论文标题
无穷大的quasiperiodic集和其分形Zeta函数的Meromorormorphic扩展
Quasiperiodic sets at infinity and meromorphic extensions of their fractal zeta functions
论文作者
论文摘要
在本文中，我们介绍了一个有趣的相对分形鼓（简称RFD）家族，并研究了它们的复杂维度，这些尺寸被定义为其相关的lapidus（距离）分形Zeta函数的杆，作者在先前的作者中引入了。 我们在无穷大处定义了管zeta函数，并获得一个功能方程，该功能方程与经典环境一样，将其连接到Infinity的距离Zeta函数。此外，在合适的假设下，我们提供了有关Minkowski可测量和不可衡量情况下无穷大的分形Zeta功能的存在的一般结果。我们还为Minkowski的可测量性以及上层Minkowski含量的上限提供了充分的条件。 我们表明，在无穷大的准二元组的复杂尺寸具有一个准碘结构，可以是代数或跨性别的。此外，我们提供了一个示例的示例，该示例是在无穷大的最大超fractal设置，并有规定的Minkowski维度，即，相应的分形Zeta函数的收敛性腹部实际上是其自然边界。
In this paper we introduce an interesting family of relative fractal drums (RFDs in short) at infinity and study their complex dimensions which are defined as the poles of their associated Lapidus (distance) fractal zeta functions introduced in a previous work by the author. We define the tube zeta function at infinity and obtain a functional equation connecting it to the distance zeta function at infinity much as in the classical setting. Furthermore, under suitable assumptions, we provide general results about existence of meromorphic extensions of fractal zeta functions at infinity in the Minkowski measurable and nonmeasurable case. We also provide a sufficiency condition for Minkowski measurability as well as an upper bound for the upper Minkowski content, both in terms of the complex dimensions of the associated RFD. We show that complex dimensions of quasiperiodic sets at infinity posses a quasiperiodic structure which can be either algebraic or transcedental. Furthermore, we provide an example of a maximally hyperfractal set at infinity with prescribed Minkowski dimension, i.e., a set such that the abscissa of convergence of the corresponding fractal zeta function is in fact its natural boundary.
