2019 Fiscal Year Research-status Report
Thermodynamic formalism for non-compact spaces with applications in conformal dynamics
| Project/Area Number |
17K14203
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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) |
2017-04-01 – 2021-03-31
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| Keywords | Transient dynamics / Multifractal analysis / Spectral gap property |
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| Outline of Annual Research Achievements |
We established new formulae for mixed multi fractal Birkhoff spectra for non-uniformly expanding one-dimensional Markov maps and applications to cusp winding spectra for Fuchsian groups with parabolic elements. The results are available as preprint. This is joint work with Hiroki Takahasi (Keio University).
We completed a paper on spectral gap property of random hyperbolic dynamics on the real-line. This has applications to fractal functions such as the Takagi function. This paper has been accepted for publication in Communications in Mathematical Physics. This is joint work with Hiroki Sumi (Kyoto University).
We obtained partial results on the thermodynamic formalism for transient intervals maps by using random walks on the real line (with and without reflective boundaries).
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| Current Status of Research Progress |
Current Status of Research Progress
4: Progress in research has been delayed.
Reason
My transfer from Shimane University to Nagoya University has delayed my research in summer/autumn 2019. Further delay has been caused by the Coronavirus outbreak in 2020. Several research visits and conference attendances had to be cancelled or postponed.
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| Strategy for Future Research Activity |
We further develop the thermodynamic formalism for dynamical systems with transience. The goal is to cover the example studied recently by Bruin and Todd in J. London Math (2012). Moreover, we complete a project on amenability of graph extensions characterizing amenability in terms of the pressure function in thermodynamic formalism. We shall also consider the spectral gap property for nicely expanding rational semigroups.
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| Causes of Carryover |
My transfer from Shimane University to Nagoya University has delayed my research in summer/autumn 2019. Further delay has been caused by the Coronavirus outbreak in 2020. Several research visits and conference attendances had to be cancelled or postponed.
Plans for the next year include: research visit to University North Texas to intensify the research collaboration with Mariusz Urbanski. Conference attendance at Mittag-Leffler Institute in Sweden. Research visit to University Bremen (Germany). Moreover, to attend various conferences in Japan, such as Annual Meetings of the Mathematical Society Japan.
We also plan to buy a license for the technical computing system Mathematica.
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| Remarks |
Website of Johannes Jaerisch
http://www.math.nagoya-u.ac.jp/~jaerisch/
Information on my Research and Preprints.
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