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On v ‐Distal Flows on 3‐Manifolds
Author(s) -
Matsumoto S.,
Nakayama H.
Publication year - 1997
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609397003081
Subject(s) - mathematics , invertible matrix , abelian group , pure mathematics , bounded function , flow (mathematics) , manifold (fluid mechanics) , parametrization (atmospheric modeling) , action (physics) , space (punctuation) , combinatorics , mathematical analysis , geometry , physics , mechanical engineering , linguistics , philosophy , quantum mechanics , engineering , radiative transfer
In [ 2 ], H. Furstenberg studied a distal action of a locally compact group G on a compact metric space X , and established a structure theorem. As a consequence, he showed that if G is abelian, then a simply connected space X does not admit a minimal distal G ‐action. In this paper we concern ourselves with a nonsingular flow φ = {φ t } on a closed 3‐manifold M . Recall that φ is called distal if for any distinct two points x , y ∈ M , the distance d (φ t x , φ t y ) is bounded away from 0. The distality depends strongly upon the time parametrization. For example, there exists a time parametrization of a linear irrational flow on T 2 which yields a nondistal flow [ 4, 6 ]. 1991 Mathematics Subject Classification 58F25, 57R30.