2015 Fiscal Year Annual Research Report
Thermodynamic formalism for conformal semigroup actions
| Project/Area Number |
15H06416
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| Research Institution | Shimane University |
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Principal Investigator |
イェーリッシュ ヨハネス 島根大学, 総合理工学研究科(研究院), 講師 (90741869)
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| Project Period (FY) |
2015-08-28 – 2017-03-31
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| Keywords | エルゴード理論 |
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| Outline of Annual Research Achievements |
Several results have been obtained which are stronger and in greater generality than expected.
The revision of the paper Dynamics of infinitely generated nicely expanding rational semigroups and the inducing method (with H. Sumi) has been completed and the paper has been accepted for publication in the Transactions of the AMS, which contributes to establishing the thermodynamic formalism for expanding rational semigroups.
Further main results: On the regularity of complex Takagi functions and random complex dynamical systems (preprint with H. Sumi), and on critical exponents of normal subgroups of free groups (joint with K.Matsuzaki, preprint submitted). Also, on critical exponents of normal subgroups of Gromov hyperbolic groups (preprint with K.Matsuzaki and Y. Yabuki).
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| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
Several results have been obtained which are stronger and in greater generality than expected (e.g. Hoelder regularity
of higher-order complex Takagi functions.)
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| Strategy for Future Research Activity |
(A) Critical exponents of subgroups of Kleinian groups and Gromov hyperbolic groups. We investigate how this exponent
changes when passing to a subgroup.To study this problem, we consider skew product extensions of topological Markov chains by infinite graphs. We study the topological pressure in for these dynamical systems, which has applications to critical exponents of subgroups of Kleinian groups. Further tasks are to investigate the multifractal formalism for cusp windings, and the Laplacian for Cayley graphs.
(B) Dynamics of semigroups of rational maps on the Riemann sphere and random compelex dynamics. We consider Takagi
functions for expanding interval maps satisfying the open set condition, and seek to generalise previous results on the Riemann sphere.
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| Remarks |
My recent preprints and publications are available on the above webpage.
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