Quantization of Random Homogeneous Self-Similar Measures

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.

Summary

This research paper delves into the quantization of random homogeneous self-similar measures, a type of probability measure generated by random iterated function systems (RIFS). The authors successfully characterize the almost sure quantization dimension and provide conditions for the positivity and finiteness of quantization coefficients, extending the existing quantization theory to the more complex random setting.

Key Insights

• •The paper provides a precise characterization of the almost sure quantization dimension (κr) for random homogeneous self-similar measures generated by RIFSs, which is a significant extension from the deterministic case.
• •The authors establish conditions, including the Uniform Extra Strong Separation Condition (UESSC) and the Strong Uniform Open Set Condition (SUOSC), under which the lower quantization coefficient is strictly positive, ensuring a lower bound on the quantization error.
• •The paper presents a sufficient condition for the finiteness of the upper quantization coefficient, providing an upper bound on the quantization error, even without imposing any separation condition.
• •The authors use the "expected pressure function" and prove Proposition 1 that establishes the existence and uniqueness of a critical exponent, <emphasis>kappa sub r</emphasis>, which then forms the basis for the main results in Theorem 2.
• •The paper uses Bernoulli measures on the symbolic space of IFSs to study properties of a *typical* attractor or measure within the continuum generated by the RIFS.

Practical Implications

• •The results have potential applications in data compression and signal processing, where efficient approximation of complex, self-similar data is crucial.
• •This work opens avenues for future research into the quantization of random self-affine measures or measures generated by more general transformations, broadening the applicability of quantization theory.
• •Future research could focus on weakening the separation conditions, particularly the UESSC and SUOSC, required for the main results, making the theory more accessible and applicable to a wider range of RIFSs.
• •The findings contribute to a deeper understanding of fractal analysis, providing tools for characterizing and approximating complex fractal structures generated by random processes.

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