Quantization coefficients in infinite systems

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.