Quantization coefficients in infinite systems

Abstract

We investigate quantization coefficients for self-similar probability measures \mu 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 the fact that the set of contraction ratios has zero infimum; another difference resides in the specific geometry of the non-compact limit set J of S. We prove that, for each r \in (0,1), there exists a unique positive number \kappa_r, so that for arbitrary \kappa < \kappa_r < \kappa', the \kappa-dimensional lower quantization coefficient of order r of \mu is positive, and we also give estimates for the \kappa'-dimensional upper quantization coefficient of order r of \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 then 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.