2022 A Cantor-Type Construction. Invariant Set and Measure
Ion Chițescu, Loredana Ioana
Author Affiliations +
Real Anal. Exchange 47(2): 333-370 (2022). DOI: 10.14321/realanalexch.47.2.1630741075


The basic idea we use throughout this paper is to generalize the construction of the Cantor set as the attractor of an iterated function system $\big(f_0,\ f_1\big)$, replacing the classical functions acting on $ [0,1]$ via $f_0(x) =\frac{x}{3}$ and $f_1(x)= \frac{2}{3} + \frac{x}{3}$ with the new functions acting via $f_0(x)= \theta x$ and $f_1(x) = 1 - \theta + \theta x$, where $\theta \in \Big[0,\ \frac{1}{2} \Big]$ is an arbitrary number. We add to the schema two positive numbers $p_0, \ p_1$ such that $p_0 + p_1 = 1$, obtaining the iterated function system with probabilities $\big(f_0,\; f_1 ; \; p_0,\; p_1\big)$ which generates the invariant set (attractor) and the invariant measure, according to J. Hutchinson. The structure of the invariant set (genera-lized Cantor set) being known, we focus on the detailed computation of the invariant measure. At the end of the paper, we study the dependence upon the parameter $\theta$ of the invariant set and of the invariant measure.


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Ion Chițescu. Loredana Ioana. "A Cantor-Type Construction. Invariant Set and Measure." Real Anal. Exchange 47 (2) 333 - 370, 2022. https://doi.org/10.14321/realanalexch.47.2.1630741075


Published: 2022
First available in Project Euclid: 10 February 2023

Digital Object Identifier: 10.14321/realanalexch.47.2.1630741075

Primary: 28A80 , 37C25 , 37C70 , 37L40
Secondary: 37B10 , ‎54E50‎

Keywords: attractor , Cantor set , contraction , fixed point , Fractal , invariant measure , invariant set , iterated function system

Rights: Copyright © 2022 Michigan State University Press


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Vol.47 • No. 2 • 2022
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