论文标题
抛物线差异轨道的分形Zeta功能
Fractal zeta functions of orbits of parabolic diffeomorphisms
论文作者
论文摘要
在本文中，我们证明，差异性抛物性细菌的轨道分形Zeta功能可以将其碎片延伸到整个复合平面。我们描述了他们的一组极（即它们的复杂维度）及其主要部分，这些部分可以理解为分形足迹。我们研究了抛物线菌的一个轨道的分形足迹，并从中提取有关该菌的内在信息，特别是其正式类别。此外，我们将复杂的维度与具有振荡的“系数”以及其动态正则化管功能的渐近膨胀相关联轨道功能的普遍渐近膨胀。有趣的是，抛物线轨道提供了具有非平凡的Minkowski（或盒）尺寸的集合的第一个示例，其管函数具有较高的振荡术语，但是，它们不具有非现实的复合尺寸，因此在Lapidus的感觉中不称为分形。
In this paper, we prove that fractal zeta functions of orbits of parabolic germs of diffeomorphisms can be meromorphically extended to the whole complex plane. We describe their set of poles (i.e. their complex dimensions) and their principal parts which can be understood as their fractal footprint. We study the fractal footprint of one orbit of a parabolic germ f and extract intrinsic information about the germ f from it, in particular, its formal class. Moreover, we relate complex dimensions to the generalized asymptotic expansion of the tube function of orbits with oscillatory "coefficients" as well as to the asymptotic expansion of their dynamically regularized tube function. Interestingly, parabolic orbits provide a first example of sets that have nontrivial Minkowski (or box) dimension and their tube function possesses higher order oscillatory terms, however, they do not posses non-real complex dimensions and are therefore not called fractal in the sense of Lapidus.
