Studies of the Palis Conjecture and new directions of dynamical systems

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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.