Quantization of Cantor-Like Set on the Real Projective Line

TL;DR

This paper studies a Cantor-like fractal set generated by an iterated function system on the real projective line, estimating its Hausdorff dimension, establishing measure existence, and analyzing quantization errors.

Contribution

It introduces a new IFS on $ ext{RP}^1$, estimates the Hausdorff dimension of its attractor, and relates quantization errors of the induced measure to the original measure.

Findings

Hausdorff dimension of the attractor is estimated.

Existence of a probability measure on the attractor is proven.

An upper bound for the quantization error of the measure is provided.

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.