Applications of periodic orbits in Hamiltonian dynamics and persistence modules

2021 Fiscal Year Final Research Report

• Project/Area Number: 20K22302
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Multi-year Fund
• Review Section: 0201:Algebra, geometry, analysis, applied mathematics,and related fields
• Research Institution: Niigata University
• Principal Investigator: Orita Ryuma 新潟大学, 自然科学系, 助教 (30874531)
• Project Period (FY): 2020-09-11 – 2022-03-31
• Keywords: シンプレクティック多様体 / フレアー理論 / ハミルトン周期軌道 / パーシステント加群 / R群

Outline of Final Research Achievements

In this research I dealt with Ginzburg-Gurel conjecture which states that "every Hamiltonian diffeomorphism of closed symplectic manifolds has infinitely many non-contractible periodic orbits, provided that the diffeomorphism has one orbit". Here a manifold is said to be symplectic if it admits a non-degenerate closed two-form. I investigated the problem by assuming some conditions on the fundamental group of the manifold and the symplectic form. Actually, I proved that the conjecture is true for spherically monotone symplectic manifolds whose fundamental group is assumed to be virtually abelian or an R-group.
During the period, I proved that the equivalence between R-groups and torsion-free group of type N. Accordingly, since torsion-free groups of type N are principal, I could apply the theory of Bredon cohomology for them.

Free Research Field

シンプレクティック幾何学

Academic Significance and Societal Importance of the Research Achievements

ハミルトン周期軌道の検知は，解析力学に端を発するシンプレクティック幾何学における基本的な問題であり，また，近年位相的データ解析にて盛んに研究されているパーシステントホモロジーとの関連の研究は学術的，社会的に意義がある。