The study of generic properties of differentiable dynamical systems

2014 Fiscal Year Final Research Report

• Project/Area Number: 22540221
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Global analysis
• Research Institution: The University of Tokyo
• Principal Investigator: HAYASHI Shuhei 東京大学, 数理(科)学研究科(研究院), 准教授 (20247208)
• Project Period (FY): 2010-04-01 – 2015-03-31
• Keywords: Palis予想 / Smale予想

Outline of Final Research Achievements

On the C1 Palis Conjecture saying that in the complement of the closure of C1 diffeomorphisms having the uniform hyperbolicity (Axiom A + no cycle condition) there exist a dense subset of those exhibiting homoclinic bifurcations, we prove that the conjecture is true if C2 ergodic measures with zero Lyapunov exponents are destroyed by C1 small perturbations. For the 2-dimensional case, if Smale Conjecture saying that uniformly hyperbolic diffeomorphisms are dense in C1 surface diffeomorphisms is not true, then (taking the inverse if necessary) in a residual subset of the complement of uniformly hyperbolic diffeomorphisms, there exist infinitely many sinks, which are either observable or pathological. Moreover, we developed a new C1 closing lemma for C2 diffeomorphisms under some condition.

Free Research Field

力学系