Quantization dimension for conformal iterated function systems (Q2782721)

Given a Borel probability measure \(P\) on \(R^d\), a number \(r\in (0,\infty)\) and a natural number \(n\), the \(n\)th quantization error of order \(r\) for \(P\) is defined by NEWLINE\[NEWLINEe_{n,r}=\inf\Bigl\{\Bigl(\int d(x,\alpha)^r dP(x)\Bigr)^{1/r}: \alpha\subset R^d,\text{ card}(\alpha)\leq n\Bigr\},NEWLINE\]NEWLINE where \(d(x,\alpha)\) denotes the distance from the point \(x\) to the set \(\alpha\) with respect to a given norm on \(R^d\). The quantization dimension of order \(r\) for \(P\) is defined to be NEWLINE\[NEWLINED_r=\lim_{n\to\infty}(\log n)/\log (1/e_{n,r})NEWLINE\]NEWLINE provided the limit exists. The authors consider conformal iterated function systems and obtain the quantization dimension function for probability measures supported on the limit set which are the Gibbs states or equilibrium measures for a Hölder potential. The main theorem indicates the relationship between the quantization dimension function and the multifractal spectrum of the measure.