2021 Fiscal Year Research-status Report
Ergodic theory for conformal dynamics with applications to fractal geometry
| Project/Area Number |
21K03269
|
|---|
| Research Institution | Nagoya University |
|---|
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)
|
|---|
| Project Period (FY) |
2021-04-01 – 2024-03-31
|
| Keywords | Ergodic theory / Fractal geometry / Multifractal analysis / Non-uniformly hyperbolic |
|---|
| Outline of Annual Research Achievements |
We developed the ergodic theory and its applications to fractal geometry for conformal dynamical systems which are non-uniformly hyperbolic or whose state space is not compact.
We completed a project on the thermodynamic formalism for transient dynamics on the real line (joint with Marc Kesseboehmer, University Bremen, Germany, and Maik Groeger University Krakov, Poland). The results are published in Nonlinearity. In particular, we investigated the geometric pressure function for a class of Markov maps not satisfying the finite irreducibility condition of Mauldin and Urbanski. We established various dimensional results (e.g., Hausdorff dimension, hyperbolic dimension) of subsets of associated limit sets. We proved criteria for dimension gaps and estimates of the gap size have been established.
We established new results on the multifractal analysis of homological growth rates for hyperbolic surfaces uniformized by finitely generated Fuchsian groups with even corners. Since these groups may have parabolic elements, the associated Bowen-Series map is non-uniformly expanding. This is a joint work with Hiroki Takahasi, Keio University. The preprint is available on the arxiv.
|
| 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, we had overall substantial progress on topics of this project.
|
| Strategy for Future Research Activity |
The results on transient dynamics on the real line shall be extended to transient dynamics on the half-line with a reflective boundary. Moreover, we shall consider infinitely branched interval maps and / or interval maps with parabolic fixed points.
Another aim is to study various multifractal spectra associated with the geodesic flow on hyperbolic surfaces. In particular, the degeneracy of spectra which appeared for backward-continued fraction expansions should be further investigated.
We plan to work also on pseudo-Markov systems which are applicable to some infinitely generated Schottky groups. For these groups we aim to study the orbital counting function by spectral methods.
|
| Causes of Carryover |
Because of covid-19 restrictions and concerns, all overseas travels have been cancelled. We expect that the pandemic situation will get better in the next year so that overseas travel will be possible again. The grant amount not used for travel in the fiscal year 2021 shall be used in the fiscal year 2022.
|
Research Products
(11 results)
