Quantization dimensions of compactly supported probability measures via R\'enyi dimensions

Abstract

We provide a full picture of the upper quantization dimension in term of the R\'enyi dimension, in that we prove that the upper quantization dimension of order $r>0$ for an arbitrary compactly supported Borel probability measure $\nu$ is given by its R\'enyi dimension at the point $q_{r}$ where the $L^{q}$-spectrum of $\nu$ and the line through the origin with slope $r$ intersect. In particular, this proves the continuity of $r\mapsto\overline{D}_{r}$ as conjectured by Lindsay (2001). This viewpoint also sheds new light on the connection of the quantization problem with other concepts from fractal geometry in that we obtain a one-to-one correspondence of the upper quantization dimension and the $L^{q}$-spectrum restricted to $\left(0,1\right)$. We give sufficient conditions in terms of the $L^{q}$-spectrum for the existence of the quantization dimension. In this way we show as a byproduct that the quantization dimension exists for every Gibbs measure with respect to a $\mathcal{C}^{1}$-self-conformal iterated function system on $\mathbb{R}^{d}$ without any assumption on the separation conditions as well as for inhomogeneous self-similar measures under the inhomogeneous open sets condition. Some known general bounds on the quantization dimension in terms of other fractal dimensions can readily be derived from our new approach, some can be improved.Comment: 17 pages, 1 figur