A study on Fatou components of transcendental entire functions and singular values

• Project/Area Number: 23540213
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kochi University
• Principal Investigator: Morosawa Shunsuke 高知大学, 教育研究部自然科学系理学部門, 教授 (50220108)
• Project Period (FY): 2011-04-28 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2015: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2014: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2013: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2012: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2011: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 複素力学系 / 超越整関数 / 特異値 / 遊走領域 / ベーカー領域 / 力学的収束 / 広義一様収束 / ハウスドルフ収束 / カラテオドリー収束 / ジーゲル円板

Outline of Final Research Achievements

I studied complex dynamics of transcendental entire functions. In particular, I investigated wandering domains and Baker domains which are Fatou components never appeared in complex dynamics of rational functions. In the case where transcendental function have those domains, the functions necessarily have infinitely many singular values. I chose a family of transcendental entire functions whose singular values are well controlled. Furthermore, some functions of the family have wandering domains or Baker domains. I gave some characterization of wandering domains and Baker domains in the family by using locally uniformly polynomial sequences.I also showed the Caratheodory convergence of components in parameter spaces.

Research Products

• [Journal Article] DYNAMICAL CONVERGENCE OF A CERTAIN POLYNOMIAL FAMILY TO fa(z) = z + e^z + a (2015)
  • Author(s): Shunsuke Morosawa
  • Journal Title: Annales Academiae Scientiarum Fennicae Volume: 40 Pages: 449-463
  • DOI: 10.5186/aasfm.2015.4028
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Meromorphic solutions of functional equations f(G(z))=R(f(z)) (2012)
  • Author(s): Katsuya Ishizaki, Shunsuke Morosawa, Mitsunori Yakou
  • Journal Title: Complex Var. Elliptic Equ. Volume: 57 Pages: 15-22
  • Peer Reviewed
• [Presentation] Dynamics of Transcendental Entire Functions (2015)
  • Author(s): 諸澤 俊介
  • Organizer: 2nd Japan - Nepal Joint Workshop Around Coding theory, Dynamical systems and related topics
  • Place of Presentation: Tribhuvan University（ネパール）
  • Year and Date: 2015-11-26
  • Int'l Joint Research / Invited
• [Presentation] Semigroups whose Julia sets are Cantor targets (2014)
  • Author(s): 諸澤 俊介
  • Organizer: RIMS Workshop on Complex Dynamics
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2014-12-09
• [Presentation] ブラシュケ積の剰余ジュリア集合について (2012)
  • Author(s): 諸澤 俊介
  • Organizer: 複素力学系の総合的研究
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] f_{a}(z)=z+e^{z}+a へのある多項式列の力学的収束について
  • Author(s): 諸澤 俊介
  • Organizer: 2013年度日本数学会秋季総合分科会
  • Place of Presentation: 愛媛大学
• [Presentation] f_{a}(z)=z+e^{z}+a へのある多項式列の力学的収束について
  • Author(s): 諸澤 俊介
  • Organizer: 「等角写像論・値分布論」合同研究集会
  • Place of Presentation: 法政大学市ヶ谷キャンパス
• [Presentation] Dynamical convergence of a certain polynomial family to f_{a}(z)=z+e^{z}+a
  • Author(s): 諸澤 俊介
  • Organizer: nternational Conference on Advances in Dynamical Systems
  • Place of Presentation: Kishangarh, India
  • Invited
• [Presentation] Caratheodory convergence to Baker domains or wandering domains
  • Author(s): 諸澤 俊介
  • Organizer: 複素力学系の新展開
  • Place of Presentation: 京都大学数理解析研究所（ 京都府）
• [Remarks] 諸澤のホームページ
  • URL: http://www.math.kochi-u.ac.jp/morosawa/indexj.html
• [Remarks] Morosawa's Home Page
  • URL: http://www.math.kochi-u.ac.jp/morosawa/index.html
• [Remarks] Morosawa's Home Page
  • URL: http://www.math.kochi-u.ac.jp/morosawa/index.html