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

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In this paper using Banach limit we have determined a Gibbs-like measure μₕ supported by a cookie-cutter set E which is generated by a single cookie-cutter mapping f. For such a measure μₕ and r∈ (0, +∞) we have shown that there exists a unique κᵣ ∈ (0, +∞) such that κᵣ is the quantization dimension function of the probability measure μₕ, 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 μₕ. In addition, we have proved that the κᵣ-dimensional lower quantization coefficient of order r of the probability measure is positive.