| Project/Area Number |
12640120
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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 |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Nara Women's University |
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Principal Investigator |
TAKAHASHI Satoshi Nara Women's University, Graduate School of Humanities and Sciences, Associate Professor, 大学院・人間文化研究科, 助教授 (70226835)
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| Co-Investigator(Kenkyū-buntansha) |
FUJIWARA Akio Osaka University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (30251359)
SUZUKI Joe Osaka University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (50216397)
KAMO Hiroyasu Nara Women's University, Faculty of Science, Assistant, 理学部, 助手 (20243355)
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| Project Period (FY) |
2000 – 2003
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| Project Status |
Completed (Fiscal Year 2003)
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| Budget Amount *help |
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2003: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2002: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2001: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2000: ¥1,200,000 (Direct Cost: ¥1,200,000)
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| Keywords | multifractal / dimension spectrum / self-affine set / 自己ファイン集合 / 特異性スペクトル / ルジャンドル変換 / ソフィック測度 |
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| Research Abstract |
Studies of multfractals so far have dealt with functions of single variables, singularity spectrum of single singularity, or free energy of single inverse tempreture parameter. However invariant sets of dynamical systems in higher dimensions have different expansion or contraction rate depending on the direction.
We considered invariant measures on self-affine sets which have different contraction rates depending on the direction, and define the singularity spectrum with two different singularities, as well as free energy with two different inverse temperatures. Under certain asumptions on the measure, the singularity spectrum with two singularities coincides with the Legendre transform of the free energy with two inverse temperatures.
We also defined dimension spectrum with two dimensions as parameter, and the corresponding free energy, for 3-dimensional self-affine sets. Applying multivariable multifractal formula for the projection of the natural measure on the self-affine sets, we have shown that the dimension spectrum with two dimension parameters coincides with the modified Legendre transform of the corresponding free energy. This result also impies that the Hausdorff dimension of 3-dimensional self-affine sets is represented by the free energy with its inverse temperatures being the ratios of the logarithmic contraction rates.
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