scholarly journals On intrinsic and extrinsic rational approximation to Cantor sets

2020 ◽  
pp. 1-30
Author(s):  
JOHANNES SCHLEISCHITZ
Keyword(s):  
Lower Bound ◽  
Rational Approximation ◽  
Iterated Function System ◽  
Rational Numbers ◽  
Cantor Sets ◽  
Algebraic Numbers ◽  
Fractal Sets ◽  
Iterated Function ◽  
Set Up ◽  
Missing Digit

We establish various new results on a problem proposed by Mahler [Some suggestions for further research. Bull. Aust. Math. Soc.29 (1984), 101–108] concerning rational approximation to fractal sets by rational numbers inside and outside the set in question. Some of them provide a natural continuation and improvement of recent results of Broderick, Fishman and Reich, and Fishman and Simmons. A key feature is that many of our new results apply to more general, multi-dimensional fractal sets and require only mild assumptions on the iterated function system. Moreover, we provide a non-trivial lower bound for the distance of a rational number $p/q$ outside the Cantor middle-third set $C$ to the set $C$ , in terms of the denominator $q$ . We further discuss patterns of rational numbers in fractal sets. We highlight two of them: firstly, an upper bound for the number of rational (algebraic) numbers in a fractal set up to a given height (and degree) for a wide class of fractal sets; and secondly, we find properties of the denominator structure of rational points in ‘missing- digit’ Cantor sets, generalizing claims of Nagy and Bloshchitsyn.

ON THE ARITHMETIC STRUCTURE OF RATIONAL NUMBERS IN THE CANTOR SET

2020 ◽  
pp. 1-6
Author(s):  
IGOR E. SHPARLINSKI
Keyword(s):  
Lower Bound ◽  
Recent Result ◽  
Ergodic Theory ◽  
Rational Approximation ◽  
Prime Factor ◽  
Rational Numbers ◽  
Cantor Set ◽  
Cantor Sets

We obtain a lower bound on the largest prime factor of the denominator of rational numbers in the Cantor set. This gives a stronger version of a recent result of Schleischitz [‘On intrinsic and extrinsic rational approximation to Cantor sets’, Ergodic Theory Dyn. Syst. to appear] obtained via a different argument.

GENERALIZATIONS OF THE KOCH CURVE

Fractals ◽  
2008 ◽  
Vol 16 (03) ◽  
pp. 267-274 ◽  
Author(s):  
R. B. DARST ◽  
J. A. PALAGALLO ◽  
T. E. PRICE
Keyword(s):  
Iterative Method ◽  
Iterated Function System ◽  
Continuous Functions ◽  
Cantor Sets ◽  
Koch Curve ◽  
Double Points ◽  
Iterated Function ◽  
Fixed Set ◽  
Equilateral Triangles ◽  
Two Parameter

We present an iterative method to define a two-parameter family of continuous functions fa,θ: I → ℂ such that f1/3,π/3 is the Koch curve. We consider the two-cases θ = π/3 and θ = π/4 of these generalized Koch curves fa,θ(I). In each case we determine the pivotal value of a, the largest value of a for which the corresponding curve is not simple. We give characterizations of the double points of the curve (points on the curve that have two inverse images). In the case where θ = π/3 double points are vertices of equilateral triangles. When θ = π/4 the double points form Cantor sets in the plane. We conclude with a more general result that proves that if the fixed set (attractor) of an iterated function system is connected, then it is a curve.

scholarly journals Resonance between Cantor sets

2009 ◽  
Vol 29 (1) ◽  
pp. 201-221 ◽  
Author(s):  
YUVAL PERES ◽  
PABLO SHMERKIN
Keyword(s):  
Hausdorff Dimension ◽  
Iterated Function System ◽  
Unit Interval ◽  
Irrational Rotation ◽  
Cantor Sets ◽  
Scaling Parameter ◽  
Iterated Function ◽  
Self Similar ◽  
Image Position ◽  
Geometric Resonance

AbstractLet Ca be the central Cantor set obtained by removing a central interval of length 1−2a from the unit interval, and then continuing this process inductively on each of the remaining two intervals. We prove that if log b/log a is irrational, then where dim is Hausdorff dimension. More generally, given two self-similar sets K,K′ in ℝ and a scaling parameter s>0, if the dimension of the arithmetic sum K+sK′ is strictly smaller than dim (K)+dim (K′)≤1 (‘geometric resonance’), then there exists r<1 such that all contraction ratios of the similitudes defining K and K′ are powers of r (‘algebraic resonance’). Our method also yields a new result on the projections of planar self-similar sets generated by an iterated function system that includes a scaled irrational rotation.

scholarly journals Fractal Interpolation Using Harmonic Functions on the Koch Curve

2021 ◽  
Vol 5 (2) ◽  
pp. 28
Author(s):  
Song-Il Ri ◽  
Vasileios Drakopoulos ◽  
Song-Min Nam
Keyword(s):  
Harmonic Functions ◽  
Iterated Function System ◽  
Function System ◽  
Fractal Interpolation ◽  
Koch Curve ◽  
Fractal Sets ◽  
Fractal Interpolation Function ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Interpolation Functions

The Koch curve was first described by the Swedish mathematician Helge von Koch in 1904 as an example of a continuous but nowhere differentiable curve. Such functions are now characterised as fractal since their graphs are in general fractal sets. Furthermore, it can be obtained as the graph of an appropriately chosen iterated function system. On the other hand, a fractal interpolation function can be seen as a special case of an iterated function system thus maintaining all of its characteristics. Fractal interpolation functions are continuous functions that can be used to model continuous signals. An in-depth discussion on the theory of affine fractal interpolation functions generating the Koch Curve by using fractal analysis as well as its recent development including some of the research made by the authors is provided. We ensure that the graph of fractal interpolation functions on the Koch Curve are attractors of an iterated function system constructed by non-constant harmonic functions.

scholarly journals VISIBILITY OF CARTESIAN PRODUCTS OF CANTOR SETS

Fractals ◽  
2020 ◽  
Vol 28 (06) ◽  
pp. 2050119
Author(s):  
TINGYU ZHANG ◽  
KAN JIANG ◽  
WENXIA LI
Keyword(s):  
Hausdorff Dimension ◽  
Iterated Function System ◽  
Cantor Sets ◽  
Function System ◽  
Topological Properties ◽  
Cartesian Products ◽  
Iterated Function

Let [Formula: see text] be the attractor of the following iterated function system(IFS): [Formula: see text] Given [Formula: see text], we say the line [Formula: see text] is visible through [Formula: see text] if [Formula: see text] Let [Formula: see text]. In this paper, we give a complete description of [Formula: see text], containing its Hausdorff dimension and topological properties.

GENERALIZED ITERATED FUNCTION SYSTEMS ON HYPERBOLIC NUMBER PLANE

Fractals ◽  
2019 ◽  
Vol 27 (04) ◽  
pp. 1950045
Author(s):  
GAMALIEL YAFTE TÉLLEZ-SÁNCHEZ ◽  
JUAN BORY-REYES
Keyword(s):  
Fixed Point ◽  
Real Line ◽  
Unique Fixed Point ◽  
Iterated Function Systems ◽  
Cantor Sets ◽  
Affine Transformations ◽  
The Real ◽  
Fractal Sets ◽  
Hyperbolic Number ◽  
Iterated Function

Iterated function systems provide the most fundamental framework to create many fascinating fractal sets. They have been extensively studied when the functions are affine transformations of Euclidean spaces. This paper investigates the iterated function systems consisting of affine transformations of the hyperbolic number plane. We show that the basics results of the classical Hutchinson–Barnsley theory can be carried over to construct fractal sets on hyperbolic number plane as its unique fixed point. We also discuss about the notion of hyperbolic derivative of an hyperbolic-valued function and then we use this notion to get some generalization of cookie-cutter Cantor sets in the real line to the hyperbolic number plane.

scholarly journals Graph directed Markov systems on Hilbert spaces

2009 ◽  
Vol 147 (2) ◽  
pp. 455-488 ◽  
Author(s):  
R. D. MAULDIN ◽  
T. SZAREK ◽  
M. URBAŃSKI
Keyword(s):  
Hausdorff Measure ◽  
Iterated Function System ◽  
Sufficient Conditions ◽  
Topological Pressure ◽  
Limit Set ◽  
Covering Theorem ◽  
Limit Sets ◽  
Markov Systems ◽  
Polish Spaces ◽  
Iterated Function

AbstractWe deal with contracting finite and countably infinite iterated function systems acting on Polish spaces, and we introduce conformal Graph Directed Markov Systems on Polish spaces. Sufficient conditions are provided for the closure of limit sets to be compact, connected, or locally connected. Conformal measures, topological pressure, and Bowen's formula (determining the Hausdorff dimension of limit sets in dynamical terms) are introduced and established. We show that, unlike the Euclidean case, the Hausdorff measure of the limit set of a finite iterated function system may vanish. Investigating this issue in greater detail, we introduce the concept of geometrically perfect measures and provide sufficient conditions for geometric perfectness. Geometrical perfectness guarantees the Hausdorff measure of the limit set to be positive. As a by–product of the mainstream of our investigations we prove a 4r–covering theorem for all metric spaces. It enables us to establish appropriate co–Frostman type theorems.

ARCLENGTH AS THE INVARIANT MEASURE FOR AN IFS WITH PROBABILITIES

Fractals ◽  
2015 ◽  
Vol 23 (04) ◽  
pp. 1550046
Author(s):  
D. LA TORRE ◽  
F. MENDIVIL
Keyword(s):  
Invariant Measure ◽  
Iterated Function System ◽  
Function System ◽  
Iterated Function

Given a continuous rectifiable function [Formula: see text], we present a simple Iterated Function System (IFS) with probabilities whose invariant measure is the normalized arclength measure on the graph of [Formula: see text].

Sign in / Sign up

Export Citation Format

Share Document