论文标题
通过rényi维度对紧凑型概率度量的量化维度
Quantization dimensions of compactly supported probability measures via Rényi dimensions
论文作者
论文摘要
我们通过证明上限$ \ \ bar {d} _ {d} _ {r} _ {r} $ r> 0 $的上限$ r> 0 $，用于任意紧凑的borel概率$ c $ q _rénynyi$ q_ r^r^给出$ q_ r^r^r^n n point $ q _ r^r^，我们通过证明上限$rénynyi$ q_ r^r^给出了$ q_ r^r^，我们在rényi的$ q _réum-q_ r l^中，我们提供了上层量化维度$ r> q _ r^r^r^，我们提供了上层量化维度的完整图片。 $ν$和通过斜率$ r $ Intersect通过原点的线。特别是，这证明了$ r \ mapsto \ bar {d} _ {r}（ν）$的连续性，如Lindsay（2001）的猜想。该观点还为量化问题与分形几何形状的其他概念的联系提供了新的启示，因为我们获得了上限量化维度的一对一对应关系，而$ l^{q} $ - 频谱 - 频谱仅限于$ \ weft（0,1 \ right）$。我们提供了足够的条件，以$ l^{q} $ - 存在量化维度的频谱。通过这种方式，我们将每个GibB的量化维度都显示为$ \ Mathcal {C}^{1} $ - 在$ \ Mathbb {r}^{d} $上，在分离条件上没有任何假设的情况下，对自校的态度却没有任何假设。量化维度的一些已知的一般界限可以很容易地从我们的新方法中得出，有些可以改善。
We provide a complete picture of the upper quantization dimension in terms of the Rényi dimension by proving that the upper quantization dimension $\bar{D}_{r}(ν)$ of order $r>0$ for an arbitrary compactly supported Borel probability measure $ν$ is given by its Rényi dimension at the point $q_{r}$ where the $L^{q}$-spectrum of $ν$ and the line through the origin with slope $r$ intersect. In particular, this proves the continuity of $r\mapsto\bar{D}_{r}(ν)$ 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.
