2003 Fiscal Year Final Research Report Summary
Research on Perron-Frobenius operator and fractals
| Project/Area Number |
14540189
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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 |
Basic analysis
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| Research Institution | Nihon University |
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Principal Investigator |
MORI Makoto Nihon University, College of Humanities and Sciences, Professor, 文理学部, 教授 (60092532)
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| Co-Investigator(Kenkyū-buntansha) |
SUZUKI Osamu Nihon University, College of Humanities and Sciences, Professor, 文理学部, 教授 (10096844)
YAMAURA Yoshihiko Nihon University, College of Humanities and Sciences, associate Professor, 文理学部, 助教授 (90255597)
FUKUDA Takuo Nihon University, College of Humanities and Sciences, Professor, 文理学部, 教授 (00009599)
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| Project Period (FY) |
2002 – 2003
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| Keywords | ergodic theory / dynamical system / Hausdorff dimension / spectrum / psudo random numbe |
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| Research Abstract |
Expanding the idea of van der Corput sequences, we construct random numbers using inverse images of piecewise linear transformations. The discrepancy of this random number has deep connection with the ergodic properties of the dynamical system generated by piecewise linear transformations. For example, it is proved that if the dynamical system is mixing, then the random number is uniformly distributed. Moreover, we can deeply study the ergodic properties of the dynamical system by The spectra of thePerron-Frobenius operator. In terms of the spectra, the random number is uniformly distributed if 1 is a simple eigenvalue., and no other eigenvalues on the unit circle. The second greatest eigenvalue of the Perron-Frobenius operator is at least the reciprocal of the slope of the transformation in modulus. We proved that the random number is of low discrepancy if the second greatest eigenvalue equals its minimum in modulus. We extend this idea to construct higher dimensional low discrepancy. sequences, and we succeeded to construct two and three dimensional low discrepancy sequences.
We also consider the Hausdorff dimensions from the view point of statistical mechanics, and proved it equals' a zero of the pressure. We also calculate the Hausdorff dimension of trees. The articles of these topics are now submitted.
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