Quantization dimension for inhomogeneous bi-Lipschitz IFS

Authors

Amit Priyadarshi
Mrinal Kanti Roychowdhury, The University of Texas Rio Grande Valley
Manuj Verma

Document Type

Article

Publication Date

2023

Abstract

Let ν be a Borel probability measure on a d-dimensional Euclidean space Rd, d≥1, with a compact support, and let (p0,p1,p2,…,pN) be a probability vector with pj>0 for 1≤j≤N. Let {Sj:1≤j≤N} be a set of contractive mappings on Rd. Then, a Borel probability measure μ on Rd such that μ=∑Nj=1pjμ∘S−1j+p0ν is called an inhomogeneous measure, also known as a condensation measure on Rd. For a given r∈(0,+∞), the quantization dimension of order r, if it exists, denoted by Dr(μ), of a Borel probability measure μ on Rd represents the speed at which the nth quantization error of order r approaches to zero as the number of elements n in an optimal set of n-means for μ tends to infinity. In this paper, we investigate the quantization dimension for such a condensation measure.

Recommended Citation

Priyadarshi, Amit, Mrinal K. Roychowdhury, and Manuj Verma. "Quantization dimension for inhomogeneous bi-Lipschitz IFS." arXiv preprint arXiv:2303.14731 (2023).