The structure of chaotic regions for area-preserving maps
| Project/Area Number |
15K13444
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| Research Category |
Grant-in-Aid for Challenging Exploratory Research
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| Allocation Type | Multi-year Fund |
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| Research Field |
Basic analysis
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| Research Institution | Kyoto University |
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Principal Investigator |
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| Project Period (FY) |
2015-04-01 – 2018-03-31
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| Project Status |
Completed (Fiscal Year 2017)
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| Budget Amount *help |
¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
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| Keywords | 力学系 / カオス / 分岐 / フラクタル / くりこみ / 面積保存 |
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| Outline of Final Research Achievements |
As a first step toward the study of area preserving 2-dimensional maps, jointly with D. Marti Pete, we studied 1-dimensional standard maps (Arnold family) z-> z+α+β sin(2πz) with complex phase variable z and complex parameters α, β. When real positive β is fixed and α is taken as the main parameter, finger-like structures have been observed near rational α in the parameter space. We used the theory of parabolic bifurcation to analyze this structure. The translation parameter in the Fatou coordinate is a new control parameter in which finger-like structure become a simple horizontal strip structure and this explains the fingers.
We also studied the family of real quadratic polynomials and introduced a new approach using Yoccoz puzzles to show that chaotic parameters have positive measure（Jakobson's theorem). This method together with complex analytic method to estimate the modulus of certain annuli gave a quantitative estimate of the parameter measure.
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Report
(4 results)
Research Products
(17 results)
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[Presentation] Fractal nature of Julia sets2016
Author(s)
Mitsuhiro Shishikura
Organizer
First Hong-Kong/Kyoto Workshop on Fractal and related areas
Place of Presentation
The Hong Kong University od Science and Technology, Hong, Kong, China
Year and Date
2016-03-21
Related Report
Int'l Joint Research / Invited
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[Presentation] Tropical Complex Dynamics2016
Author(s)
宍倉光広
Organizer
Mexican-American Workshop in Holomorphic Dynamics
Place of Presentation
Fiesta Americana Condesa hotel in Cancun, Mexico
Related Report
Int'l Joint Research / Invited
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