Clarification on topological models for ergodic transformation groups having infinite invariant measures

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K04900
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Osaka Kyoiku University
• Principal Investigator: YUASA HISATOSHI 大阪教育大学, 教育学部, 准教授 (50363346)
• Research Collaborator: Hama Masaki
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: 保測変換 / 無限ルベーグ空間 / 狭義エルゴード性 / 代入 / S-進性

Outline of Final Research Achievements

I could prove the following three facts. Any factor map between ergodic measure-preserving transformations on infinite Lebesgue spaces is realized up to a measure theoretical isomorphism as a topological semi-conjugacy between strictly ergodic, locally compact Cantor systems. In the category of ergodic measure-preserving systems on infinite Lebesgue spaces, any diagram without any extension of two common transformations is realized up to a measure theoretical isomorphism as that of strictly ergodic, locally compact Cantor systems. The S-adic property is possessed by Bratteli-Vershik representations associated with almost minimal subshifts arising from non-primitive substitutions with a certain property between the maximal and second maximal eigenvalues of their incidence matrices.

Free Research Field

エルゴード理論

Academic Significance and Societal Importance of the Research Achievements

狭義エルゴード的同型表現定理は，分野の歴史と照らし合わせると，大変に重要である．事実，確率保測エルゴード変換に対する同型表現定理はW.Kriegerによって一般的な形で解決され，エルゴード理論の教科書にはよく掲載されている．一方で，殆ど原始的代入に付随する殆ど極小サブシフトのS-進性については，全く未開の領域であり，さらには非定常性を呈することが解明できたことは，原始的代入に付随するサブシフトにはなかった新しい現象である．