On quantization dimensions of idempotent probability measures

Abstract

The concept of quantization dimensions (upper and lower) of idempotent probability measures on a metric compactum is introduced. These dimensions, which are defined using a general functorial scheme, are analogous to the quantization dimensions of classical probability distributions. It is proved that the quantization dimensions of an idempotent measure do not exceed the corresponding box dimensions of its support. At the same time, for any non-negative number b not exceeding the upper box dimension of a metric compactum X, there exists an idempotent probability measure μb with support equal to X and the upper quantization dimension equal to b. For the lower quantization dimension, a similar intermediate value theorem is valid under some additional restrictions on the compactum X.