The research on the uniqueness of the density functions for the existence probability of orbits

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K04902
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kumamoto University
• Principal Investigator: Sumi Naoya 熊本大学, 大学院先端科学研究部(理), 教授 (50301411)
• Research Collaborator: HIRAYAMA Michihiro 筑波大学, 数理物質科学研究科, 准教授 (50452735) ; CHUNG Yong Moo 広島大学, 工学研究科, 准教授 (20314734) ; YAMAMOTO Kenichiro 長岡技術科学大学, 工学部, 准教授 (30635181)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: 解析学 / 力学系 / エルゴード理論

Outline of Final Research Achievements

In this study, we proved that in the case when the ambient space is at most 3-dimensional, under the local condition that for every pair of orbits, there exists another orbit which transfers from one orbit to the other, the following global dynamical property holds: if the density function for the existence probability of orbits exists for every orbit, then it is uniquely determined. Moreover, if the dynamical system satisfies the specification property, we obtained a similar result.
On the other hands, concerning the space of all dynamical systems satisfying the specification property, the following holds: if a partially hyperbolic diffeomorphism has two saddles with different indices, and the central direction is one-dimensional, then it does not satisfy the specification property.

Free Research Field

数物系科学

Academic Significance and Societal Importance of the Research Achievements

軌道の存在状態を表す確率密度は，力学系理論の初期から扱われている基本的な研究対象である．存在確率密度の一意性は，力学系が個々の軌道に依らない１つの平衡状態をもつことを保証する．本研究により，相空間の次元が3次元以下の場合に，平衡状態の一意性という大域的な性質は，局所的な軌道同士の条件から導かれることが結論付けられた．