Empirical multifractal moment measures and moment scaling functions of self-similar multifractals

2002 ◽  
Vol 133 (3) ◽  
pp. 459-485 ◽  
Author(s):  
L. OLSEN
Keyword(s):  
Lipschitz Constant ◽  
The Self ◽  
Probability Vector ◽  
Periodic Functions ◽  
Open Set Condition ◽  
Similar Measure ◽  
Open Set ◽  
Moment Measures ◽  
Self Similar ◽  
Exact Rate Of Convergence

Let Si: ℝd → ℝd for i = 1, …, n be contracting similarities, and let (p1, …, pn) be a probability vector. Let K and μ be the self-similar set and the self-similar measure associated with (Si,pi)i. For q ∈ ℝ and r > 0, define the qth covering moment and the qth packing moment of μ by[formula here]where the infimum is taken over all r-spanning subsets E of K, and the supremum is taken over all r-separated subsets F of K. If the Open Set Condition (OSC) is satisfied then it is well known that[formula here]where β(q) is defined by [sum ]ipqirβi(q) = 1 (here ri denotes the Lipschitz constant of Si). Assuming the OSC, we determine the exact rate of convergence in (*): there exist multiplicatively periodic functions πq, Πq: (0,∞) → ℝ such that[formula here]where ε(r) → 0 as r[searr ]0. As an application of (**) we show that the empirical multi-fractal moment measures converges weakly:[formula here]where, for each positive r, Er is a (suitable) minimal r-spanning subset of K and Fr is a (suitable) maximal r-separated subset of K, and [Hscr ]q,β(q)μ and [Pscr ]q,β(q)μ are the multifractal Hausdorff measure and the multifractal packing measure, respectively.

scholarly journals The mutual singularity of the relative multifractal measures

2021 ◽  
Vol 8 (1) ◽  
pp. 18-26
Author(s):  
Zied Douzi ◽  
Bilel Selmi
Keyword(s):  
General Framework ◽  
The Self ◽  
Open Set Condition ◽  
Open Set ◽  
Self Similar ◽  
Bernoulli Measures ◽  
Multifractal Measures

Abstract M. Das proved that the relative multifractal measures are mutually singular for the self-similar measures satisfying the significantly weaker open set condition. The aim of this paper is to show that these measures are mutually singular in a more general framework. As examples, we apply our main results to quasi-Bernoulli measures.

scholarly journals The quantization for in-homogeneous self-similar measures with in-homogeneous open set condition

2015 ◽  
Vol 26 (05) ◽  
pp. 1550030 ◽  
Author(s):  
Sanguo Zhu
Keyword(s):  
Open Set Condition ◽  
Similar Measure ◽  
Quantization Dimension ◽  
Open Set ◽  
Quantization Coefficient ◽  
Self Similar ◽  
Quantization Problem

Let [Formula: see text] be a family of contractive similitudes satisfying the open set condition. Let ν be a self-similar measure associated with [Formula: see text]. We study the quantization problem for the in-homogeneous self-similar measure μ associated with a condensation system [Formula: see text]. Assuming a version of in-homogeneous open set condition for this system, we prove the existence of the quantization dimension for μ of order r ∈ (0, ∞) and determine its exact value ξr. The finiteness and positivity of the ξr-dimensional upper and lower quantization coefficient are also explored.

INTERSECTIONS OF TRANSLATION OF A CLASS OF SIERPINSKI CARPETS

Fractals ◽  
2018 ◽  
Vol 26 (03) ◽  
pp. 1850034
Author(s):  
JIAN LU ◽  
BO TAN ◽  
YURU ZOU
Keyword(s):  
Iterated Function System ◽  
The Self ◽  
Open Set Condition ◽  
Moran Set ◽  
Open Set ◽  
Iterated Function ◽  
Directed Set ◽  
Self Similar ◽  
Sierpinski Carpets ◽  
Intersection Formula

For [Formula: see text], a middle-[Formula: see text] Sierpinski carpet [Formula: see text] is defined as the self-similar set generated by the iterated function system (IFS) [Formula: see text], where [Formula: see text] is defined by [Formula: see text] Here, [Formula: see text]. In this paper, for [Formula: see text], we investigated the equivalent characterizations of the intersection [Formula: see text] being a generalized Moran set. Furthermore, under some conditions, we show that [Formula: see text] can be represented as a graph-directed set satisfying the open set condition (OSC), and then the Hausdorff dimension can be explicitly calculated.

Non-spectral Problem for Some Self-similar Measures

2019 ◽  
Vol 63 (2) ◽  
pp. 318-327
Author(s):  
Ye Wang ◽  
Xin-Han Dong ◽  
Yue-Ping Jiang
Keyword(s):  
Spectral Problem ◽  
The Self ◽  
Exponential Functions ◽  
Similar Measure ◽  
Self Similar

AbstractSuppose that $0<|\unicode[STIX]{x1D70C}|<1$ and $m\geqslant 2$ is an integer. Let $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}$ be the self-similar measure defined by $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}(\cdot )=\frac{1}{m}\sum _{j=0}^{m-1}\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}(\unicode[STIX]{x1D70C}^{-1}(\cdot )-j)$. Assume that $\unicode[STIX]{x1D70C}=\pm (q/p)^{1/r}$ for some $p,q,r\in \mathbb{N}^{+}$ with $(p,q)=1$ and $(p,m)=1$. We prove that if $(q,m)=1$, then there are at most $m$ mutually orthogonal exponential functions in $L^{2}(\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m})$ and $m$ is the best possible. If $(q,m)>1$, then there are any number of orthogonal exponential functions in $L^{2}(\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m})$.

Open set condition for graph directed self-similar structure

2013 ◽  
Vol 276 (1-2) ◽  
pp. 243-260 ◽  
Author(s):  
Tian-jia Ni ◽  
Zhi-ying Wen
Keyword(s):  
Open Set Condition ◽  
Open Set ◽  
Self Similar

scholarly journals SPACE-FILLING CURVES OF SELF-SIMILAR SETS (III): SKELETONS

Fractals ◽  
2020 ◽  
Vol 28 (02) ◽  
pp. 2050028
Author(s):  
HUI RAO ◽  
SHU-QIN ZHANG
Keyword(s):  
Finite Type ◽  
Type Condition ◽  
Space Filling ◽  
Open Set Condition ◽  
Substitution Rule ◽  
Space Filling Curves ◽  
Neighbor Graph ◽  
Open Set ◽  
Self Similar

Skeleton is a new notion designed for constructing space-filling curves of self-similar sets. In a previous paper by Dai and the authors [Space-filling curves of self-similar sets (II): Edge-to-trail substitution rule, Nonlinearity 32(5) (2019) 1772–1809] it was shown that for all the connected self-similar sets with a skeleton satisfying the open set condition, space-filling curves can be constructed. In this paper, we give a criterion of existence of skeletons by using the so-called neighbor graph of a self-similar set. In particular, we show that a connected self-similar set satisfying the finite-type condition always possesses skeletons: an algorithm is obtained here.

scholarly journals Lattice self-similar sets on the real line are not Minkowski measurable

2018 ◽  
Vol 40 (1) ◽  
pp. 221-232
Author(s):  
SABRINA KOMBRINK ◽  
STEFFEN WINTER
Keyword(s):  
Real Line ◽  
Iterated Function System ◽  
Function System ◽  
Open Set Condition ◽  
The Real ◽  
Open Set ◽  
Puzzle Piece ◽  
Iterated Function ◽  
Self Similar ◽  
Special Classes

We show that any non-trivial self-similar subset of the real line that is invariant under a lattice iterated function system (IFS) satisfying the open set condition (OSC) is not Minkowski measurable. So far, this has only been known for special classes of such sets. Thus, we provide the last puzzle-piece in proving that under the OSC a non-trivial self-similar subset of the real line is Minkowski measurable if and only if it is invariant under a non-lattice IFS, a 25-year-old conjecture.

scholarly journals A CONVEX SURFACE WITH FRACTAL CURVATURE

Fractals ◽  
2020 ◽  
Vol 28 (04) ◽  
pp. 2050059
Author(s):  
IANCU DIMA ◽  
RACHEL POPP ◽  
ROBERT S. STRICHARTZ ◽  
SAMUEL C. WIESE
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Convex Surface ◽  
The Self ◽  
The Finite Element Method ◽  
Similar Measure ◽  
Entire Spectrum ◽  
Self Similar ◽  
Polyhedral Approximations ◽  
Element Method

We construct a surface that is obtained from the octahedron by pushing out four of the faces so that the curvature is supported in a copy of the Sierpinski gasket (SG) in each of them, and is essentially the self similar measure on SG. We then compute the bottom of the spectrum of the associated Laplacian using the finite element method on polyhedral approximations of our surface, and speculate on the behavior of the entire spectrum.

ECCENTRIC DISTANCE SUM OF SIERPIŃSKI GASKET AND SIERPIŃSKI NETWORK

Fractals ◽  
2019 ◽  
Vol 27 (02) ◽  
pp. 1950016 ◽  
Author(s):  
JIN CHEN ◽  
LONG HE ◽  
QIN WANG
Keyword(s):  
Asymptotic Formula ◽  
Complex Networks ◽  
Sierpinski Gasket ◽  
The Self ◽  
Self Similarity ◽  
Sierpiński Gasket ◽  
Similar Measure ◽  
Self Similar ◽  
Eccentric Distance

The eccentric distance sum is concerned with complex networks. To obtain the asymptotic formula of eccentric distance sum on growing Sierpiński networks, we study some nonlinear integral in terms of self-similar measure on the Sierpiński gasket and use the self-similarity of distance and measure to obtain the exact value of this integral.

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