Research on random complex dynamics and the dynamics of semigroups of holomorphic maps

2014 Fiscal Year Final Research Report

• Project/Area Number: 24540211
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Global analysis
• Research Institution: Osaka University
• Principal Investigator: SUMI Hiroki 大阪大学, 理学(系)研究科(研究院), 准教授 (40313324)
• Co-Investigator(Kenkyū-buntansha): SATO Yuzuru (30342794)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: 複素力学系 / ランダム複素力学系 / フラクタル / エルゴード理論 / 協調原理 / ジュリア集合 / 複素解析 / 有理半群

Outline of Final Research Achievements

I investigated the random dynamics of rational maps and the dynamics of semigroups of rational maps on the Riemann sphere. I showed that regarding random complex dynamics of polynomials, generically, the chaos of the averaged system disappears at any point in the Riemann sphere, due to the automatic cooperation of the generators. I investigated the iteration and spectral properties of transition operators acting on the space of (Hoelder) continuous functions on the Riemann sphere. I showed that the set of stable systems is open and dense in the space of random dynamical systems of polynomials. Also, I investigated the limit state functions. By taking a partial derivative of the function of probability of tending to a minimal set with respect to a probability parameter, I introduced a complex analogue of the Takagi function. I am developing multifractal analysis for limit state funtions and the complex analogue of the Takagi function.

Free Research Field

複素力学系