Typical Rényi dimensions of measures. The cases: \(q=1\) and \(q=\infty\)
The typical Rényi dimensions of measures are studied. For a Borel probability measure \(\mu\) on \(\mathbb{R}^d\), for \(q\in [-\infty,+\infty]\), the lower and upper \(q\) Rényi dimensions are defined by \[ \begin{aligned} \underline D_\mu(q)&= \liminf_{r\to 0}\,{1\over q-1} {\log\int_U \mu(B(x, r))\, q^{-1} \,d\mu(x)\over\log r},\quad q\in\mathbb{R}\setminus\{1\},\\ \overline D_\mu(q)&= \limsup_{r\to 0}\,{1\over 1-q} {\ln\int_U \mu(B(x, r))\, q^{-1}\,d\mu(x)\over \ln r},\quad q\in\mathbb{R}\setminus\{1\}, \end{aligned} \] where \(U= \text{supp\,}\mu\), and \[ \begin{alignedat}{2} \underline D_\mu(1)&= \liminf_{r\to 0}\,{\log \int_U\mu(B(x, r))\,d\mu(x)\over \log r},&\qquad \overline D_\mu(1)&= \limsup_{r\to 0}\,{\log \int_U \mu(B(x, r))\,d\mu(x)\over \log r},\\ \underline D_\mu(-\infty)&= \liminf_{r\to 0}\, {\log\inf_{x\in U}\,\mu(B(x, r))\over\log r},&\qquad \overline D_\mu(-\infty)&= \limsup_{r\to 0}\, {\log\inf_{x\in U}\,\mu(B(x,r))\over \log r},\\ \underline D_\mu(\infty)&= \liminf_{r\to 0}\,{\log \sup_{x\in U}\,\mu(x, r))\over\log r},&\qquad \overline D_\mu(\infty)&= \limsup_{r\to 0}\,{\log \sup_{x\in U}\,\mu(B(x, r))\over\log r}. \end{alignedat} \] The main result of the note is the following Theorem. Let \(K\subset\mathbb{R}^d\) be a compact subset, and denote \[ \underline s= \underset{x\in K}{}{\text{inf}}\,\underline{\dim}_{B,\text{loc}}(x, K),\quad \overline s= \underset{x\in K}{} {\text{inf}}\overline{\dim}_{B,\text{loc}}(x, K), \quad s=\overline{\dim}_B(x, K), \] and \(\mathbb{P}(K)\) the Borel probability measure family on \(K\) with weak topology. Then for the case \(q= \infty\), (1) for all \(\mu\in \mathbb{P}(K)\), we have \(0\leq \underline D_\mu(\infty)\leq\overline D_\mu(\infty)\leq s\); (2) a typical measure \(\mu\in \mathbb{P}(K)\), satisfies \(\underline D_\mu(\infty)= 0\) and \(\underline s\leq\overline D_\mu(\infty)\leq\overline s\). In particular, if \(\underline s= s=\overline s\), then a typical measure \(\mu\in \mathbb{P}(K)\) satisfies \(\underline D_\mu(\infty)= 0\) and \(\overline D_\mu(\infty)= s\). And for the case \(q= 1\), (1\('\)) for \(\mu\in\mathbb{P}(K)\), we have \(0\leq\underline D_\mu(1)\leq\overline D_\mu(1)\leq s\); (2\('\)) a typical measure \(\mu\in \mathbb{P}(K)\) satisfies \(\underline s\leq\overline D_\mu(1)\leq s\). In particular, if \(\underline s= s=\overline s\), then a typical measure \(\mu\in \mathbb{P}(K)\) satisfies \(\overline D_\mu(1)= s\).
scientific article; zbMATH DE number 5146749
scientific article