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

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 exact convergence order of (en,0(μ))n=1∞; namely, for D0(μ)=:d0, there exists a constant D>0, such thatD−1n−1d0≤en,0(μ)≤Dn−1d0,n≥1.