2022 Fiscal Year Final Research Report
Higher-order asymptotic analysis of nonconformal iterative function systems with infinite graphs by asymptotic theory construction of transfer operators
| Project/Area Number |
20K03636
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 12010:Basic analysis-related
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| Research Institution | Wakayama Medical University |
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Principal Investigator |
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| Project Period (FY) |
2020-04-01 – 2023-03-31
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| Keywords | 転送作用素 / 漸近摂動 / 反復関数系 / 漸近分散 / 擬コンパクト |
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| Outline of Final Research Achievements |
If a transfer operator is asymptotically perturbed, then we give the asymptotic expansions of the eigenvalues, of the corresponding eigenfunctions, and of the corresponding eigenvectors of the dual operator. In particular, by a method of recursively giving the coefficients of the asymptotic expansion, it possible to make the uniform spectral gap condition of the eigenvalues unnecessary or weak. As an application, we give a high-order asymptotic expansion for the Hausdorff dimension of the limit set of non-conformal iterated function systems with an infinite directed graph. As another application, we obtained some splitting phenomena of Gibbs measures for open-type perturbed Markov systems with countable states.
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| Free Research Field |
数学
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| Academic Significance and Societal Importance of the Research Achievements |
関数を漸近展開することにより，近似値を得ることができるという考え方は，線形作用素の固有値および固有ベクトルにも適用することができる．また，漸近展開を行うためだけであれば，可微分性の条件を緩めることもできる．本研究では，これらを転送作用素の中で定式化し，反復関数系から生成される極限集合のHausdorff次元の近似値を求めることで，その有用性を実証した．今後はランダム化や非自励系版などにも適用し，漸近理論の可能性をさらに広げていくことが期待される．
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