Rigidity problem on group actions with an invariant geometric structure

• Project/Area Number: 26400085
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Kyoto University
• Principal Investigator: Asaoka Masayuki 京都大学, 理学研究科, 准教授 (10314832)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000); Fiscal Year 2016: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2015: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2014: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
• Keywords: 群作用 / 力学系 / 周期軌道 / 群作用の剛性 / 野生的力学系

Outline of Final Research Achievements

We obtained a local rigidity result for certain group action related to the rank-1 Lie group SO(n,1) with n>1
and a description of deformation of certain group action on the torus which preserves a geometric structure. The latter was an application of a method to show a rigidity result for a conformal action on the sphere.
We also proved the abundance of super-exponential growth of the number of periodic points and existence of universal dynamics for one-dimensional iterated function systems which satisfies some mild conditions. One-dimensional iterated function systems are toy models of partially hyperbolic dynamical systems. Hence, our result is a step to understand wild behavior of partially hyperbolic systems.

Research Products

• [Int'l Joint Research] Imperial College London(英国)
• [Int'l Joint Research] Imperial College London(英国)
• [Journal Article] Degenerate behavior in non-hyperbolic semigroup actions on the interval: fast growth of periodic points and universal dynamics (2016)
  • Author(s): M. Asaoka, K. Shinohara, D.Turaev
  • Journal Title: Mathematische Annalen Volume: - Issue: 3-4 Pages: 1277-1309
  • DOI: 10.1007/s00208-016-1468-0
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Presentation] Fast growth of the number of periodic points and unversal dynamics in semigroup actions on the interval (2015)
  • Author(s): 浅岡 正幸
  • Organizer: ランダム力学系理論とその応用
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2015-09-30
  • Invited
• [Presentation] Arbitrary growth of the number of periodic points and universal dynamics in one-dimensional semigroups (2014)
  • Author(s): Masayuki Asaoka
  • Organizer: Workshop on Theory and Applications of Random/Non-autonomous Dynamical Systems
  • Place of Presentation: Imperial College London
  • Year and Date: 2014-09-09
  • Invited
• [Presentation] Rigidity of certain solvable actions on the tori (2014)
  • Author(s): Masayuki Asaoka
  • Organizer: Workshop on Geometry and Ddynamics of Foliations
  • Place of Presentation: ICMAT, Madrid, Spain
  • Year and Date: 2014-09-04
• [Presentation] Arbitrary growth of the number of periodic points in one-dimensional semigroups (2014)
  • Author(s): Masayuki Asaoka
  • Organizer: ICM satellite conference on Dynamical Systems and Related Topics
  • Place of Presentation: Chungnam National University, Daejeon, Korea
  • Year and Date: 2014-08-11
  • Invited