| Project/Area Number |
14540214
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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 | Keio University |
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Principal Investigator |
NAKADA Hitoshi KEIO UNIVERSITY, Fac. of Sci. and Tech., Professor, 理工学部, 教授 (40118980)
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| Co-Investigator(Kenkyū-buntansha) |
ISHIKAWA Shiro KEIO UNIVERSITY, Fac. of Sci. and Tech., Associate Professor, 理工学部, 助教授 (10051913)
MAEJIMA Makoto KEIO UNIVERSITY, Fac. of Sci. and Tech., Professor, 理工学部, 教授 (90051846)
SHIOKAWA Iekata KEIO UNIVERSITY, Fac. of Sci. and Tech., Professor, 理工学部, 教授 (00015835)
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| Project Period (FY) |
2002 – 2003
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| Project Status |
Completed (Fiscal Year 2003)
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| Budget Amount *help |
¥2,900,000 (Direct Cost: ¥2,900,000)
Fiscal Year 2003: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 2002: ¥1,500,000 (Direct Cost: ¥1,500,000)
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| Keywords | Ergodic Theory / Invariant Measure / 連分数 |
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| Research Abstract |
The following three are the main results of this project:
An unite measure preserving transformation can be reduced to a finite measure preserving one as an induced transformation with a set of finite measure. In this case, the ceiling function represents the return time to this set and it is non-integrable. In this sense, we studied the stochastic processes with infinite expectations. In particular, we studied continued fraction mixing stochastic processes with barely infinite expectations. We have proved the strong law of large numbers after light trimming under some conditions on the distribution of random variables.
We also studied Maharum extension of ergodic nonsingular transformations. We discussed locally finite invariant measures for Maharam extensions associated to irrational rotations and subshifts. In the case of finite Markov subshifts, we characterized such measures by conformal measures for subshifts. We extend such results to countable Markov subshifts.
As an application of infinite ergodic theory, we studied the theory of arithmetic progressions in particular, concerning to Erods conjecture. In this point of view, we considered the multiple recurrence property of infinite measure preserving transformations. We gave an lower estimate of the multiplicity by Kakutani type. We also studied metric number theory as an application of ergodic theory. Some of main results are the following (1) We proved a property of the arithmetic distribution of convergents arising from the Jacobi-Perron algorithm. (2) We constructed Farey maps associated to Rosen's continued fractions. here the Farey maps are 1-dimensional maps which induces mediant convergents of continued fractions.
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