Studies of the Palis Conjecture and new directions of dynamical systems

在本研究中，我研究了以下两个主题；1.在高维空间中求解Palis猜想；2.为Palis猜想解后的动力系统研究找到了新的方向。对于第一个问题，我在几个国际会议上谈到了结果，解决方案已经在准备的论文中公布。但是，这一证据并没有被当局承认是正确的。整个证明可分为三个部分。当一个微分同胚不能被Palis猜想中的同二分岔所近似时，我们证明了它是公理a:(1)利用遍历闭引理的推广，在每一个遍历测度的支持上构造一个支配分裂；(2)在上述遍历测度支持下的支配分裂中，Lyapunov指数为零所对应的子空间为零；(3)遍历测度的支点的闭包实际上是一个双曲集，并且通过双曲性的一致性（在微分同态的邻域内），可以推广到非游走集。在这段时间的研究中，我为(1)写了一篇论文作为预印本，(2)和(3)我正在准备论文。第二个科目，我还没有结果。但我对Smale提出的“复杂性”和计算机科学很感兴趣，并考虑了在动力系统理论中应用的可能性。不仅是在京都举行的国际会议“动力系统新方向”，我还参加了几个研讨会，并在日本的非专业期刊上发表了来自这项研究的文章。

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.