Canonical Sequences of Optimal Quantization for Condensation Measures

We consider condensation measures of the form P:=13P∘S−11+13P∘S−12+13ν associated with the system (S,(13,13,13),ν), where S={Si}2i=1 are contractions and ν is a Borel probability measure on R with compact support. Let D(μ) denote the quantization dimension of a measure μ if it exists. In this paper, we study self-similar measures ν satisfying D(ν)\u3eκ, D(ν)κ, the D(P)-dimensional lower and upper quantization coefficients are finite, positive and unequal; and for D(ν)≤κ, the D(P)-dimensional lower quantization coefficient is infinity. We consider condensation measures of the form P:=13P∘S−11+13P∘S−12+13ν associated with the system (S,(13,13,13),ν), where S={Si}2i=1 are contractions and ν is a Borel probability measure on R with compact support. Let D(μ) denote the quantization dimension of a measure μ if it exists. In this paper, we study self-similar measures ν satisfying D(ν)\u3eκ, D(ν)κ, the D(P)-dimensional lower and upper quantization coefficients are finite, positive and unequal; and for D(ν)≤κ, the D(P)-dimensional lower quantization coefficient is infinity