| 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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| 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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