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

2025/03/14 by Dubey, Shivam, Roychowdhury, Mrinal Kanti, Verma, Saurabh
#28A80 #60Exx #94A34 #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2503.11105

Abstract

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 \mathbb 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 D_r(μ) does not exists, we call \underline D_r(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 (\f_i\_i=1^N, (p_i)_i=0^N, ν), where the mappings f_i 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.

Related

• Quantization dimensions for inhomogeneous bi-Lipschitz Iterated Function Systems
• Quantization for a condensation system
• Quantization dimensions for the bi-Lipschitz recurrent Iterated function systems
• Asymptotics of the quantization errors for condensation measures
• Canonical sequences of optimal quantization for condensation measures