2018 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 | Shimane University |
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Principal Investigator |
イェーリッシュ ヨハネス 島根大学, 学術研究院理工学系, 講師 (90741869)
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| Project Period (FY) |
2017-04-01 – 2020-03-31
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| Keywords | Spectral gap property / Multifractal analysis / Transience |
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| Outline of Annual Research Achievements |
We established the spectral gap property for random iteration of certain diffeomorphisms on the real line which do not have a common fixed point. Using this, generalized Takagi functions have been introduced and analyzed using thermodynamic formalism (Preprint on Arxiv, Joint with H. Sumi).
The results motivated a refinement of a well-known result on local dimension spectra of measures invariant under a hyperbolic dynamical system. (Preprint on Arxiv, Joint with H. Sumi).
We also found that the thermodynamic formalism for Z-Extensions, which was developed in the first year of the funding period, gives new insights to a well-studied interval map with transient behavior introduced by van Strien, and studied recently by Bruin/Todd in J. London Math 2012 from viewpoint of thermodynamics.
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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
We had good progress on the spectral gap property for transfer operators associated with the random iteration of hyperbolic maps of the interval. This has naturally motivated another general result in multifractal Analysis, which provides a positive answer to a question by Pieter Allaart (Adv. Math. Differentiability and Hoelder spectra of a class of self-affine functions, 2018).
Regarding our formalism for Z extensions, it seems possible to extend our formalism so that it covers in particular an example studied recently by Bruin/Todd in J. London Math 2012.
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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/Todd in J. London Math 2012.
We aim to complete a project on amenability of graph extensions. The goal is to characterize amenability in terms of the pressure function in thermodynamic formalism.
We also consider the spectral gap property for nicely expanding rational semigroups.This can be used to derive analyticity of the pressure function and stochastic laws such as the central limit theorem.
Further, we work on multifractal analysis of birkhoff spectra in the non-uniformly hyperbolic setting.
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| Causes of Carryover |
The purchase of computer was delayed because the desired new model was not available.
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| Remarks |
Information on my Research and Preprints.
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Research Products
(8 results)
