Asymptotic order of the quantization error for a class of self-similar measures with overlaps

Let $\{f_i\}_{i=1}^N$ be a set of equi-contractive similitudes on $\mathbb{R}^1$ satisfying the finite-type condition. We study the asymptotic quantization error for self-similar measures $\mu$ associated with $\{f_i\}_{i=1}^N$ and a positive probability vector. With a verifiable assumption, we prove that the upper and lower quantization coefficient for $\mu$ are both bounded away from zero and infinity. This can be regarded as an extension of Graf and Luschgy's result on self-similar measures with the open set condition. Our result is applicable to a significant class of self-similar measures with overlaps, including Erd\"{o}s measure, the $3$-fold convolution of the classical Cantor measure and the self-similar measures on some $\lambda$-Cantor sets.