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

2012/07/24 by Roychowdhury, Mrinal Kanti
#28A80 #60Exx #94A34 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1207.5842

Abstract

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

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