Some Remarks on the Fractal Structure of Irrigation Balls
Some Remarks on the Fractal Structure of Irrigation Balls
复制标题
关于灌溉球分形结构的一些评论
DOI：
10.1515/ans-2018-2035
复制
发表时间：
2018
影响因子：
1.8
通讯作者：
S. Solimini
中科院分区：
文献类型：
--
作者：
G. Devillanova;S. Solimini
摘要 本文与 Pegon、Santambrogio 和 Xia 的关于某些集合（我们称之为“灌溉球”）的边界维数的猜想有关。我们提出了子球和规定半径的子球体的概念，并证明，一般来说，子球体唯一可能的闵可夫斯基维数就是猜想中预期的维数。同时，除了一些重要集合的尺度转换特性和维数估计之外，我们提出了第三种方法来研究分形规律，该方法依赖于景观函数的较低振荡估计，结果表现为 Weierstrass 型函数。
Abstract The paper is related to a conjecture by Pegon, Santambrogio and Xia concerning the dimension of the boundary of some sets which we are calling “irrigation balls”. We propose a notion of sub-balls and sub-spheres of prescribed radius and we prove that, generically, the only possible Minkowski dimension of sub-spheres is the one expected in the conjecture. At the same time, beside the scale transition properties and the dimension estimates on some significant sets, we propose a third approach to study the fractal regularity which relies on lower oscillation estimates on the landscape function, which turns out to behave as a Weierstrass-type function.