Asymptotics of the geometric mean error for in-homogeneous self-similar measures

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Let (fᵢ)ᵢ₌₁ⁿ be a family of contractive similitudes on Rq satisfying the open set condition. Let (pᵢ)ᵢ₌₀ⁿ be a probability vector with pᵢ>0 for all i=0,1,,N. We study the asymptotic geometric mean errors eₙ,₀(μ),n≥ 1, in the quantization for the in-homogeneous self-similar measure μ associated with the condensation system ((fᵢ)ᵢ₌₁ⁿ,(pᵢ)ᵢ₌₀ⁿ,ν). We focus on the following two independent cases: (I) ν is a self-similar measure on Rq associated with (fᵢ)ᵢ₌₁ⁿ; (II) ν is a self-similar measure associated with another family of contractive similitudes (gᵢ)ᵢ₌₁ᵐ on Rq satisfying the open set condition and ((fᵢ)ᵢ₌₁ⁿ,(pᵢ)ᵢ₌₀ⁿ,ν) satisfies a version of in-homogeneous open set condition. We show that, in both cases, the quantization dimension D₀(μ) of μ of order zero exists and agrees with that of ν, which is independent of the probability vector (pᵢ)ᵢ₌₀ⁿ. We determine the convergence order of (eₙ,₀(μ))ₙ₌₁∞; namely, for D₀(μ)=:d₀, there exists a constant D>0, such that D⁻¹n⁻¹/ᵈ⁰≤ eₙ,₀(μ)≤ D n⁻¹/ᵈ⁰, n≥ 1.