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 Space-filling curves of self-similar sets (II): edge-to-trail substitution rule

Nonlinearity ◽  
2019 ◽  
Vol 32 (5) ◽  
pp. 1772-1809 ◽  
Author(s):  
Xin-Rong Dai ◽  
Hui Rao ◽  
Shu-Qin Zhang
Keyword(s):  
Space Filling ◽  
Substitution Rule ◽  
Space Filling Curves ◽  
Self Similar

HAUSDORFF DIMENSION OF SELF-SIMILAR SETS WITH OVERLAPS

2001 ◽  
Vol 63 (3) ◽  
pp. 655-672 ◽  
Author(s):  
SZE-MAN NGAI ◽  
YANG WANG
Keyword(s):  
Hausdorff Dimension ◽  
Finite Type ◽  
Iterated Function Systems ◽  
Open Set Condition ◽  
Open Set ◽  
Iterated Function ◽  
Self Similar

The notion of ‘finite type’ iterated function systems of contractive similitudes is introduced, and a scheme for computing the exact Hausdorff dimension of their attractors in the absence of the open set condition is described. This method extends a previous one by Lalley, and applies not only to the classes of self-similar sets studied by Edgar, Lalley, Rao and Wen, and others, but also to some new classes that are not covered by the previous ones.

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 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.

Open set condition and post-critically finite self-similar sets

Nonlinearity ◽  
2008 ◽  
Vol 21 (6) ◽  
pp. 1227-1232 ◽  
Author(s):  
Qi-Rong Deng ◽  
Ka-Sing Lau
Keyword(s):  
Open Set Condition ◽  
Open Set ◽  
Self Similar

Exact Hausdorff and packing measures of Cantor sets with overlaps

2014 ◽  
Vol 35 (8) ◽  
pp. 2632-2668 ◽  
Author(s):  
HUA QIU
Keyword(s):  
Cantor Sets ◽  
Packing Measure ◽  
Type Condition ◽  
Elementary Functions ◽  
Open Set Condition ◽  
Open Set ◽  
Density Theorems ◽  
Iterated Function ◽  
Finite Set ◽  
Image Position

Let $K$ be the attractor of a linear iterated function system (IFS) $S_{j}(x)={\it\rho}_{j}x+b_{j},j=1,\ldots ,m$, on the real line $\mathbb{R}$ satisfying the generalized finite type condition (whose invariant open set ${\mathcal{O}}$ is an interval) with an irreducible weighted incidence matrix. This condition was recently introduced by Lau and Ngai [A generalized finite type condition for iterated function systems. Adv. Math.208 (2007), 647–671] as a natural generalization of the open set condition, allowing us to include many important overlapping cases. They showed that the Hausdorff and packing dimensions of $K$ coincide and can be calculated in terms of the spectral radius of the weighted incidence matrix. Let ${\it\alpha}$ be the dimension of $K$. In this paper, we state that $$\begin{eqnarray}{\mathcal{H}}^{{\it\alpha}}(K\cap J)\leq |J|^{{\it\alpha}}\end{eqnarray}$$ for all intervals $J\subset \overline{{\mathcal{O}}}$, and $$\begin{eqnarray}{\mathcal{P}}^{{\it\alpha}}(K\cap J)\geq |J|^{{\it\alpha}}\end{eqnarray}$$ for all intervals $J\subset \overline{{\mathcal{O}}}$ centered in $K$, where ${\mathcal{H}}^{{\it\alpha}}$ denotes the ${\it\alpha}$-dimensional Hausdorff measure and ${\mathcal{P}}^{{\it\alpha}}$ denotes the ${\it\alpha}$-dimensional packing measure. This result extends a recent work of Olsen [Density theorems for Hausdorff and packing measures of self-similar sets. Aequationes Math.75 (2008), 208–225] where the open set condition is required. We use these inequalities to obtain some precise density theorems for the Hausdorff and packing measures of $K$. Moreover, using these density theorems, we describe a scheme for computing ${\mathcal{H}}^{{\it\alpha}}(K)$ exactly as the minimum of a finite set of elementary functions of the parameters of the IFS. We also obtain an exact algorithm for computing ${\mathcal{P}}^{{\it\alpha}}(K)$ as the maximum of another finite set of elementary functions of the parameters of the IFS. These results extend previous ones by Ayer and Strichartz [Exact Hausdorff measure and intervals of maximum density for Cantor sets. Trans. Amer. Math. Soc.351 (1999), 3725–3741] and by Feng [Exact packing measure of Cantor sets. Math. Natchr.248–249 (2003), 102–109], respectively, and apply to some new classes allowing us to include Cantor sets in $\mathbb{R}$ with overlaps.

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 quasi-Assouad dimension of stochastically self-similar sets

2019 ◽  
Vol 150 (1) ◽  
pp. 261-275 ◽  
Author(s):  
Sascha Troscheit
Keyword(s):  
Hausdorff Dimension ◽  
Iterated Function Systems ◽  
Random Sets ◽  
Open Set Condition ◽  
Fractal Percolation ◽  
Open Set ◽  
Assouad Dimension ◽  
Iterated Function ◽  
Self Similar

AbstractThe class of stochastically self-similar sets contains many famous examples of random sets, for example, Mandelbrot percolation and general fractal percolation. Under the assumption of the uniform open set condition and some mild assumptions on the iterated function systems used, we show that the quasi-Assouad dimension of self-similar random recursive sets is almost surely equal to the almost sure Hausdorff dimension of the set. We further comment on random homogeneous and V -variable sets and the removal of overlap conditions.

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