Construction and application of a theory of dynamical systems on complex surfaces

• Project/Area Number: 24740096
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Global analysis
• Research Institution: Saga University (2014-2015); Niigata University (2012-2013)
• Principal Investigator: Uehara Takato 佐賀大学, 工学(系)研究科(研究院), 准教授 (40613261)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2012: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 複素曲面 / 力学系 / 有理曲面 / エントロピー / ディターミナント / ジーゲル領域 / ジーゲル円板 / 自己同型写像

Outline of Final Research Achievements

We study dynamical systems of biholomorphic automorphisms on rational surfaces with positive topological entropy. We determine explicitly the determinant values, which are defined for automorphisms preserving anticanonical curves, and show the existence of an abundance of rational surface automorphisms with positive entropy. Moreover, we show the existence of automorphisms on complex surfaces with positive entropy having a given number of Siegel domains, which are observed around fixed points.

Research Products

• [Journal Article] Rational Surface Automorphisms with Positive Entropy (2016)
  • Author(s): Takato Uehara
  • Journal Title: Ann. Inst. Fourier Volume: 65 Issue: 1 Pages: 377-432
  • DOI: 10.5802/aif.3014
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Presentation] ジーゲル領域をもつ有理曲面上の自己同型写像 (2016)
  • Author(s): Takato Uehara
  • Organizer: アクセサリー・パラメーター研究会
  • Place of Presentation: 熊本大学, 熊本
  • Year and Date: 2016-03-24
  • Invited
• [Presentation] On the topological entropies for automorphisms on rational surfaces (2016)
  • Author(s): Takato Uehara
  • Organizer: 複素領域の微分方程式、漸近解析とその周辺
  • Place of Presentation: 広島大学, 東広島
  • Year and Date: 2016-03-09
  • Invited
• [Presentation] Siegel domains for rational surface automorphisms with positive entropy (2015)
  • Author(s): Takato Uehara
  • Organizer: RIMS Workshop on Complex Dynamics
  • Place of Presentation: 京都大学数理解析研究所, 京都
  • Year and Date: 2015-12-11
  • Int'l Joint Research / Invited
• [Presentation] 有理曲面上の自己同型写像とエントロピーについて (2015)
  • Author(s): Takato Uehara
  • Organizer: 大域解析セミナー
  • Place of Presentation: 熊本大学, 熊本
  • Year and Date: 2015-12-03
  • Invited
• [Presentation] Entropy values of automorphisms on rational surfaces (2015)
  • Author(s): Takato Uehara
  • Organizer: Mini-workshop on moduli spaces and its related topics
  • Place of Presentation: KIAS, Seoul
  • Year and Date: 2015-05-13
  • Int'l Joint Research / Invited
• [Presentation] Isolated periodic points for area-preserving surface mappings (2015)
  • Author(s): Takato Uehara
  • Organizer: Differential and Complex Geometry Seminar
  • Place of Presentation: KIAS, Seoul
  • Year and Date: 2015-03-30
  • Invited
• [Presentation] Rational Surface Automorphisms with Siegel Disks (2014)
  • Author(s): Takato Uehara
  • 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, 京都
  • Year and Date: 2014-12-10
  • Invited
• [Presentation] 有理曲面上の自己同型写像のエントロピー (2014)
  • Author(s): Takato Uehara
  • Organizer: 九州大学組み合せ数学セミナー
  • Place of Presentation: 福岡工業大学サテライト福岡天神，福岡
  • Year and Date: 2014-10-18
  • Invited
• [Presentation] On rational surface automorphisms with positive entropy (2014)
  • Author(s): Takato Uehara
  • Organizer: Symmetries of Kahler manifolds, dynamics and moduli spaces
  • Place of Presentation: SISSA, Trieste
  • Year and Date: 2014-09-22
  • Invited
• [Presentation] 尖点反標準曲線を保つ自己同型写像について (2014)
  • Author(s): Takato Uehara
  • Organizer: 代数学セミナー
  • Place of Presentation: 広島大学，広島
  • Year and Date: 2014-05-23
  • Invited
• [Presentation] 有理曲面上の自己同型写像のエントロピー
  • Author(s): Takato Uehara
  • Organizer: 複素解析幾何セミナー
  • Place of Presentation: 東京大学，東京
  • Invited
• [Presentation] On rational surface automorphisms
  • Author(s): Takato Uehara
  • Organizer: 射影多様体の幾何とその周辺2013
  • Place of Presentation: 高知大学，高知
  • Invited
• [Presentation] On automorphism groups of rational surfaces
  • Author(s): Takato Uehara
  • Organizer: 第三回若手代数複素幾何研究集会
  • Place of Presentation: 松藤プラザ，長崎
  • Invited
• [Presentation] On rational surface automorphisms preserving cuspidal anticanonical curves
  • Author(s): Takato Uehara
  • Organizer: Moduli spaces and self-maps
  • Place of Presentation: Kyoto University, Kyoto
  • Invited
• [Presentation] Constructing automorphisms of rational surfaces
  • Author(s): Takato Uehara
  • Organizer: Interactions Between Continuous and Discrete Holomorphic Dynamical Systems
  • Place of Presentation: Banff International Research Station, Banff
• [Presentation] On automorphisms of rational surfaces
  • Author(s): Takato Uehara
  • Organizer: Korea-Japan Joint Conference in Algebraic Geometry
  • Place of Presentation: Ritz-Plaza Hotel, Gunsan
• [Presentation] Construction of automorphisms on rational surfaces
  • Author(s): Takato Uehara
  • Organizer: 第10 回アフィン代数幾何学研究集会
  • Place of Presentation: 関西学院大学, 大阪
• [Presentation] The entropy values of automorphisms on rational surfaces
  • Author(s): Takato Uehara
  • Organizer: Various Aspects on the Painleve Equations
  • Place of Presentation: RIMS Kyoto University, Kyoto
• [Presentation] Siegel disks on rational surfaces
  • Author(s): Takato Uehara
  • Organizer: New Developments in Complex Dynamical Systems
  • Place of Presentation: RIMS Kyoto University, Kyoto