Extensions of minimal flows on manifolds

TLDR: In this paper, it was shown that a minimal vector field flow admits a minimal extension beyond a toral extension on compact manifolds and compact Lie groups, where the minimal sets form a partition into imbedded submanifolds.

Abstract: Group extensions of vector field flows for which the orbit closure map forms a fiber bundle are constructed for the case of minimal flows on compact manifolds and compact Lie groups. Conditions for which minimal nontoral extensions exist are studied. The existence question for minimal sets and minimal extensions of minimal sets has been studied extensively in topological dynamics. In [5], Ellis considers the group extension question for minimal homeomorphisms on compact metric spaces. For real flows on compact manifolds it is well known that the torus admits a minimal flow but the Klein bottle does not. In fact, Markley [7] shows that recurrent orbits on the Klein bottle are circles or points. It is also known that a minimal vector field flow admits a minimal extension through any torus [1, p. 58]. The purpose of this paper is to prove that a minimal vector field flow admits a minimal extension beyond a toral extension. It will follow that any Lie group admits a vector field flow which is minimal on the product of two distinct maximal tori and that any minimal vector field flow admits a minimal compact group extension in which the group is a semidirect product with a torus. It is also shown that a minimal vector field flow can always be extended so that the minimal sets form a partition into imbedded submanifolds and the projection onto the orbit closure spaces forms a fiber bundle. 0. Preliminaries. Definitions and proofs omitted here can be found in [1], [3] and [4]. All manifolds, maps, and transformation groups will be assumed C X unless otherwise specified. If (M, G) is right transformation group, the set {n * gln E N C M, g E G} will be denoted by N G and the space {n. GIn C N C M} with the quotient topology will be denoted by N/ G. If G is the real line and X is the vector field generating (M, G), we denote (M, G) by (M, X), N G by N X and N/ G by N/X. The symbols G N and G \ N will be used in case G acts on the left. If (G, P, M, v) is a principal G-bundle with base M, bundle space P and projection v; (H, P, H \ P, rl) will denote the induced principal H-bundle in which 1: P -H \ P takesp to H p. (H \ G, H \ P, M, v2) will denote the induced fiber bundle with standard fiber H \ G and projection r2: H \ P -M which takes H p to r (p). Note that 7 = o2 ? 7 1 For any mapf, Received by the editors September 28, 1975. AMS (MOS) subject classifications (1970). Primary 54H20, 34H35; Secondary 58F99.