Kyoto Journal of Mathematics
- Kyoto J. Math.
- Volume 55, Number 4 (2015), 857-873.
Quantization coefficients in infinite systems
We investigate quantization coefficients for probability measures on limit sets, which are generated by systems 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 and of its noncompact limit set . We prove that, for each , there exists a unique positive number , so that for any , the -dimensional lower quantization coefficient of order for is positive, and we give estimates for the -upper quantization coefficient of order for . In particular, it follows that the quantization dimension of order of exists, and it is equal to . The above results allow one to estimate the asymptotic errors of approximating the measure in the -Kantorovich–Wasserstein metric, with discrete measures supported on finitely many points.
Kyoto J. Math., Volume 55, Number 4 (2015), 857-873.
Received: 27 January 2014
Accepted: 15 August 2014
First available in Project Euclid: 25 November 2015
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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