2022 Fiscal Year Research-status Report
Ergodic theory for conformal dynamics with applications to fractal geometry
| Project/Area Number |
21K03269
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| Research Institution | Nagoya University |
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Principal Investigator |
イェーリッシュ ヨハネス 名古屋大学, 多元数理科学研究科, 准教授 (90741869)
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| Project Period (FY) |
2021-04-01 – 2024-03-31
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| Keywords | Fuchsian groups / Hausdorff dimension / Large deviations |
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| Outline of Annual Research Achievements |
We studied the interplay of dynamical, geometric and stochastic properties of conformal dynamical systems from the viewpoint of ergodic theory. We focused mainly on the dynamics of Fuchsian groups admitting a Dirichlet fundamental domain with even corners. For the associated hyperbolic surface, we obtained new results on the Hausdorff dimension spectrum of homological growth rates associated with oriented geodesics. In particular, we are able to express the dimension in terms of a generalized Poincare exponent associated with a given inverse temperature.
As a stochastic counterpart we studied the probability to observe a certain homological growth rate. It turns out that the growth rate satisfies large deviations with a rate function closely related to the Hausdorff dimension spectrum. We have combined distortion arguments based on some geometric properties of the geodesic flow with the symbolic thermodynamic formalism for countable Markov shits. This is a joint work with Hiroki Takahasi (Keio U.).
We also had some progress on transient dynamics on the real line with a reflective boundary. In particular, we obtained formulas for the topological pressure function which indicate a possible phase transition which arises from the reflective boundary.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
Although some travel plans had to be cancelled because of covid-19 restrictions in summer, we had overall substantial progress on topics of this project.
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| Strategy for Future Research Activity |
The large deviation results for Fuchsian groups shall be completed within a few weeks. The project on transient dynamics on the real line with a reflective boundary shall be completed when visiting University Bremen in summer 2023. A first preprint on thermodynamic formalism for infinitely generated Schottky groups shall be completed in autumn 2023.
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| Causes of Carryover |
Again, in summer 2022, overseas travel had to be postponed because of the pandemic. We plan to use the remaining funds to visit Germany and USA in summer/autumn 2023.
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Research Products
(5 results)
