论文标题
Gromov双曲度度空间中非扩展地图的向后动力学
Backward dynamics of non-expanding maps in Gromov hyperbolic metric spaces
论文作者
论文摘要
我们研究了适当的Geodesic Gromov双曲度度空间$ x $的非扩展自图$ f $的向后动力学之间的相互作用与Gromov边界中$ f $的边界常规固定点之间的相互作用。为此，我们在Gromov边界的边界常规固定点上介绍了稳定扩张的概念，其值与固定点的动力学行为有关。该理论特别适用于有界域的全态自图$ω\ subset \ subset \ subset \ mathbb {c}^q $，其中$ω$要么是强烈的pseudoconvex，convex有限型类型，或pseudoconvex有限类型，带有$ q = 2 $，并解决了来自文献的几个开放问题。我们扩展了由Bracci和Poggi-Corradini获得的光盘$ \ mathbb {d} \ subset \ Mathbb {C} $的全态自图的结果。特别是，通过我们的几何方法，我们能够回答一个问题，即使对于单位球$ \ mathbb {b}^q \ subset \ subset \ mathbb {c}^q $，也就是说，对于全体形态抛物面自图逐步升级的任何带有有界步骤的向后级别的升级始终将其转换为边界上的点。
We study the interplay between the backward dynamics of a non-expanding self-map $f$ of a proper geodesic Gromov hyperbolic metric space $X$ and the boundary regular fixed points of $f$ in the Gromov boundary. To do so, we introduce the notion of stable dilation at a boundary regular fixed point of the Gromov boundary, whose value is related to the dynamical behaviour of the fixed point. This theory applies in particular to holomorphic self-maps of bounded domains $Ω\subset\subset \mathbb{C}^q$, where $Ω$ is either strongly pseudoconvex, convex finite type, or pseudoconvex finite type with $q=2$, and solves several open problems from the literature. We extend results of holomorphic self-maps of the disc $\mathbb{D}\subset \mathbb{C}$ obtained by Bracci and Poggi-Corradini. In particular, with our geometric approach we are able to answer a question, open even for the unit ball $\mathbb{B}^q\subset \mathbb{C}^q$, namely that for holomorphic parabolic self-maps any escaping backward orbit with bounded step always converges to a point in the boundary.
