2017 Fiscal Year Final Research Report
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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| Keywords | 力学系 / カオス / 分岐 / フラクタル |
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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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| Free Research Field |
力学系理論
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