2002 Fiscal Year Final Research Report Summary
Research on the dynamics of cubic polynomials (on the topological structure of the parameter space)
| Project/Area Number |
11640218
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | TOKYO INSTITUTE OF POLYTECHNICS |
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Principal Investigator |
NAKANE Shizuo TOKYO INSTITUTE OF POLYTECHNICS, 工学部, 教授 (50172359)
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| Project Period (FY) |
1999 – 2002
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| Keywords | stretching rays / parabolic implosion / Boettcher vector / Fatou vector / radial Julia set |
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| Research Abstract |
We have investigated the landing property of the stretching rays for the family of real cubic polynomials. Stretching rays are defined by a quasi-conformal deformation in the escape locus and are generalizations of external rays of the Mandelbrot set for quadratic polynomials. The Boettcher vector is an invariant on each stretching ray in the shift locus.
We have proved that the rays above the parabolic arc with non-integral Boettcher vectors do not land, i.e. they have non-trivial accumulation sets. The proof is done by applying the theory of parabolic implosion. In case of rational Boettcher vectors, we also use the notion of radial Julia sets. Those rays with integral Boettcher vectors land at points on the parabolic arc with the same Fatou vectors.
Below the parabolic arc, all the stretching rays land and we have characterized their landing points. The rays in the shift locus land at critically prefixed maps. In the proof for rays in the remaining locus, we have to use the theorem on density of hyperbolicity in the real quadratic polynomials.
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