Noise on the triadic Cantor set

Abstract

The effect of noise on the triadic 1D Cantor set is studied. The iterative procedure used to generate the Cantor set is modified by adding a stochastic variable to the interval length at each iteration. It is found that the presence of this ‘noise’ truncates the self-similar structure of the set after a finite number of iterations. Either the noise causes the closest spaced intervals to merge together, or it causes all the intervals to collapse to a set of a finite number of points. Distribution functions are found describing the probability that self-similarity was destroyed on the nth iteration. A program using a random number generator was written that simulates our modified Cantor procedure. The results of 100 000 simulations are compared with the predictions of the probability distribution functions. Good agreement is found.