Fractal dimension: Limit capacity or Hausdorff dimension?

1990 ◽  
Vol 58 (10) ◽  
pp. 986-988 ◽  
Author(s):  
Christopher Essex ◽  
M. A. H. Nerenberg
Keyword(s):  
Fractal Dimension ◽  
Hausdorff Dimension ◽  
Limit Capacity

scholarly journals Discontinuity of Hausdorff dimension and limit capacity on arcs of diffeomorphisms

1989 ◽  
Vol 9 (3) ◽  
pp. 403-425 ◽  
Author(s):  
Lorenzo J. Diaz ◽  
Marcelo Viana
Keyword(s):  
Hausdorff Dimension ◽  
Equilibrium Measures ◽  
Open Set ◽  
Limit Capacity ◽  
Nonwandering Set

AbstractWe consider one-parameter families of torus diffeomorphisms that bifurcate from global hyperbolic maps (Anosov) to DA maps (derived from Anosov). For an open set of these families, we show that the Hausdorff dimension and limit capacity of the nonwandering set are not continuous across the bifurcation. We also study the behaviour of equilibrium measures near the bifurcation.

ON THE MENDÈS FRANCE FRACTAL DIMENSION OF STRANGE ATTRACTORS: THE HÉNON CASE

2002 ◽  
Vol 12 (07) ◽  
pp. 1549-1563 ◽  
Author(s):  
M. PIACQUADIO ◽  
R. HANSEN ◽  
F. PONTA
Keyword(s):  
Fractal Dimension ◽  
Hausdorff Dimension ◽  
Strange Attractors ◽  
Planar Curves ◽  
Box Counting ◽  
Box Counting Dimension

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.

THE CANTOR SET’S MULTI-FRACTAL SPECTRUM FORMED BY DIFFERENT PROBABILITY FACTORS IN MATHEMATICAL EXPERIMENT

Fractals ◽  
2017 ◽  
Vol 25 (01) ◽  
pp. 1750002 ◽  
Author(s):  
XUEZAI PAN ◽  
XUDONG SHANG ◽  
MINGGANG WANG ◽  
ZUO-FEI
Keyword(s):  
Fractal Dimension ◽  
Hausdorff Dimension ◽  
Statistical Physics ◽  
Computer Calculation ◽  
Fractal Dimensions ◽  
The Other ◽  
Mathematical Experiment ◽  
Maximal Width ◽  
Fractal Spectrum ◽  
Vertical Height

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.

FRACTAL DIMENSION OF A SPECIAL CONTINUOUS FUNCTION

Fractals ◽  
2018 ◽  
Vol 26 (04) ◽  
pp. 1850048 ◽  
Author(s):  
NING LIU ◽  
KUI YAO
Keyword(s):  
Fractal Dimension ◽  
Continuous Function ◽  
Hausdorff Dimension ◽  
Fractional Integral ◽  
Fractal Function

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.

scholarly journals Adsorption on Fractal Surfaces: A Non Linear Modeling Approach of a Fractional Behavior

2021 ◽  
Vol 5 (3) ◽  
pp. 65
Author(s):  
Vincent Tartaglione ◽  
Jocelyn Sabatier ◽  
Christophe Farges
Keyword(s):  
Fractal Dimension ◽  
Hausdorff Dimension ◽  
Nonlinear Model ◽  
Fractional Dynamics ◽  
Random Sequential Adsorption ◽  
Linear Modeling ◽  
High Coverage ◽  
Fractal Surfaces ◽  
Affine Model ◽  
Sequential Adsorption

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.

scholarly journals Calculating Hausdorff Dimension in Higher Dimensional Spaces

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 564 ◽  
Author(s):  
Manuel Fernández-Martínez ◽  
Juan Luis García García Guirao ◽  
Miguel Ángel Sánchez-Granero
Keyword(s):  
Fractal Dimension ◽  
Hausdorff Dimension ◽  
Fractal Structure ◽  
General Setting ◽  
Higher Dimensional

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.

The 2-adic derivatives and fractal dimension of Takagi-like function on 2-series field

2020 ◽  
Vol 23 (3) ◽  
pp. 875-885
Author(s):  
Bo Wu
Keyword(s):  
Fractal Dimension ◽  
Hausdorff Dimension ◽  
Box Dimension ◽  
Derivatives Of

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.

scholarly journals A new fractal dimension: The topological Hausdorff dimension

2015 ◽  
Vol 274 ◽  
pp. 881-927 ◽  
Author(s):  
Richárd Balka ◽  
Zoltán Buczolich ◽  
Márton Elekes
Keyword(s):  
Fractal Dimension ◽  
Hausdorff Dimension
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