Quantization of Cantor-Like Set on the Real Projective Line

Abstract

In this article, an iterated function system (IFS) is considered on the real projective line ${\mathbb{RP}}^1$ so that the attractor is a Cantor-like set. Hausdorff dimension of this attractor is estimated. The existence of a probability measure associated with this IFS on ${\mathbb{RP}}^1$ is also demonstrated. It is shown that the $n$-th quantization error of order $r$ for the push-forward measure is a constant multiple of the $n$-th quantization error of order $r$ of the original measure. Finally, an upper bound for the $n$-th quantization error of order $2$ for this measure is provided.

16 pages, 5 figures