Quantization dimension for a generalized inhomogeneous bi-Lipschitz iterated function system

Authors

• Shivam Dubey, Indian Institute of Information Technology, Allahabad
• Mrinal Kanti Roychowdhury, The University of Texas Rio Grande ValleyFollow
• Saurabh Verma, Indian Institute of Information Technology, Allahabad

Document Type

Article

Publication Date

7-1-2026

Abstract

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 R^d represents the speed how fast 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. If Dr(μ) does not exists, we call D̲r(μ) and D¯r(μ), the lower and upper quantization dimensions of μ of order r. In this paper, we estimate the quantization dimension of condensation measures associated with condensation systems ({fi}i=1^N,(pi)i=0^N,ν), where the mappings fi are bi-Lipschitz and the measure ν is an image measure of an ergodic measure with bounded distortion supported on a conformal set. In addition, we determine the optimal quantization for an infinite discrete distribution, and give an example which shows that the quantization dimension of a Borel probability measure can be positive with zero quantization coefficient.

Recommended Citation

Dubey, Shivam, Mrinal Kanti Roychowdhury, and Saurabh Verma. "Quantization dimension for a generalized inhomogeneous bi-Lipschitz iterated function system." Chaos, Solitons & Fractals 208 (2026): 118321. https://doi.org/10.1016/j.chaos.2026.118321

Publication Title

Chaos Solitons and Fractals

DOI

10.1016/j.chaos.2026.118321