Quantization coefficients for ergodic measures on infinite symbolic space

Abstract

In this paper we consider an ergodic measure with bounded distortion on a symbolic space generated by an infinite alphabet, and showed that for each $r\in (0, +\infty)$ there exists a unique $k_r \in (0, +\infty)$ such that both the $k_r$-dimensional lower and upper quantization coefficients for its image measure $m$ with the support lying on the limit set generated by an infinite conformal iterated function system satisfying the strong open set condition are finite and positive. In addition, it shows that $k_r$ can be expressed by a simple formula involving the temperature function of the system. The result extends and generalizes a similar result ofRoychowdhury established for a finite conformal iterated function system [Bull. Polish Acad. Sci. Math. 57 (2009)].