Quantization dimensions for inhomogeneous bi-Lipschitz iterated function systems

Authors

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

Document Type

Article

Publication Date

3-12-2025

Abstract

Let ν be a Borel probability measure on a d-dimensional Euclidean space R d , d ≥ 1 , with a compact support, and let ( p 0 , p 1 , p 2 , … , p N ) be a probability vector with p j > 0 for 0 ≤ j ≤ N . Let { S j : 1 ≤ j ≤ N } be a set of contractive mappings on R d . Then, a Borel probability measure μ on R d such that μ = ∑ N j = 1 p j μ ∘ S − 1 j + p 0 ν is called an inhomogeneous measure, also known as a condensation measure on R d . For a given r ∈ ( 0 , + ∞ ) , the quantization dimension of order r, if it exists, denoted by D r ( μ ) , of a Borel probability measure μ on R d 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.

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Recommended Citation

Priyadarshi, Amit, Mrinal K. Roychowdhury, and Manuj Verma. 2025. “Quantization Dimensions for Inhomogeneous Bi-Lipschitz Iterated Function Systems.” Monatshefte Für Mathematik 207 (1): 125–40. https://doi.org/10.1007/s00605-025-02063-4

Publication Title

Monatshefte für Mathematik

DOI

10.1007/s00605-025-02063-4