Shear distality and equicontinuity

Dennis F. De Riggi and Nelson Groh Markley

Vol. 70 (1977), No. 2, 337–345

DOI: 10.2140/pjm.1977.70.337

Abstract

Let X be a compact Hausdorff space, let R^p be p-dimensional Euclidean space, and let (X,R^p) be a minimal transformation group. It may happen that xH will always contain flow lines in at least one direction in R^p for any discretr syndetic subgroup H no matter how sparce. We interpret this phenomena as some intrinsic shearing motion in the minimal transformation group. This is quantified in Section 1 and it turns out that equicontinuous minimal sets have as little shear as possible. Since distality is also a rigidity condition, it is natural to investigate the shear of a distal minimal set. We show by example in Section 2 that distal minimal sets can contain more shear than equicontinuous ones.

In Section 3 we show how the topology of xH is locally determined by local sections and subspaces of R^ρ. Using this result we prove in Section 4 that a distal minimal action of Rⁿ⁻¹ with trivial isotropy on a compact n-dimensional manifold is equicontinuous.

This paper contains portions of the first author’s dissertation [1] and generalizations of some results in an unpublished preprint [6] by the second author.

Mathematical Subject Classification 2000

Primary: 54H20

Milestones

Received: 14 September 1976

Revised: 31 January 1977

Published: 1 June 1977