Canonical Sequences of Optimal Quantization for Condensation Measures

We consider condensation measures of the form P:=1/3 P∘ S_1^-1+ 1/3 P∘ S_2^-1+ 1/3ν associated with the system (𝒮, (1/3, 1/3, 1/3), ν ) , where 𝒮={S_i}_i=1^2 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 )>κ , D(ν )<κ , and D(ν )=κ , respectively, where κ is the unique number satisfying [1/3 (1/5)^2]^κ/2+κ=1/2 . For each case we construct two sequences a ( n ) and F ( n ), which are utilized in determining the optimal sets of F ( n )-means and the F ( n )th quantization errors for P . We also show that for each measure ν the quantization dimension D ( P ) of P exists and satisfies D(P)=max{κ , D(ν )} . Moreover, we show that for 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.