2009 Fiscal Year Final Research Report
Tropical geometry and integrable systems
| Project/Area Number |
19740086
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
Global analysis
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| Research Institution | Osaka University |
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Principal Investigator |
NOBE Atsushi Osaka University, 教育学部, 准教授 (80397728)
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| Project Period (FY) |
2007 – 2009
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| Keywords | 解析学 / 関数方程式論 / 数理物理学 / トロピカル幾何 / 可積分系 |
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| Research Abstract |
We construct an 8-parameter family of two-dimensional bi-piecewise linear maps in terms of the addition of the points on tropical elliptic curves, and obtain the general solution for each member of the family. Through the ultradiscretization procedure, we associate the family with an 18-parameter family of two-dimensional birational maps called the QRT map including its general solutions. Similarly, applying such technique to solvable chaotic maps induced from the duplication of points on tropical elliptic curves, we obtain their general solution and clarify the correspondence to the rational maps induced from the duplication of points on elliptic curves. Moreover, we show that there exists a family of cellular automata each of which has a property called the linearizability. Then we obtain a formula concerning the fundamental period with respect to the time evolution of the family imposing periodic boundary conditions.
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