Studies of the Palis Conjecture and new directions of dynamical systems

在这项研究中，我研究了以下两个主题：1.高维Palis猜想的求解；2.在Palis猜想的解之后，为研究寻找动力系统的新方向。对于第一个主题，我在几个国际会议上谈到了结果，解决方案已经在一篇论文中公布了准备。但是，当局并没有承认这一证据是正确的。整个证明可以分为三个部分。当Palis猜想中的微分同态不能近似为同态分支时，证明它是公理A的过程如下：(1)利用遍历闭合引理的推广，在每个遍历测度的支集上构造一个受控分裂：(2)在上述遍历测度支撑下的受控分裂中，对应于零Lyapunov指数的子空间为零；也就是说，它是双曲遍历测度(3)遍历测度的支集的闭包实际上是一个双曲集，通过双曲性(在微分同胚的邻域内)的一致性，它可以推广到非游荡集。在这个研究阶段，我为(1)写了一篇论文，作为预印本，对于(2)和(3)，我现在正在准备论文，对于第二个主题，我还没有结果。但我感兴趣的是斯梅尔提出的“复杂性”和计算机科学，并考虑了在动力系统理论中应用的可能性。我不仅参加了在京都举行的“动力系统的新方向”国际会议，还参加了几个研讨会，我在日本期刊上为非专业人士撰写的文章都来自于这项研究。

In this research, I studied the following two subjects ; 1.solving the Palis Conjecture in higher dimension, and 2.finding a new direction of dynamical systems for the research after the solution of the Palis Conjecture.For the first one, I talked results in several international conferences, and the solution has been announced in a paprer in preparation. But, the proof has not been admitted to be correct by authorities. The whole proof can be divided into three parts. When a diffeomorphism can not be approximated by one exhibiting a homodinic bifurcations in the Palis Conjecture, we have process of proving that it is Axiom A as follows :(1)Constructing a dominated splitting on the support of every ergodic measure using an extension of the ergodic closing lemma ;(2)In the dominated splitting on the support of the ergodic measure above, the subspace corresponding the zero Lyapunov exponent is zero ; that is, it is a hyperbolic ergodic measure(3)The closure of the supports of the ergodic measures is, in fact, a hyperbolic set, and by the uniformity of the hyperbolicity (in a neighborhood of the diffeomorphisim), it can be extended to the nonwandering set.In this period of the research, I wrote a paper for (1) as a preprint, and for (2) and (3), I am preparing the paper now.For the second subject, I don't have results yet. But I am interested in the "complexity" and computer science that Smale proposed, and considered a possibility to applications in the theory of dynamical systems. Not only the international conference "New directions in Dynamical Systems" held in Kyoto, I participated in several workshops and the articles I wrote in Japanese journals for non-specialists were come from this research.