The embedding of homeomorphisms of the plane in continuous flows

Gary Douglas Jones

Vol. 41 (1972), No. 2, 421–436

DOI: 10.2140/pjm.1972.41.421

Abstract

A study of fundamental regions of the plane under an orientation preserving, fixed point free, self-homeomorphism of the plane is made under the conditions that there are finitely many fundamental regions R_i under f, if x ∈ R_i − IntR_i, then x ∈ C ⊂ R_i − IntR_i where C is a proper flowline, and if x₁ and x₂ are in Int R_i, then x₁ ∼ x₂ mod Int R_i. The topological structure of the fundamental regions is determined. Using these results, it is shown that in certain cases the embedding problem can be reduced to a problem of extending a continuous flow defined on an open set to the closure of the set.

In the last section, sufficient conditions for self-homeomorphisms of the plane and the closed unit disc with one fixed point to be embedded in continuous flows are given.

Mathematical Subject Classification 2000

Primary: 54H20

Milestones

Received: 24 October 1969

Revised: 19 October 1971

Published: 1 May 1972