Research on parameter spaces in complex dynamics

• Project/Area Number: 22740105
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Global analysis
• Research Institution: Kyoto University
• Principal Investigator: INOU Hiroyuki 京都大学, 理学(系)研究科(研究院), 講師 (00362434)
• Project Period (FY): 2010-04-01 – 2014-03-31
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000); Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2010: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 複素力学系 / Mandelbrot集合 / tricorn / straightening map / くりこみ / フラクタル

Research Abstract

Computer pictures has been playing an important role in studying complicated phoenomena in complex dynamics. Recent development in computer envirionment and virtual reality technology enabled us to immerse into 3D virtual reality space. We visualize objects in complex 2D (real 4D) space with help of such a technology.
We proved that homeomorphic copies of Mandelbrot set and so on are naturally embedded into 1-parameter spaces of cubic polynomials with a periodic critical points.
We proved that the tricorn, an analogue of the Mandelbrot set in the anti-holomorphic quadratic family does not have the self-similar property which the Mandelbrot set has.

Research Products

• [Journal Article] Combinatorics and topology of straightening maps I : Compactness and bijectivity (2012)
  • Author(s): Hiroyuki Inou
  • Journal Title: Advances in Mathematics Volume: 231 Issue: 5 Pages: 2666-2733
  • DOI: 10.1016/j.aim.2012.07.014
  • Peer Reviewed
• [Journal Article] Extending local analytic conjugacies (2011)
  • Author(s): Inou, Hiroyuki
  • Journal Title: Trans.Amer.Math.Soc. Volume: 363 Issue: 01 Pages: 331-343
  • DOI: 10.1090/s0002-9947-2010-05049-4
  • Peer Reviewed
• [Journal Article] Holomorphic index and domains and ranges of parabolic renormalizations (2011)
  • Author(s): 稲生啓行
  • Journal Title: 数理解析研究所講究録1762「複素力学系とその周辺分野の研究」 Pages: 39-45
  • URL: http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/1762.html
• [Journal Article] Holomorphic index and domains and ranges of parabolic renormalizations (2011)
  • Author(s): 稲生啓行
  • Journal Title: 数理解析研究所講究録「複素力学系とその周辺分野の研究」 Volume: 1,762 Pages: 39-45
• [Journal Article] Extending local holomorphic conjugacies (2010)
  • Author(s): 稲生啓行
  • Journal Title: 数理解析研究所講究録1699「複素力学系とその関連分野の総合的研究」 Pages: 62-67
  • URL: http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/1699.html
• [Journal Article] Extending local holomorphic conjugacies (2010)
  • Author(s): 稲生啓行
  • Journal Title: 数理解析研究所講究録 Volume: 1699 Pages: 62-67
• [Presentation] A baby tricorn is not homeomorphic to the tricorn (2013)
  • Author(s): Hiroyuki Inou
  • Organizer: 第21回有限無限次元複素解析国際会議
  • Place of Presentation: 南京大学
  • Year and Date: 2013-06-16
• [Presentation] On self-similarity of the tricorn (2013)
  • Author(s): Hiroyuki Inou
  • Organizer: Mathematics Colloquium
  • Place of Presentation: Jacobs University Bremen
  • Year and Date: 2013-02-18
• [Presentation] Tricorn の(非)自己相似性について (2013)
  • Author(s): 稲生 啓行
  • Organizer: 2012年度冬の力学系研究集会
  • Place of Presentation: 日本大学軽井沢研修所
  • Year and Date: 2013-01-12
• [Presentation] Are baby tricorns homeomorphic to the tricorn? (2012)
  • Author(s): 稲生啓行
  • Organizer: 複素力学系の新展開
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2012-12-12
• [Presentation] On domains and ranges of near-parabolic renormalizations (2012)
  • Author(s): Hiroyuki Inou
  • Organizer: Workshop on Non-uniformly Hyperbolic and Neutral One-dimensional Dynamics
  • Place of Presentation: National University of Singapore
  • Year and Date: 2012-04-24
• [Presentation] 複素力学系のパラメータ空間における相似性について (2012)
  • Author(s): 稲生 啓行
  • Organizer: 東北大学大学院理学研究科数学専攻談話会
  • Place of Presentation: 東北大学大学院理学研究科
  • Year and Date: 2012-04-16
• [Presentation] "Cut-open" rabbit in parameter spaces (2012)
  • Author(s): 稲生啓行
  • 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
  • Year and Date: 2012-01-26
• [Presentation] Parameter spaces in complex dynamics (2011)
  • Author(s): Hiroyuki Inou
  • Organizer: Doctorial Forum of Mathematics between Fudan and Kyoto Universities
  • Place of Presentation: Fudan University
  • Year and Date: 2011-11-04
• [Presentation] Parameter spaces in complex dynamics (2011)
  • Author(s): Hiroyuki Inou
  • Organizer: Doctorial Forum of Mathematics between Fudan and Kyoto Universities
  • Place of Presentation: Fudan university (China)(招待講演)
  • Year and Date: 2011-11-04
• [Presentation] Bifurcation locus of the cubic polynomial family (2011)
  • Author(s): Hiroyuki Inou
  • Organizer: Frontiers in Complex Dynamics
  • Place of Presentation: カナダ、バンフ国際研究センター(招待講演)
  • Year and Date: 2011-02-24
• [Presentation] Holomorphic index and parabolic renormalization (2010)
  • Author(s): 稲生啓行
  • Organizer: 複素力学系とその周辺分野の研究
  • Place of Presentation: 京都大学大学院理学研究科
  • Year and Date: 2010-12-07
• [Presentation] Straightening maps (2010)
  • Author(s): Hiroyuki Inou
  • Organizer: Workshop on Holomorphic Dynamics around Thurston's Theorem
  • Place of Presentation: デンマーク、ロスキレ大学
  • Year and Date: 2010-10-01
• [Presentation] On a generalization of the self-similar property of the Mandelbrot set (2010)
  • Author(s): Hiroyuki Inou
  • Organizer: ICM Satellite Conference on Various Aspects of Dynamical Systems
  • Place of Presentation: インド、Maharaja Sayajirao University of Baroda(招待講演)
  • Year and Date: 2010-08-29
• [Presentation] Straightening maps (2010)
  • Author(s): Hiroyuki Inou
  • Organizer: Bicentennial Workshop on Dynamical Systems
  • Place of Presentation: チリ, 北カトリック大学考古学博物館
  • Year and Date: 2010-06-01
• [Presentation] Straightening maps (2010)
  • Author(s): Hiroyuki Inou
  • Organizer: Bicentennial Workshop on Dynamical Systems
  • Place of Presentation: チリ、北カトリック大学考古学博物館
  • Year and Date: 2010-06-01
• [Presentation] Are baby tricorns homeomorphic to the tricorn?
  • Author(s): 稲生 啓行
  • Organizer: 複素力学系の新展開
  • Place of Presentation: 京都大学数理解析研究所
• [Remarks]
  • URL: http://www.math.kyoto-u.ac.jp/~inou/