## Kyoto Journal of Mathematics

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

### Quantization coefficients in infinite systems

Eugen Mihailescu and Mrinal Kanti Roychowdhury

#### Abstract

We investigate quantization coefficients for probability measures $\mu $ on limit sets, which are generated by systems $\mathcal{S}$ 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 $\mathcal{S}$ and of its noncompact limit set $J$. We prove that, for each $r\in (0,\infty )$, there exists a unique positive number ${\kappa}_{r}$, so that for any $\kappa <{\kappa}_{r}<\kappa \text{'}$, the $\kappa $-dimensional lower quantization coefficient of order $r$ for $\mu $ is positive, and we give estimates for the $\kappa \text{'}$-upper quantization coefficient of order $r$ for $\mu $. In particular, it follows that the quantization dimension of order $r$ of $\mu $ exists, and it is equal to ${\kappa}_{r}$. The above results allow one to estimate the asymptotic errors of approximating the measure $\mu $ in the ${L}_{r}$-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