A MATRIX METHOD FOR APPROXIMATING FRACTAL MEASURES

1992 ◽  
Vol 02 (01) ◽  
pp. 167-175 ◽  
Author(s):  
A. BOYARSKY ◽  
Y.S. LOU
Keyword(s):  
Iterated Function System ◽  
Markov Operator ◽  
Measurable Subset ◽  
Function System ◽  
Probability Measures ◽  
Bounded Subset ◽  
Step Functions ◽  
Iterated Function ◽  
Fractal Measures ◽  
Borel Probability

Let X be a bounded subset of Rn and let A be the Lebesgue measure on X. Let {X:τ1,…, τN} be an iterated function system (IFS) with attractor S. We associate probabilities p1,…, pN with τ1,…, τN, respectively. Let M(X) be the space of Borel probability measures on X, and let M: M(X)→M(X) be the Markov operator associated with the IFS and its probabilities given by: [Formula: see text] where A is a measurable subset of X. Then there exists a unique µ∈M (A) such that Mµ=µ; µ is referred to as the measure invariant under the iterated function system with the associated probabilities. The support of μ is the attractor S. We prove the existence of a sequence of step functions {fi}, which are the eigenvectors of matrices {Mi}, such that the measures {fidλ} converge weakly to µ. An algorithm is presented for the construction of Mi and an example is given.

scholarly journals (In)homogeneous invariant compact convex sets of probability measures

2020 ◽  
Vol 12 (4) ◽  
pp. 60-68
Author(s):  
Natalia Mazurenko ◽  
Mykhailo Zarichnyi
Keyword(s):  
Metric Space ◽  
Compact Support ◽  
Convex Sets ◽  
Iterated Function System ◽  
Complete Metric Space ◽  
Compact Convex ◽  
Function System ◽  
Probability Measures ◽  
Iterated Function

It is proved that for any iterated function system of contractions on a complete metric space there exists an invariant compact convex sets of probability measures of compact support on this space. A similar result is proved for the inhomogeneous  compact convex sets of probability measures of compact support.

scholarly journals Approximating integrals with respect to stationary probability measures of iterated function systems

2020 ◽  
pp. 1-17
Author(s):  
ITALO CIPRIANO ◽  
NATALIA JURGA
Keyword(s):  
Lyapunov Exponents ◽  
Iterated Function System ◽  
Iterated Function Systems ◽  
Unit Interval ◽  
Function System ◽  
Probability Measures ◽  
Stationary Probability ◽  
Iterated Function ◽  
Wasserstein Distances

We study fast approximation of integrals with respect to stationary probability measures associated to iterated function systems on the unit interval. We provide an algorithm for approximating the integrals under certain conditions on the iterated function system and on the function that is being integrated. We apply this technique to estimate Hausdorff moments, Wasserstein distances and Lyapunov exponents of stationary probability measures.

scholarly journals Dimensions of random affine code tree fractals

2013 ◽  
Vol 34 (3) ◽  
pp. 854-875 ◽  
Author(s):  
ESA JÄRVENPÄÄ ◽  
MAARIT JÄRVENPÄÄ ◽  
ANTTI KÄENMÄKI ◽  
HENNA KOIVUSALO ◽  
ÖRJAN STENFLO ◽  
...  
Keyword(s):  
Hausdorff Dimension ◽  
General Class ◽  
Classical Result ◽  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Probability Measures ◽  
Box Counting ◽  
Iterated Function ◽  
Natural Measures

AbstractWe study the dimension of code tree fractals, a class of fractals generated by a set of iterated function systems. We first consider deterministic affine code tree fractals, extending to the code tree fractal setting the classical result of Falconer and Solomyak on the Hausdorff dimension of self-affine fractals generated by a single iterated function system. We then calculate the almost sure Hausdorff, packing and box counting dimensions of a general class of random affine planar code tree fractals. The set of probability measures describing the randomness includes natural measures in random $V$-variable and homogeneous Markov constructions.

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].

Stochastic Fractal Modeling and Process Planning for Machining Using Two-Dimensional Iterated Function System

2008 ◽  
Vol 392-394 ◽  
pp. 575-579
Author(s):  
Yu Hao Li ◽  
Jing Chun Feng ◽  
Y. Li ◽  
Yu Han Wang
Keyword(s):  
Path Planning ◽  
Tool Path ◽  
Iterated Function System ◽  
Numerical Control ◽  
Function System ◽  
Tool Path Planning ◽  
Ongoing Research ◽  
Iterated Function ◽  
Planning Algorithm ◽  
Path Planning Algorithm

Self-affine and stochastic affine transforms of R2 Iterated Function System (IFS) are investigated in this paper for manufacturing non-continuous objects in nature that exhibit fractal nature. A method for modeling and fabricating fractal bio-shapes using machining is presented. Tool path planning algorithm for numerical control machining is presented for the geometries generated by our fractal generation function. The tool path planning algorithm is implemented on a CNC machine, through executing limited number of iteration. This paper describes part of our ongoing research that attempts to break through the limitation of current CAD/CAM and CNC systems that are oriented to Euclidean geometry objects.

Fractal dimension and iterated function system (IFS) for speech recognition

1992 ◽  
Vol 28 (15) ◽  
pp. 1382 ◽  
Author(s):  
E.L.J. Bohez ◽  
T.R. Senevirathne ◽  
J.A. van Winden
Keyword(s):  
Fractal Dimension ◽  
Speech Recognition ◽  
Iterated Function System ◽  
Function System ◽  
Iterated Function
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