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Canonical sequences of optimal quantization for condensation measures

We consider condensation measures of the form $P:=\frac 13 P\circ S_1^{-1}+ \frac 13 P\circ S_2^{-1}+ \frac 13 ν$ associated with the system $(\mathcal{S}, (\frac 13, \frac 13, \frac 13), ν) , $ where $\mathcal{S}=\{S_i\}_{i=1}^2 $ are contractions and $ ν$ is a Borel probability measure on $\mathbb R$ 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 $[\frac13 (\frac{1}{5})^2]^{\fracκ{2+κ}}=\frac 12. $ 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 $D(ν)\leq κ$, the $D(P)$-dimensional lower quantization coefficient is infinity.