Quantization dimension for inhomogeneous bi-Lipschitz IFS

Let \(\nu\) be a Borel probability measure on a \(d\)-dimensional Euclidean space \(\mathbb{R}^d\), \(d\geq 1\), with a compact support, and let \((p_0, p_1, p_2, \ldots, p_N)\) be a probability vector with \(p_j>0\) for \(1\leq j\leq N\). Let \(\{S_j: 1\leq j\leq N\}\) be a set of contractive ma...

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Bibliographic Details
Published in:arXiv.org 2023-03
Main Authors: Priyadarshi, Amit, Roychowdhury, Mrinal K, Verma, Manuj
Format: Article
Language:English
Subjects:
Euclidean geometry
Measurement
Online Access:Get full text
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Summary:Let \(\nu\) be a Borel probability measure on a \(d\)-dimensional Euclidean space \(\mathbb{R}^d\), \(d\geq 1\), with a compact support, and let \((p_0, p_1, p_2, \ldots, p_N)\) be a probability vector with \(p_j>0\) for \(1\leq j\leq N\). Let \(\{S_j: 1\leq j\leq N\}\) be a set of contractive mappings on \(\mathbb R^d\). Then, a Borel probability measure \(\mu\) on \(\mathbb R^d\) such that \(\mu=\sum_{j=1}^N p_j\mu\circ S_j^{-1}+p_0\nu\) is called an inhomogeneous measure, also known as a condensation measure on \(\mathbb R^d\). For a given \(r\in (0, +\infty)\), the quantization dimension of order \(r\), if it exists, denoted by \(D_r(\mu)\), of a Borel probability measure \(\mu\) on \(\mathbb R^d\) represents the speed at which the \(n\)th quantization error of order \(r\) approaches to zero as the number of elements \(n\) in an optimal set of \(n\)-means for \(\mu\) tends to infinity. In this paper, we investigate the quantization dimension for such a condensation measure.
ISSN:2331-8422