A Cantor-Type Construction. Invariant Set and Measure

Abstract

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.