The Hausdorff dimension of self-affine fractals

1988 ◽  
Vol 103 (2) ◽  
pp. 339-350 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Hausdorff Dimension ◽  
Value Function ◽  
Singular Values ◽  
Singular Value ◽  
Compact Set ◽  
Linear Transformations ◽  
Box Counting ◽  
Box Counting Dimension ◽  
Finite Sequences ◽  
Almost All

AbstractIf T is a linear transformation on ℝn with singular values α1 ≥ α2 ≥ … ≥ αn, the singular value function øs is defined by where m is the smallest integer greater than or equal to s. Let T1, …, Tk be contractive linear transformations on ℝn. Let where the sum is over all finite sequences (i1, …, ir) with 1 ≤ ij ≤ k. Then for almost all (a1, …, ak) ∈ ℝnk, the unique non-empty compact set F satisfying has Hausdorff dimension min{d, n}. Moreover the ‘box counting’ dimension of F is almost surely equal to this number.

The dimension of self-affine fractals II

1992 ◽  
Vol 111 (1) ◽  
pp. 169-179 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Exact Expression ◽  
Singular Value ◽  
Compact Set ◽  
Value Functions ◽  
Affine Transformations ◽  
Box Counting ◽  
Box Counting Dimension ◽  
Almost All

AbstractA family {S1, ,Sk} of contracting affine transformations on Rn defines a unique non-empty compact set F satisfying . We obtain estimates for the Hausdorff and box-counting dimensions of such sets, and in particular derive an exact expression for the box-counting dimension in certain cases. These estimates are given in terms of the singular value functions of affine transformations associated with the Si. This paper is a sequel to 4, which presented a formula for the dimensions that was valid in almost all cases.

Exceptional sets for self-affine fractals

2008 ◽  
Vol 145 (3) ◽  
pp. 669-684 ◽  
Author(s):  
KENNETH FALCONER ◽  
JUN MIAO
Keyword(s):  
Hausdorff Dimension ◽  
Value Function ◽  
Singular Value ◽  
Exceptional Sets ◽  
Fourier Dimension ◽  
Almost All

AbstractUnder certain conditions the ‘singular value function’ formula gives the Hausdorff dimension of self-affine fractals for almost all parameters in a family. We show that the size of the set of exceptional parameters is small both in the sense of Hausdorff dimension and Fourier dimension.

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.

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 On the carrying dimension of occupation measures for self-affine random fields

2019 ◽  
Vol 39 (2) ◽  
pp. 459-479
Author(s):  
Peter Kern ◽  
Ercan Sönmez
Keyword(s):  
Hausdorff Dimension ◽  
Large Class ◽  
General Formula ◽  
Random Fields ◽  
Value Function ◽  
Singular Value ◽  
Occupation Measure ◽  
Occupation Measures ◽  
Alternative Approach ◽  
Path Properties

Hausdorff dimension results are a classical topic in the study of path properties of random fields. This article presents an alternative approach to Hausdorff dimension results for the sample functions of a large class of self-affine random fields. The aim is to demonstrate the following interesting relation to a series of articles by U. Zähle 1984, 1988, 1990, 1991. Under natural regularity assumptions, we prove that the Hausdorff dimension of the graph of self-affine fields coincides with the carrying dimension of the corresponding self-affine random occupation measure introduced by U. Zähle. As a remarkable consequence we obtain a general formula for the Hausdorff dimension given by means of the singular value function.

ON THE RELATIVE FREQUENCY OF RESIDUE CLASSES IN PASCAL’S TRIANGLE MODULO A PRIME

Fractals ◽  
2019 ◽  
Vol 27 (06) ◽  
pp. 1950098
Author(s):  
DAVID M. BRADLEY ◽  
DANIEL D’ALESSIO ◽  
ANDRÉ KHALIL ◽  
ROBERT G. NIEMEYER ◽  
ELLIOT OSSANNA ◽  
...  
Keyword(s):  
Relative Frequency ◽  
Linear Functions ◽  
Compact Set ◽  
Pascal’S Triangle ◽  
Residue Classes ◽  
Residue Formula ◽  
Residue Modulo ◽  
Box Counting ◽  
Box Counting Dimension ◽  
Pascal's Triangle

When the entries of Pascal’s triangle which are congruent to a given nonzero residue modulo a fixed prime are mapped to corresponding locations of the unit square, a fractal-like structure emerges. In a previous publication, Bradley, Khalil, Niemeyer and Ossanna [The box-counting dimension of Pascal’s triangle [Formula: see text] mod [Formula: see text], Fractals 26(5) (2018) 1850071] showed that this mapping yields a nonempty compact set which can be realized as a limit of a sequence of sets representing incrementally refined approximations. Moreover, it was shown therein that for any fixed prime, the sequence converges to the same set, regardless of the nonzero residue or combination of nonzero residues considered. Consequently, the fractal (box-counting) dimension of the limiting set is independent of the residue. To study the relative frequency of various residue classes in the sequence of approximating sets, it would be desirable to have a closed-form formula for the number of entries in the first [Formula: see text] rows of Pascal’s triangle which are congruent to a given nonzero residue [Formula: see text] modulo the prime [Formula: see text]. Unfortunately, the numerical evidence presented in this paper supports the contention that there is no such formula. Nevertheless, the evidence indicates that for sufficiently large primes [Formula: see text], the number of entries congruent to [Formula: see text] for [Formula: see text], [Formula: see text], and [Formula: see text] is well approximated by the respective linear functions [Formula: see text], [Formula: see text], and [Formula: see text]. In particular, for large primes [Formula: see text] there are approximately six times as many occurrences of the residue [Formula: see text] in the first [Formula: see text] rows of Pascal’s triangle reduced modulo [Formula: see text] than there are of any other residue [Formula: see text] in the range [Formula: see text], and three times as many as [Formula: see text]. On the other hand, if we let the nonnegative integer [Formula: see text] vary while keeping the prime [Formula: see text] fixed, and look at the relative frequency of various residue classes that occur in the first [Formula: see text] rows, the seemingly substantial differences in frequency between [Formula: see text], [Formula: see text], and [Formula: see text] when [Formula: see text] are increasingly dissipated as [Formula: see text] grows without bound. We show that in the limit as [Formula: see text] tends to infinity, all nonzero residues are equally represented with asymptotic proportion [Formula: see text].

Fractal Properties of Generalized Sierpiński Triangles

Fractals ◽  
1998 ◽  
Vol 06 (01) ◽  
pp. 31-41 ◽  
Author(s):  
Kenneth J. Falconer ◽  
Birger Lammering
Keyword(s):  
Singular Values ◽  
Affine Transformations ◽  
Sierpinski Triangle ◽  
Box Counting ◽  
Fractal Properties ◽  
Box Counting Dimension

We calculate the box-counting dimension of a self-affine version of the Sierpiński triangle. This is done by investigating the singular values of the affine transformations. We also investigate multifractal features of self-affine measures supported by certain generalized Sierpiński triangles.

scholarly journals Sample Path Properties of Generalized Random Sheets with Operator Scaling

2020 ◽  
Author(s):  
Ercan Sönmez
Keyword(s):  
Hausdorff Dimension ◽  
Random Fields ◽  
Sample Path ◽  
Polar Coordinates ◽  
Box Counting ◽  
Sample Path Properties ◽  
Box Counting Dimension ◽  
Uniform Modulus Of Continuity ◽  
Stable Random Fields ◽  
Path Properties

Abstract We consider operator scaling $$\alpha $$ α -stable random sheets, which were introduced in Hoffmann (Operator scaling stable random sheets with application to binary mixtures. Dissertation Universität Siegen, 2011). The idea behind such fields is to combine the properties of operator scaling $$\alpha $$ α -stable random fields introduced in Biermé et al. (Stoch Proc Appl 117(3):312–332, 2007) and fractional Brownian sheets introduced in Kamont (Probab Math Stat 16:85–98, 1996). We establish a general uniform modulus of continuity of such fields in terms of the polar coordinates introduced in Biermé et al. (2007). Based on this, we determine the box-counting dimension and the Hausdorff dimension of the graph of a trajectory over a non-degenerate cube $$I \subset {\mathbb {R}}^d$$ I ⊂ R d .

scholarly journals Generalised Cantor sets and the dimension of products

2015 ◽  
Vol 160 (1) ◽  
pp. 51-75 ◽  
Author(s):  
ERIC J. OLSON ◽  
JAMES C. ROBINSON ◽  
NICHOLAS SHARPLES
Keyword(s):  
Hausdorff Dimension ◽  
Large Class ◽  
Cantor Sets ◽  
Length Scales ◽  
Box Counting ◽  
Box Counting Dimension ◽  
The Relationship

AbstractIn this paper we consider the relationship between the Assouad and box-counting dimension and how both behave under the operation of taking products. We introduce the notion of ‘equi-homogeneity’ of a set, which requires a uniformity in the cardinality of local covers at all length-scales and at all points, and we show that a large class of homogeneous Moran sets have this property. We prove that the Assouad and box-counting dimensions coincide for sets that have equal upper and lower box-counting dimensions provided that the set ‘attains’ these dimensions (analogous to ‘s-sets’ when considering the Hausdorff dimension), and the set is equi-homogeneous. Using this fact we show that for any α ∈ (0, 1) and any β, γ ∈ (0, 1) such that β + γ ⩾ 1 we can construct two generalised Cantor sets C and D such that dimBC = αβ, dimBD = α γ, and dimAC = dimAD = dimA (C × D) = dimB (C × D) = α.

scholarly journals Exact relation between singular value and eigenvalue statistics

2016 ◽  
Vol 05 (04) ◽  
pp. 1650015 ◽  
Author(s):  
Mario Kieburg ◽  
Holger Kösters
Keyword(s):  
Singular Values ◽  
Singular Value ◽  
Specific Class ◽  
Rotation Invariant ◽  
Exact Relation ◽  
Finite Matrix ◽  
Eigenvalue Statistics ◽  
Probability Densities ◽  
Analytical Relations ◽  
Matrix Spaces

We use classical results from harmonic analysis on matrix spaces to investigate the relation between the joint densities of the singular values and the eigenvalues for complex random matrices which are bi-unitarily invariant (also known as isotropic or unitary rotation invariant). We prove that each of these joint densities determines the other one. Moreover, we construct an explicit formula relating both joint densities at finite matrix dimension. This relation covers probability densities as well as signed densities. With the help of this relation we derive general analytical relations among the corresponding kernels and biorthogonal functions for a specific class of polynomial ensembles. Furthermore, we show how to generalize the relation between the singular value and eigenvalue statistics to certain situations when the ensemble is deformed by a term which breaks the bi-unitary invariance.

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