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

In this paper using 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 $r\in (0,+\mathrm{\infty })$ we have shown that there exists a unique ${\kappa }_r\in (0,+\mathrm{\infty })$ such that ${\kappa }_r$ is the quantization dimension function of the probability measure ${\mu }_h$, and 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.