| 研究実績の概要 |
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.
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| 今後の研究の推進方策 |
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.
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