On the homomorphic and isomorphic embeddings of a semiflow into a radial flow

M. Edelstein

Vol. 91 (1980), No. 2, 281–291

DOI: 10.2140/pjm.1980.91.281

Abstract

It is the main purpose of this paper to prove the following two theorems.

Theorem I. (Isomorphism) Let (X,R⁺,f) be a semiflow on a separable metric space (X,d), having the properties:

1. there is an ω ∈ X such that, for each neighborhood U of ω, there is a T ∈ R⁺ with f[X,t] ⊂ U for all t ≧ T;
2. for each t ∈ R⁺, f(⋅,t) is a homeomorphism of X onto a closed subspace of X.

Then (X,R⁺,f) is isomorphic to a radial semiflow on a subset of the Hilbert Cube in l².

Theorem II. (Homomorphism) If (X,R⁺,f) satisfies the hypotheses of Theorem I, with (i) replaced by

1. ∩{f[X,t] : t ≧ 0} = {ω} for some ω ∈ X, then (X,R⁺,f) is homomorphic to a radial semiflow on a subset of the Hilbert Cube C and the subsemiflow induced on X∕{ω} is isomorphic to a radial semiflow in C.

Mathematical Subject Classification 2000

Primary: 54H20

Milestones

Received: 27 August 1979

Published: 1 December 1980