Asymptotics of the quantization errors for Markov-type measures with complete overlaps

02/15/2022

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by   Sanguo Zhu, et al.

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Let 𝒢 be a directed graph with vertices 1,2,…, 2N. Let 𝒯=(T_i,j)_(i,j)∈𝒢 be a family of contractive similitudes. For every 1≤ i≤ N, let i^+:=i+N. For 1≤ i,j≤ N, we define ℳ_i,j={(i,j),(i,j^+),(i^+,j),(i^+,j^+)}∩𝒢. We assume that T_i,j=T_i,j for every (i,j)∈ℳ_i,j. Let K denote the Mauldin-Williams fractal determined by 𝒯. Let χ=(χ_i)_i=1^2N be a positive probability vector and P a row-stochastic matrix which serves as an incidence matrix for 𝒢. We denote by ν the Markov-type measure associated with χ and P. Let Ω={1,…,2N} and G_∞={σ∈Ω^ℕ:(σ_i,σ_i+1)∈𝒢, i≥ 1}. Let π be the natural projection from G_∞ to K and μ=ν∘π^-1. We consider the following two cases: 1. 𝒢 has two strongly connected components consisting of N vertices; 2. 𝒢 is strongly connected. With some assumptions for 𝒢 and 𝒯, for case 1, we determine the exact value s_r of the quantization dimension D_r(μ) for μ and prove that the s_r-dimensional lower quantization coefficient is always positive, but the upper one can be infinite; we establish a necessary and sufficient condition for the upper quantization coefficient for μ to be finite; for case 2, we determine D_r(μ) in terms of a pressure-like function and prove that D_r(μ)-dimensional upper and lower quantization coefficient are both positive and finite.