2018 Fiscal Year Final Research Report
Combinatorics study of zero-dimensional systems - beyond Bratteli-Vershik systems
| Project/Area Number |
16K05185
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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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| Research Field |
Basic analysis
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| Research Institution | Nagoya Keizai University |
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Principal Investigator |
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| Project Period (FY) |
2016-04-01 – 2019-03-31
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| Keywords | Bratteli-Vershik system / zero-dimensional system / subshift / minimal / topological rank / residually scrambled |
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| Outline of Final Research Achievements |
My plan was to extend the case study of minimal homeomorphic zero-dimensional systems, and to construct chaotic cases.I could extend the notion of topological rank, showing that the natural extensions have no greater ranks. I could characterize substitution maps that create minimal subshifts. Rank 2 proximal Cantor systems were residually scrambled.
Apart from minimality, finite rank Bratteli-Vershik systems were expansive if they do not have odometers. More importantly, all homeomorphic zero-dimensional systems had non-trivial Bratteli-Vershik representation, including the basic sets.
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| Free Research Field |
力学系，組み合わせ論的零次元系
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| Academic Significance and Societal Importance of the Research Achievements |
組み合わせ論的零次元系の理論は，C*環のある程度計算可能な具体的議論として，とても重要なものです．何故なら，C*環の理論は，量子場の理論と深く関連しており，その量子場の理論はもはや研究の枠を越え，量子コンピュータの実現という，極めて重要な社会的目標を持つものだからです．
もっとも，私の研究成果は，基礎的なものに留まります．Bratteli--Vershik表現というものは，位相的ベルヌーイ系を含む，とてつもなく広大な零次元系達の class を，上記C*環の計算可能な構造と深く関連させることができるものです．私の研究は，このBratteli--Vershik表現についての，基礎的な考察です．
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