On the quantization for self-affine measures on Bedford–McMullen carpets

Abstract

For a self-affine measure on a Bedford-McMullen carpet we prove that its quantization dimension exists and determine its exact value. Further, we give various sufficient conditions for the corresponding upper and lower quantization coefficient to be both positive and finite. Finally, we compare the quantization dimension with corresponding quantities derived from the multifractal temperature function and show that -different from conformal systems -they in general do not coincide.1991 Mathematics Subject Classification. 28A75, 28A80, 94A15.

Asymptotic Order of the Geometric Mean Error for Self-Affine Measures on Bedford–mcmullen Carpets

“…We refer to [4,6] for rigorous mathematical foundations of quantization. One can see [5,6,7,8,12,15,16,18,20]) for related results. By [6], we have, e k,r (ν) → e k,0 (ν) as r → 0, provided that |x| s dν(x) < ∞ for some s > 0.…”

“…Remark 1.2. For r > 0, Kessböhmer and Zhu [12] proved that D r (µ) = D r (µ) and determined the exact value; they also proved the finiteness and positivity of the upper and lower quantization coefficient for µ of order r in some special cases and Zhu [25] proved this fact in general by associating subsets of E with those of the product coding set W := G N × G N y and considering an auxiliary measure which is supported on W .…”

Some Recent Developments in Quantization of Fractal Measures

“…allowing us to find explicit formulae for the quantization dimension for a given problem (see [15] for an instance of this). Typically, for a non-self-similar measure such as a self-affine measures on Bedford-McMullen carpets, this requires a detailed analysis of the asymptotic quantization errors.…”

“…Typically, this question is much harder to answer than finding the quantization dimension. So far, the quantization coefficient has been studied for absolutely continuous probability measures ( [6]) and several classes of singular measures, including self-similar and self-conformal [19,29,39,41] measures, Markov-type measures [16,33,29] and self-affine measures on Bedford-McMullen carpets [15,38].…”

“…(3) for certain measures arising from dynamical systems, the quantization dimension can be expressed within the thermodynamic formalism in terms of appropriate temperature functions (see [19,15,27,28]). (4) The upper and lower quantization dimension of order zero are closely connected with the upper and lower local dimension.…”