2007 Fiscal Year Final Research Report Summary
Aspects of Mathematics on Fractals
| Project/Area Number |
17340026
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Kyoto University |
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Principal Investigator |
KIGAMI Jun Kyoto University, Graduate School of Informatics, Professor (90202035)
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| Co-Investigator(Kenkyū-buntansha) |
SHISHIKURA Mitsuhiro Kyoto University, Department of Mathematics, Faculty of Science, Professor (70192606)
KUMAGAI Takashi Kyoto University, Department of Mathematics, Faculty of Science, Professor (90234509)
ITO Shunji Kanazawa University, Guraduate School of Natural Science & Thehnology, Professor (30055321)
AIKAWA Hiroaki Hokkaido University, Department of Mathematics, Professor (20137889)
OSADA Hirofumi Kyushi University, Faculty of Mathematics, Professor (20177207)
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| Project Period (FY) |
2005 – 2007
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| Keywords | Fractal / heat kerenl / self-similar set / percolation / harmonic function / energy measure / tiling |
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| Research Abstract |
We have obtained the following two results on relations between analysis and geometry on fractals. The first result concerns a heat kernel associated with a time change of a process associated with a self-similar Dirichlet form on a self-similar set. We show an sufficient and necessary condition on the existence of a distance under which the heat kernel satisfy the on-diagonal Li-Yau type sub-Gaussian estimate. The second result is on the measurable Riemannian Geometry on the Sierpinski gasket. We show that the geodesic distance coincides with the shortest path metric on the harrmonic Sierpinski gasket and establilsh Li-Yau type Gaussian estimate of the heat kernel under the geodesic metric.
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