| Co-Investigator(Kenkyū-buntansha) |
MUTO Hideo Yamanashi University, Associate professor, 教育人間科学部, 助教授 (20143646)
NAKAI Yoshiobu Yamanashi University, professor, 教育人間科学部, 教授 (40022652)
SUZUKI Toshio Yamanashi University, professor, 教育人間科学部, 教授 (20020472)
KUBO Izumu Graduate school of Science, Hiroshima University, Professor, 大学院・理学研究科, 教授 (70022621)
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| Research Abstract |
In this research, with the aim to establish the foundation of the theory of dimensions and measures of fractals such as Cantor sets, we tried to apply the probabilistic methods to their analysis. Concretely obtained results are the following.
Firstly, on statistically self-similar Cantor sets, we developed mathematically rigorous theory of the Hausdorff dimension, packing dimension and the measures associated with these dimensions. Here we utilized the statistical mechanical notions of the thermodynamic formalism such as transfer operators or Gibbs measures, and characterized the dimensions through them. Namely, on statistically perturbed Cantor sets, we showed that Hausdorff dimensions and packing dimensions are both equal to the zeroes of the pressures defined by the logarithmic potential of the derivatives of generating maps on symbolic dynamics.
Secondly, in deterministic cases we analyzed the perturbed Cantor sets. We constructed examples of Cantor sets which Hausdorff dimensions do not coincide with packing ones. We also showed that the dimensions of measures are equal to the entropy of the corresponding symbolic dynamics divided by the Lyapunov exponents with respect to the considered measures. Further-more we treated self-similar sets such that the derivatives of generating maps are non-Holder continuous, and showed that a dimension formula of the pressure holds.
Associated with the above, we studied the spectra of the transfer operators, or as their extension, the quasi-transfer operators, which play a crucial role in the analysis of self-similar sets. Here we treated the operators whose potentials do not satisfy the condition of the Holder continuity. Defining the pressure of the operators appropriately, we proved that the spectral sets of the opearators are equal to the disks in the complex plane of which radii are equal to their pressures.
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