOUS FUNCTIONS WITH UNBOUNDED VARIATION

Fractals ◽  
2017 ◽  
Vol 25 (01) ◽  
pp. 1730001 ◽  
Author(s):  
JUN WANG ◽  
KUI YAO
Keyword(s):  
Hausdorff Dimension ◽  
Fractal Dimensions ◽  
Continuous Functions ◽  
Packing Dimension ◽  
Box Dimension ◽  
Box Counting ◽  
Dimension Analysis ◽  
Box Counting Dimension ◽  
Self Similar

In this paper, we mainly discuss fractal dimensions of continuous functions with unbounded variation. First, we prove that Hausdorff dimension, Packing dimension and Modified Box-counting dimension of continuous functions containing one UV point are [Formula: see text]. The above conclusion still holds for continuous functions containing finite UV points. More generally, we show the result that Hausdorff dimension of continuous functions containing countable UV points is [Formula: see text] also. Finally, Box dimension of continuous functions containing countable UV points has been proved to be [Formula: see text] when [Formula: see text] is self-similar.

scholarly journals Relative multifractal box-dimensions

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2841-2859 ◽  
Author(s):  
Najmeddine Attia ◽  
Bilel Selmi
Keyword(s):  
Hausdorff Dimension ◽  
Multifractal Spectrum ◽  
Packing Dimension ◽  
Probability Measures ◽  
Multifractal Formalism ◽  
The Relationship ◽  
Box Dimensions

Given two probability measures ? and ? on Rn. We define the upper and lower relative multifractal box-dimensions of the measure ? with respect to the measure ? and investigate the relationship between the multifractal box-dimensions and the relative multifractal Hausdorff dimension, the relative multifractal pre-packing dimension. We also, calculate the relative multifractal spectrum and establish the validity of multifractal formalism. As an application, we study the behavior of projections of measures obeying to the relative multifractal formalism.

Geometric measures for parabolic rational maps

1992 ◽  
Vol 12 (1) ◽  
pp. 53-66 ◽  
Author(s):  
M. Denker ◽  
M. Urbański
Keyword(s):  
Hausdorff Dimension ◽  
Hausdorff Measure ◽  
Julia Set ◽  
Rational Maps ◽  
Packing Measure ◽  
Rational Map ◽  
Dimensional Hausdorff Measure ◽  
Conformal Measure

AbstractLet h denote the Hausdorff dimension of the Julia set J(T) of a parabolic rational map T. In this paper we prove that (after normalisation) the h-conformal measure on J(T) equals the h-dimensional Hausdorff measure Hh on J(T), if h ≥ 1, and equals the h-dimensional packing measure Πh on J(T), if h ≤ 1. Moreover, if h < 1, then Hh = 0 and, if h > 1, then Πh(J(T)) = ∞.

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