scholarly journals A new fractal dimension: The topological Hausdorff dimension

2015 ◽  
Vol 274 ◽  
pp. 881-927 ◽  
Author(s):  
Richárd Balka ◽  
Zoltán Buczolich ◽  
Márton Elekes
2002 ◽  
Vol 12 (07) ◽  
pp. 1549-1563 ◽  
Author(s):  
M. PIACQUADIO ◽  
R. HANSEN ◽  
F. PONTA

We consider the Hénon attractor ℋ as a curve limit of continuous planar curves H(n), as n → ∞. We describe a set of tools for studying the Hausdorff dimension dim H of a certain family of such curves, and we adapt these tools to the particular case of the Hénon attractor, estimating its Hausdorff dimension dim H (ℋ) to be about 1.258, a number smaller than the usual estimates for the box-counting dimension of the attractor. We interpret this discrepancy.


Fractals ◽  
2017 ◽  
Vol 25 (01) ◽  
pp. 1750002 ◽  
Author(s):  
XUEZAI PAN ◽  
XUDONG SHANG ◽  
MINGGANG WANG ◽  
ZUO-FEI

With the purpose of researching the changing regularities of the Cantor set’s multi-fractal spectrums and generalized fractal dimensions under different probability factors, from statistical physics, the Cantor set is given a mass distribution, when the mass is given with different probability ratios, the different multi-fractal spectrums and the generalized fractal dimensions will be acquired by computer calculation. The following conclusions can be acquired. On one hand, the maximal width of the multi-fractal spectrum and the maximal vertical height of the generalized fractal dimension will become more and more narrow with getting two probability factors closer and closer. On the other hand, when two probability factors are equal to 1/2, both the multi-fractal spectrum and the generalized fractal dimension focus on the value 0.6309, which is not the value of the physical multi-fractal spectrum and the generalized fractal dimension but the mathematical Hausdorff dimension.


Fractals ◽  
2018 ◽  
Vol 26 (04) ◽  
pp. 1850048 ◽  
Author(s):  
NING LIU ◽  
KUI YAO

In this paper, we mainly construct a special fractal function defined on [Formula: see text] of unbounded variation by method of iteration, and prove that both Box and Hausdorff dimension of this function be 1. Further, we discuss the Riemann–Liouville fractional integral of this function. Finally, some numerical and graphic results are provided to characterize this special fractal function.


2021 ◽  
Vol 5 (3) ◽  
pp. 65
Author(s):  
Vincent Tartaglione ◽  
Jocelyn Sabatier ◽  
Christophe Farges

This article deals with the random sequential adsorption (RSA) of 2D disks of the same size on fractal surfaces with a Hausdorff dimension 1<d<2. According to the literature and confirmed by numerical simulations in the paper, the high coverage regime exhibits fractional dynamics, i.e., dynamics in t−1/d where d is the fractal dimension of the surface. The main contribution this paper is that it proposes to capture this behavior with a particular class of nonlinear model: a driftless control affine model.


Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 564 ◽  
Author(s):  
Manuel Fernández-Martínez ◽  
Juan Luis García García Guirao ◽  
Miguel Ángel Sánchez-Granero

In this paper, we prove the identity dim H(F) = d dim H(a?1(F)), where dim H denotesHausdorff dimension, F Rd, and a : [0, 1] ! [0, 1]d is a function whose constructive definition isaddressed from the viewpoint of the powerful concept of a fractal structure. Such a result standsparticularly from some other results stated in a more general setting. Thus, Hausdorff dimension ofhigher dimensional subsets can be calculated from Hausdorff dimension of 1-dimensional subsets of[0, 1]. As a consequence, Hausdorff dimension becomes available to deal with the effective calculationof the fractal dimension in applications by applying a procedure contributed by the authors inprevious works. It is also worth pointing out that our results generalize both Skubalska-Rafajłowiczand García-Mora-Redtwitz theorems.


2020 ◽  
Vol 23 (3) ◽  
pp. 875-885
Author(s):  
Bo Wu

AbstractIn this paper, we consider a Takagi-like function on 2-series field and give its 2-adic derivatives by applying Vladimirov operator. The 2-adic derivatives of Takagi-like function with order 0 < α < 1 exist and show some fractal feature. Furthermore, both box dimension and Hausdorff dimension of the graph of its derivatives are obtained and equal to 1 + α.


2017 ◽  
Vol 31 (09) ◽  
pp. 1750060
Author(s):  
Harumi Hikita ◽  
Hirohisa Ishikawa ◽  
Kazuo Morigaki

Hydrogen diffusion in a-Si:H with exponential distribution of the states in energy exhibits the fractal structure. It is shown that a probability [Formula: see text] of the pausing time [Formula: see text] has a form of [Formula: see text] ([Formula: see text]: fractal dimension). It is shown that the fractal dimension [Formula: see text]/[Formula: see text] ([Formula: see text]: hydrogen temperature, [Formula: see text]: a temperature corresponding to the width of exponential distribution of the states in energy) is in agreement with the Hausdorff dimension. A fractal graph for the case of [Formula: see text] is like the Cantor set. A fractal graph for the case of [Formula: see text] is like the Koch curves. At [Formula: see text], hydrogen migration exhibits Brownian motion. Hydrogen diffusion in a-Si:H should be the fractal process.


Fractals ◽  
1995 ◽  
Vol 03 (04) ◽  
pp. 747-754 ◽  
Author(s):  
M. ZÄHLE

In this note we introduce the concept of local average dimension of a measure µ, at x∈ℝn as the unique exponent where the lower average density of µ, at x jumps from zero to infinity. Taking the essential infimum or supremum over x we obtain the lower and upper average dimensions of µ, respectively. The average dimension of an analytic set E is defined as the supremum over the upper average dimensions of all measures supported by E. These average dimensions lie between the corresponding Hausdorff and packing dimensions and the inequalities can be strict. We prove that the local Hausdorff dimensions and the local average dimensions of µ at almost all x are invariant under orthogonal projections onto almost all m- dimensional linear subspaces of higher dimension. The corresponding global results for µ and E (which are known for Hausdorff dimension) follow immediately.


1990 ◽  
Vol 58 (10) ◽  
pp. 986-988 ◽  
Author(s):  
Christopher Essex ◽  
M. A. H. Nerenberg

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