Recent zbMATH articles in MSC 28A78https://www.zbmath.org/atom/cc/28A782021-05-28T16:06:00+00:00WerkzeugDifferentiability of \(L^{q}\)-spectrum and multifractal decomposition by using infinite graph-directed IFSs.https://www.zbmath.org/1459.280042021-05-28T16:06:00+00:00"Deng, Guotai"https://www.zbmath.org/authors/?q=ai:deng.guotai"Ngai, Sze-Man"https://www.zbmath.org/authors/?q=ai:ngai.sze-manFor some self-similar measures \(\mu\) defined by iterated function systems which do not satisfy the open set condition, the authors proved the differentiability of \(L^q\) spectrum, i.e., \[ \tau_\mu(q)=\liminf\limits_{\delta\to0+}\frac{\log(\sup\sum_i\mu(B_\delta(x_i))^q)}{\log \delta},\ \ q\in\mathbb{R}. \] Here the supremum is taken over all families of disjoint balls \(B_\delta(x_i)\) with \({x_i\in\mathrm{supp}\,\mu}.\) The proof is based on a new approach of construction of an associated infinite graph-directed iterated function system.
Reviewer: Ivan Podvigin (Novosibirsk)Dimensional coincidence does not imply measure-theoretic tameness.https://www.zbmath.org/1459.030522021-05-28T16:06:00+00:00"Thamrongthanyalak, Athipat"https://www.zbmath.org/authors/?q=ai:thamrongthanyalak.athipatSummary: We show that there is a compact \(C^0\) submanifold \(M\) such that the Hausdorff measure of \(M\) is \(\infty\) and if \(\mathfrak R\) is an o-minimal expansion of the real field that is exponentially bounded, then \((\mathfrak R,M)\) does not define \(\mathbb Z\).Exact Hausdorff and packing measures for random self-similar code-trees with necks.https://www.zbmath.org/1459.280022021-05-28T16:06:00+00:00"Troscheit, Sascha"https://www.zbmath.org/authors/?q=ai:troscheit.saschaThe author considers the problem of finding appropriate gauge functions for (positive and finite) Hausdorff and packing measures. After establishing some useful bounds on these gauge functions, the author shows that, unexpectedly, self-similar code-trees do not admit gauge functions that simultaneously give positive and finite Hausdorff measure almost surely.
Reviewer: George Stoica (Saint John)Uniform continuity of fractal interpolation function.https://www.zbmath.org/1459.280072021-05-28T16:06:00+00:00"Pan, Xuezai"https://www.zbmath.org/authors/?q=ai:pan.xuezai"Wang, Minggang"https://www.zbmath.org/authors/?q=ai:wang.minggang"Shang, Xudong"https://www.zbmath.org/authors/?q=ai:shang.xudongSummary: In order to research analysis properties of fractal interpolation function generated by the iterated function system defined by affine transformation, the continuity of fractal interpolation function is proved by the continuous definition of function and the uniform continuity of fractal interpolation function is proved by the definition of uniform continuity and compactness theorem of sequence of numbers or finite covering theorem in this paper. The result shows that the fractal interpolation function is uniformly continuous in a closed interval which is from the abscissa of the first interpolation point to that of the last one.