Quantization of Random Homogeneous Self-Similar Measures
Akash Banerjee, Alamgir Hossain, Md. Nasim Akhtar
TL;DR
This work extends quantization theory to random homogeneous self-similar measures by introducing a κ_r determined via an expected pressure and proving its almost-sure equality to the quantization dimension under a uniform extra strong separation condition. The authors develop a symbolic framework with a Bernoulli shift, prove existence and uniqueness of μ_ω, and derive both lower and upper quantization coefficient results under separation-type conditions and a probabilistic non-separation criterion. Key techniques include ergodic theorems on the shift space, maximal antichain analysis, and precise control of scale fluctuations across the random IFS. The results are illustrated with examples showing the applicability beyond deterministic OSC/SOSC regimes.
Abstract
In this article, we study a class of invariant measures generated by a random homogeneous self-similar iterated function system. Unlike the deterministic setting, the random quantization problem requires controlling distortion errors across non-uniform scales. For $r>0$, under a suitable separation condition, we precisely determine the almost sure quantization dimension $κ_r$ of this class, by utilizing the ergodic theory of the shift map on the symbolic space. By imposing an additional separation condition, we establish almost sure positivity of the $κ_r$-dimensional lower quantization coefficient. Furthermore, without assuming any separation condition, we provide a sufficient condition that guarantees almost sure finiteness of the $κ_r$-dimensional upper quantization coefficient. We also include some illustrative examples.