摘    要

本论文研究与流的C¹稳定性猜测（结构稳定性猜测，Ω稳定性猜测和星号猜测）有关的问题。

全文共分为三章。第一章为引言部分。首先介绍了与稳定性猜测有关的一些历史。在§1.2中，在介绍了一些动力系统的必要知识后，我们对流引入了双曲性与公理A的概念，并且不加证明地罗列了一些与此有关的结果。因为我们在后面需要大量运用廖山涛先生的正常集，阻碍集和极小歧变集的理论，在§1.3中，我们对这一理论作了一简要介绍。在本章的最后，我们讨论了一般紧致度量空间上的流的流等价与连续流等价的关系。我们证明了：如果两个流是流等价的，并且不含奇点，则它们也是连续流等价的。同时，我们还举了一个简单的例子说明无奇点这一条件不能去掉。这只是一个孤立的结果，与随后的内容并无太大关系。

第二章为本论文的主体部分。在陈述了一些必要的扰动技巧后(§2.1)，在§2.2中，对星号系统我们讨论了准双曲分解以及无环性条件。所谓星号系统指的是该系统及其附近的系统的奇点以及周期轨皆为双曲的。在这里，我们仔细地分析了准双曲分解中的子空间与流的稳定子空间与不稳定子空间之间的关系。这一节的主要结论为：如果S为公理A系统的内部，即S有一C¹邻域U使得U中的任一向量场皆为公理A系统，则S满足无环性条件。这一结论将Ω-稳定性猜测约化为从稳定性推公理A。Palis曾在1970年对微分同胚得到过类似的结论。但是他的原文的证明中存在一漏洞，并且其陈述对流不成立。在§2.3中，我们研究无奇星号猜测。是否所有的无奇星号系统皆满足公理 A 和无环性条件就是所谓的无奇星号猜测。无奇星号猜测在三维情形由廖先生用阻碍集的方法证明。而无奇星号猜测对于三维以上的流是否成立则是廖山涛先生的一个著名问题。至今依然未获解决。我们证明了在无奇星号系统中有一开稠集满足公理 A 和无环性条件。这意味着在通有的意义下，无奇星号猜测成立。证明中的主要工具有Mañé，文兰教授的流的C¹遍历封闭引理，廖山涛先生的关于星号系统的一个基本性质定理以及经文兰和夏志宏改进后的C¹连接引理。我们的贡献在于在无奇星号系统中找到了一Baire子集，使得上述方法在现在的情形下正好可以使用。作为上述方法的一个运用，马上可以得出在Ω稳定(结构稳定)系统的集合中有一开稠集满足公理 A 和无环性条件(对应的，公理 A 和强横截性条件)。我们在§2.4节给出了流的C¹结构稳定性猜测的另外两个不同证明。首先，我们证明了如果系统及其附近的系统皆为Kupka-Smale系统，则该系统的阻碍集为空。作为一个推论，这解决了廖山涛先生的一个猜测，即下述四个条件等价：1) 系统为结构稳定的，2)系统及其附近的系统皆为Kupka-Smale系统，3)系统的阻碍集为空集，4) 系统为公理A且满足横截性条件。3)→4)，4)→1)，1)→4)已经分别由廖山涛，Smale、 Anosov、 Robbin、 Robinson以及Robinson完成。从而我们的结论就给出了C¹结构稳定性猜测的一个证明。需要指出的是稳定性猜测本身只肯定了1）与4）两个条件等价，而不能蕴涵四个条件等价。在这个意义上，我们所证明的结果比稳定性猜测更强。在证明中，我们大量地运用了阻碍集的理论和强有力的连接引理。这一证明的基本思路非常清晰: 廖先生将极小歧变集分为简单与非简单两类，我们的目标就是将其逐一排除掉。这一方法与已有的证明有着很大的不同。第二种证明方法本质上则与已有的方法大体一样。只是在证明的最后才有些不同。如上所述，由前一节的结论我们有结构稳定的系统的集合中有一开稠集满足公理A及强横截条件。然后，我们再利用结构稳定性得出所有的系统皆满足公理A及强横截条件。此时，我们在证明中用到了正常集以及极小歧变集的理论。而已有的方法则是直接去证明每一结构稳定的系统皆满足公理A。

在第三章我们对离散系统引入了阻碍集的概念，并探讨了在此情形下的正常集的一些理论。这建立了廖山涛先生的正常集理论与西方关于拟Anosov理论的某种联系。在§3.2中我们得到了在定义阻碍集这一概念中起着关键作用的Ψ映射与原来映射在其余切丛上所诱导映射之间仅仅只相差一个非线性因子这一重要关系。仿照流的情形，在§3.3中，引入了微分同胚的阻碍集的概念。运用上面的重要关系以及Mañé关于拟Anosov映射的结果，我们得到了如下结论：对于微分同胚，如下四条等价1） 系统满足公理A和强横截条件，2） 满足线性横截条件，3） 系统在余切丛上诱导的映射为拟Anosov，4） 系统的阻碍集为空。最后，作为上述重要关系的一个简单应用，我们给出了流的正常集的另一刻划。

Abstract

In the thesis, we study some problems related to the C¹ stability conjectures, i.e., C¹ structural stability conjecture, C¹ Ω-stability conjecture and non-singular C¹ star conjecture for flows.

The thesis consists of three chapters. The first chapter is an introduction. Firstly, we survey the history of stability conjectures. In§1.2, after some necessary preparation on dynamical systems, we introduce the concepts of hyperbolicity and Axiom A for flows. In §1.3, we give a brief introduction to the Liao's theory on normal sets, obstruction sets and minimal rambling sets since they will be used extensively in the following. In the last section of this chapter, we discuss the relation between flow equivalence and continuous flow equivalence for flows on a compact metric space. The main result is: If two flows are flow equivalent and contains no singularity, then they are also continuous flow equivalent. On the other hand, a simple example is given to show that the condition“no singularity”can not be dropped. But this result is relatively isolated to the rest of the thesis.

The second chapter is the main part of the thesis. After stating some necessary perturbation techniques (§2.1), in §2.2, we discuss in detail the dominated splittings and the no-cycle condition for star systems. We analyze in detail the relation of the subspaces in dominated splitting and the stable and unstable subspaces of the flow. The main result of this section is: If S is in the interior of the set of Axiom A systems, i.e., S has a C¹ neighbourhood U such that every system in U satisfies Axiom A, then S satisfies the no-cycle condition. With the help of this  result, one only needs to prove the system satisfies Axiom A in order to prove the Ω-stability conjecture. In 1970, Palis proved a similar result for diffeomorphisms. But there is a gap in his article and his statement is no longer correct for flows. In §2.3, we study star systems without singularity. A system is called a star system if the system has a C¹ neighbourhood such that the critical elements (singularities and periodic orbits) of every system in the neighbourhood are all hyperbolic. Whether every star system satisfies Axiom A and the no-cycle condition is the so-called non-singular star conjecture, which is a famous problem raised by Prof. Liao. Liao proved the conjecture in 3-dimensional case. It is still a big open problem for higher dimensional case. We proved that there is an open and dense set in the set of star systems with no singularity such that every system in the set satisfies Axiom A and the no-cycle condition, which means that non-singular star conjecture is correct in the generic sense. The method used here is similar to the method of Wen in proving structural stability conjecture for flows. This method goes back to that of Mañé in proving the structural stability conjecture for diffeomorphisms. The main tools in the proving are the C¹ ergodic closing lemma for flows, a basic property theorem on star systems of Liao and the improved version of C¹ connecting lemma of Wen and Xia. Our contribution is that we find a Baire set in the set of star systems with no singularity and the method of Wen can be used for the systems in the Baire set. As an application of the above proof, we obtain at once that there is an open and dense set in the set of Ω-stable systems (structural stable systems) such that every system in the set satisfies Axiom A and the no-cycle condition (Axiom A and the strong transversality condition, respectively). In the last section, we give two different proofs for the structural stability conjecture for flows and they are different more or less from the original one of Wen. First we prove that if a system and its nearby systems are all Kupka-Smale systems, i.e., the system is in the interior of the set of Kupka-Smale systems, then the obstruction set of the system is empty. As a corollary, we solve a conjecture of Liao, i.e., the following four conditions are all equivalent: 1) the system is structurally stable, 2) the system is in the interior of the set of Kupka-Smale systems, 3) the obstruction set and the interior of singularities of the system are empty, 4) the system satisfies Axiom A and the strong transversality condition. 3)→4)，4)→1)，1)→4)have been proved by Liao, Smale, Anosov, Robbin and  Robinson. Combining with their results, this gives a proof for the structural stability conjecture. We should note that the equivalence of 1) and 4) i.e., the structural stability conjecture, does not imply the whole 4 conditions are equivalent. In this sense, our result is stronger than those of Wen and Hayashi. In the proof, we use a great deal of the obstruction set theory of Liao. Of course, we have to use the powerful C¹ connecting lemma.  The basic idea of the proof is very clear. Liao classified rambling sets into two classes, simple one and non-simple one. Then we rule out the two cases one by one. This method is very different to that of Wen. The second method is same as that of Wen in principal. The difference occurs in the last part of the proof. As stated before, we have proved that there is an open and dense set in the set of structurally stable systems such that every system in the set satisfies Axiom A and the strong transversality condition. Then, using structural stability condition again, we obtain that every structurally stable system satisfies Axiom A and the strong transversality condition. Here, we analyze the minimal rambling set again. But Wen solved the problem by directly showing that every system satisfies Axiom A.

In the last chapter, we introduce the concept of obstruction set for diffeomorphism and obtain some results on normal sets. We do not prove these results directly. In fact, we obtain them by showing an important relation between the obstruction set theory and the classical theory of quasi-Anosov. More precisely, we obtain an important relation between the mapping Ψ, which plays an important role in defining obstruction set, and the mapping on the cotangent bundle induced by the original diffeomorphism. We show that the only difference between them is a nonlinear factor. We define obstruction set for diffeomorphism corresponding to the flow case. By using the above important relation and some results on quasi-Anosov of Mañé, we obtain that the following conditions are equivalent for a diffeomorphism: 1) the system satisfies Axiom A and the strong transversality condition, 2) the system satisfies the so-called linear transversality condition, 3) the mapping on the cotangent bundle induced by the system is quasi-Anosov and 4) the obstruction set of the system is empty. At last, as another simple application of the above important relation, we give another characterization for normal sets of flows.