Canonical sequences of optimal quantization for condensation measures

Abstract

We consider condensation measures of the form $P:=\frac 13 P\circ S_1^{-1}+ \frac 13 P\circ S_2^{-1}+ \frac 13 \nu $ associated with the system $(\mathcal{S}, (\frac 13, \frac 13, \frac 13), \nu) , $ where $\mathcal{S}=\{S_i\}_{i=1}^2 $ are contractions and $ \nu$ is a Borel probability measure on $\mathbb R$ with compact support. Let $D(\mu)$ denote the quantization dimension of a measure $\mu$ if it exists. In this paper, we study self-similar measures $\nu$ satisfying $D(\nu)>\kappa$, $D(\nu)<\kappa$, and $D(\nu)=\kappa, $ respectively, where $\kappa $ is the unique number satisfying $[\frac13 (\frac{1}{5})^2]^{\frac{\kappa}{2+\kappa}}=\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 $\nu$ the quantization dimension $D(P)$ of $P$ exists and satisfies $D(P)=\max\{\kappa, D(\nu)\}. $ Moreover, we show that for $D(\nu)>\kappa$, the $D(P)$-dimensional lower and upper quantization coefficients are finite, positive and unequal; and for $D(\nu)\leq \kappa$, the $D(P)$-dimensional lower quantization coefficient is infinity.