AI 中文总结
本文证明存在无穷多仅含一个几何块的图流形，每个可容许任意多个两两不等价的传递Anosov流，其构造方法与此前基于粘合测地流的方式完全不同。
AI 中文摘要
Anosov流拥有悠久且丰富的历史，最初由Anosov和Sinai研究负曲率曲面上的测地流所推动。由于众所周知的原因，并非每个闭流形都容许Anosov流：容许Anosov流的三维流形\textit{M}的基本群必须具有指数增长，且\textit{M}必须被\textit{R}^3普遍覆盖。不过，存在足够多的机制来在可容许的三维流形上构造不同的Anosov流，例如Dehn-Goodman-Fried手术或利用双曲构件。该领域的一个核心问题是确定单个流形可支撑的Anosov流数量。至今仍未解决的问题是，是否存在三维流形上的无穷多组两两不等价的Anosov流。已有若干论文证明，对任意自然数\textit{n}，存在流形\textit{M}_\textit{n}容许\textit{n}个两两不等价的Anosov流。在所有已知例子中，流形\textit{M}_\textit{n}由多个几何块构成。本文中，我们证明存在可数多个图流形\textit{M}_{\textit{k},\textit{n}}（\textit{k}为自然数），它们仅含一个几何块，每个都容许\textit{n}个两两不等价的传递Anosov流。此前已知的同一图流形上不同流的所有构造均基于粘合测地流，而本文构造的流的性质完全不同；它们由单个双曲插件构造而成，该插件是曲面上Morse-Smale微分同胚的 suspension（悬挂）。
英文摘要
Anosov flows have a long and rich history, firstly motivated by the study of geodesic flows in negative curvature surface by Anosov and Sinai. Not every closed manifold admits an Anosov flow for well-known reasons: the fundamental group of a 3-manifold \(M\) admitting an Anosov flow must have exponential growth, and \(M\) must be universally covered by \(\mathbb{R}^{3}\). Nevertheless, there are sufficient mechanisms for constructing distinct Anosov flows on admissible 3-manifolds, such as Dehn-Goodman-Fried surgery or playing with hyperbolic building blocks. A central problem in the field has been to determine the number of Anosov flows that can be supported by a single manifold. The question of whether there exists an infinite set of pairwise non-equivalent Anosov flows on a 3-manifold remains open to this day. However, there are several papers proving the existence of a manifold $M_n$ that admits $n$ pairwise inequivalent Anosov flows for any natural number $n$. In all known examples, the manifolds $M_n$ are composed of several geometric pieces. In the present paper, we prove the existence of a countable number of graph manifolds $M_{k,n}$, $k \in \mathbb{N}$ with a single geometric piece, each of which admits $n$ pairwise non-equivalent transitive Anosov flows. All previously known constructions of different flows on the same graph manifold were based on gluing geodesic flows. The nature of the flows constructed in this paper is completely different; they are constructed from a single hyperbolic plug, which is a suspension over a Morse-Smale diffeomorphism on a surface.
Comments16 pages, 13 figures
