Kyoto Journal of Mathematics

Quantization coefficients in infinite systems

Eugen Mihailescu and Mrinal Kanti Roychowdhury

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Abstract

We investigate quantization coefficients for probability measures μ on limit sets, which are generated by systems S of infinitely many contractive similarities and by probabilistic vectors. The theory of quantization coefficients for infinite systems has significant differences from the finite case. One of these differences is the lack of finite maximal antichains, and another is the fact that the set of contraction ratios has zero infimum; another difference resides in the specific geometry of S and of its noncompact limit set J. We prove that, for each r(0,), there exists a unique positive number κr, so that for any κ<κr<κ', the κ-dimensional lower quantization coefficient of order r for μ is positive, and we give estimates for the κ'-upper quantization coefficient of order r for μ. In particular, it follows that the quantization dimension of order r of μ exists, and it is equal to κr. The above results allow one to estimate the asymptotic errors of approximating the measure μ in the Lr-Kantorovich–Wasserstein metric, with discrete measures supported on finitely many points.

Article information

Source
Kyoto J. Math., Volume 55, Number 4 (2015), 857-873.

Dates
Received: 27 January 2014
Accepted: 15 August 2014
First available in Project Euclid: 25 November 2015

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1448460082

Digital Object Identifier
doi:10.1215/21562261-3089118

Mathematical Reviews number (MathSciNet)
MR3479313

Zentralblatt MATH identifier
1378.60013

Subjects
Primary: 28A32 28A80: Fractals [See also 37Fxx] 28A25: Integration with respect to measures and other set functions 60B05: Probability measures on topological spaces

Keywords
Self-similar measures on limit sets quantization for infinite iterated function systems quatization dimension convergence of probability measures $L_{r}$-Kantorovich–Wasserstein metric

Citation

Mihailescu, Eugen; Roychowdhury, Mrinal Kanti. Quantization coefficients in infinite systems. Kyoto J. Math. 55 (2015), no. 4, 857--873. doi:10.1215/21562261-3089118. https://projecteuclid.org/euclid.kjm/1448460082


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