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 Some Dimensional Results of a Class of Homogeneous Moran Sets

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Jingru Zhang ◽  
Yanzhe Li ◽  
Manli Lou
Keyword(s):  
Hausdorff Dimension ◽  
Packing Dimension ◽  
Box Dimension ◽  
Maximum Value

In this paper, we construct a class of special homogeneous Moran sets: m k -quasi-homogeneous perfect sets, and obtain the Hausdorff dimension of the sets under some conditions. We also prove that the upper box dimension and the packing dimension of the sets can get the maximum value of the homogeneous Moran sets under the condition sup k ≥ 1 m k < ∞ and estimate the upper box dimension of the sets in two cases.

scholarly journals Dimensions of Fractional Brownian Images

2021 ◽  
Author(s):  
Stuart A. Burrell
Keyword(s):  
Brownian Motion ◽  
Hausdorff Dimension ◽  
Stochastic Processes ◽  
Fractional Brownian Motion ◽  
Box Dimension ◽  
General Strategy ◽  
Borel Sets ◽  
Exceptional Sets ◽  
Explicit Condition ◽  
Box Dimensions

AbstractThis paper concerns the intermediate dimensions, a spectrum of dimensions that interpolate between the Hausdorff and box dimensions. Potential-theoretic methods are used to produce dimension bounds for images of sets under Hölder maps and certain stochastic processes. We apply this to compute the almost-sure value of the dimension of Borel sets under index-$$\alpha $$ α fractional Brownian motion in terms of dimension profiles defined using capacities. As a corollary, this establishes continuity of the profiles for Borel sets and allows us to obtain an explicit condition showing how the Hausdorff dimension of a set may influence the typical box dimension of Hölder images such as projections. The methods used propose a general strategy for related problems; dimensional information about a set may be learned from analysing particular fractional Brownian images of that set. To conclude, we obtain bounds on the Hausdorff dimension of exceptional sets, with respect to intermediate dimensions, in the setting of projections.

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.

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.

Local dimension for piecewise monotonic maps on the interval

1995 ◽  
Vol 15 (6) ◽  
pp. 1119-1142 ◽  
Author(s):  
Franz Hofbauer
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measures ◽  
Positive Characteristic ◽  
Characteristic Exponent ◽  
Local Dimension ◽  
Pressure Function ◽  
Piecewise Monotonic ◽  
Almost Everywhere ◽  
Conformal Measures

AbstractThe local dimension of invariant and conformal measures for piecewise monotonic transformations on the interval is considered. For ergodic invariant measures m with positive characteristic exponent χm we show that the local dimension exists almost everywhere and equals hm/χm For certain conformal measures we show a relation between a pressure function and the Hausdorff dimension of sets, on which the local dimension is constant.

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&lt;$\sqrt2/2$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 RELATIONSHIP BETWEEN FRACTAL DIMENSIONS OF BESICOVITCH FUNCTION AND THE ORDER OF HADAMARD FRACTIONAL INTEGRAL

Fractals ◽  
2020 ◽  
Vol 28 (07) ◽  
pp. 2050128
Author(s):  
BIN WANG ◽  
WENLONG JI ◽  
LEGUI ZHANG ◽  
XUAN LI
Keyword(s):  
Fractional Integral ◽  
Fractal Dimensions ◽  
Packing Dimension ◽  
Box Dimension ◽  
Dimension Formula ◽  
The Relationship ◽  
Hadamard Fractional Integral

In this paper, we mainly research on Hadamard fractional integral of Besicovitch function. A series of propositions of Hadamard fractional integral of [Formula: see text] have been proved first. Then, we give some fractal dimensions of Hadamard fractional integral of Besicovitch function including Box dimension, [Formula: see text]-dimension and Packing dimension. Finally, relationship between the order of Hadamard fractional integral and fractal dimensions of Besicovitch function has also been given.

On the Hausdorff dimension of general Cantor sets

1965 ◽  
Vol 61 (3) ◽  
pp. 679-694 ◽  
Author(s):  
A. F. Beardon
Keyword(s):  
Hausdorff Dimension ◽  
Cantor Set ◽  
Cantor Sets ◽  
Local Dimension ◽  
Underlying Space

Introduction and notation. In this paper a generalization of the Cantor set is discussed. Upper and lower estimates of the Hausdorff dimension of such a set are obtained and, in particular, it is shown that the Hausdorff dimension is always positive and less than that of the underlying space. The concept of local dimension at a point is introduced and studied as a function of that point.

Metric properties of some fractal sets and applications of inverse pressure

2009 ◽  
Vol 148 (3) ◽  
pp. 553-572 ◽  
Author(s):  
EUGEN MIHAILESCU
Keyword(s):  
Hausdorff Dimension ◽  
General Setting ◽  
Topological Pressure ◽  
Box Dimension ◽  
Stable Manifolds ◽  
Fractal Sets ◽  
Metric Properties ◽  
Basic Set ◽  
Unstable Set ◽  
Unstable Manifolds

AbstractWe consider iterations of smooth non-invertible maps on manifolds of real dimension 4, which are hyperbolic, conformal on stable manifolds and finite-to-one on basic sets. The dynamics of non-invertible maps can be very different than the one of diffeomorphisms, as was shown for example in [4,7,12,17,19], etc. In [13] we introduced a notion of inverse topological pressureP−which can be used for estimates of the stable dimension δs(x) (i.e the Hausdorff dimension of the intersection between the local stable manifoldWsr(x) and the basic set Λ,x∈ Λ). In [10] it is shown that the usual Bowen equation is not always true in the case of non-invertible maps. By using the notion of inverse pressureP−, we showed in [13] that δs(x) ≤ts(ϵ), wherets(ϵ) is the unique zero of the functiont→P−(tφs, ϵ), for φs(y):= log|Dfs(y)|,y∈ Λ and ϵ > 0 small. In this paper we prove that if Λ is not a repellor, thents(ϵ) < 2 for any ϵ > 0 small enough. In [11] we showed that a holomorphic s-hyperbolic map on2has a global unstable set with empty interior. Here we show in a more general setting than in [11], that the Hausdorff dimension of the global unstable setWu() is strictly less than 4 under some technical derivative condition. In the non-invertible case we may have (infinitely) many unstable manifolds going through a point in Λ, and the number of preimages belonging to Λ may vary. In [17], Qian and Zhang studied the case of attractors for non-invertible maps and gave a condition for a basic set to be an attractor in terms of the pressure of the unstable potential. In our case the situation is different, since the local unstable manifolds may intersect both inside and outside Λ and they do not form a foliation like the stable manifolds. We prove here that the upper box dimension ofWsr(x) ∩ Λ is less thants(ϵ) for any pointx∈ Λ. We give then an estimate of the Hausdorff dimension ofWu() by a different technique, using the Holder continuity of the unstable manifolds with respect to their prehistories.

Sign in / Sign up

Export Citation Format

Share Document