TOPOLOGICAL STRUCTURE OF SELF-SIMILAR SETS

Fractals ◽  
2002 ◽  
Vol 10 (02) ◽  
pp. 223-227 ◽  
Author(s):  
JUN LUO ◽  
HUI RAO ◽  
BO TAN
Keyword(s):  
Unit Disk ◽  
Topological Structure ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Open Set Condition ◽  
Open Set ◽  
Self Similar

We consider the attractor T of injective contractions f1, …, fm on R2 which satisfy the Open Set Condition. If T is connected, then T's interior T° is either empty or has no holes, and T's boundary ∂T is connected; if further T° is non-empty and connected, then ∂T is a simple closed curve, thus T is homeomorphic to the unit disk {x∈R2: |x|≤1}.

A NOTE ON A SELF-SIMILAR TILING GENERATED BY THE MINIMAL PISOT NUMBER

Fractals ◽  
2002 ◽  
Vol 10 (03) ◽  
pp. 335-339 ◽  
Author(s):  
JUN LUO
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Pisot Number ◽  
Self Similar

This note considers a self-similar tiling [Formula: see text] of the complex plane generated by the minimal Pisot number β. It will show that the boundary of every tile in [Formula: see text] is a simple closed curve, hence each tile is homeomorphic to the unit disk.

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 Handel’s fixed point theorem revisited

2012 ◽  
Vol 33 (5) ◽  
pp. 1584-1610
Author(s):  
JULIANA XAVIER
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Unit Disk ◽  
Point Theorem ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Open Unit ◽  
Open Unit Disk ◽  
Closed Disk ◽  
Oriented Cycle

AbstractMichael Handel proved in [A fixed-point theorem for planar homeomorphisms. Topology38 (1999), 235–264] the existence of a fixed point for an orientation-preserving homeomorphism of the open unit disk that can be extended to the closed disk, provided that it has points whose orbits form an oriented cycle of links at infinity. Later, Patrice Le Calvez gave a different proof of this theorem based only on Brouwer theory and plane topology arguments in [Une nouvelle preuve du théorème de point fixe de Handel. Geom. Topol.10(2006), 2299–2349]. These methods improved the result by proving the existence of a simple closed curve of index 1. We give a new, simpler proof of this improved version of the theorem and generalize it to non-oriented cycles of links at infinity.

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

A 1-ALG Simple Closed Curve in E3 is Tame

1973 ◽  
Vol 25 (3) ◽  
pp. 646-656
Author(s):  
W. S. Boyd ◽  
A. H. Wright
Keyword(s):  
Fundamental Group ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Open Set

Let J be a simple closed curve in a 3-manifold M3. We say M — J is 1-ALG at p ∈ J (or has locally abelian fundamental group at p) if and only if for each sufficiently small open set U containing p, there is an open set V such that p ∈ V ⊂ U and each loop in V — J which bounds in U — J is contractible to a point in U — J.

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.

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