Quantization dimension for Gibbs-like measures on cookie-cutter sets

Abstract

In this paper using the Banach limit we have determined a Gibbs-like measure ${\mu }_h$ supported by a cookie-cutter set $E$ which is generated by a single cookie-cutter mapping $f$. For such a measure ${\mu }_h$ and $r\in (0,+\infty )$ we have shown that there exists a unique ${\kappa }_r\in (0,+\infty )$ such that ${\kappa }_r$ is the quantization dimension function of the probability measure ${\mu }_h$, and we established its functional relationship with the temperature function of the thermodynamic formalism. The temperature function is commonly used to perform the multifractal analysis, in our context of the measure ${\mu }_h$. In addition, we have proved that the ${\kappa }_r$-dimensional lower quantization coefficient of order $r$ of the probability measure is positive.