Asymptotics of the quantization errors for in-homogeneous self-similar measures supported on self-similar sets

Abstract

We study the quantization for in-homogeneous self-similar measures µ supported on self-similar sets. Assuming the open set condition for the corresponding iterated function system, we prove the existence of the quantization dimension for µ of order r ∈ (0,∞) and determine its exact value ξr. Furthermore, we show that, the ξr-dimensional lower quantization coefficient for µ is always positive and the upper one can be infinite. A sufficient condition is given to ensure the finiteness of the upper quantization coefficient.