Derivations on the line and flows along orbits

Charles James Keith Batty

Vol. 126 (1987), No. 2, 209–225

DOI: 10.2140/pjm.1987.126.209

Abstract

The closure of the derivation λD : C_c¹(ℝ) → C₀(ℝ) defined by (λD)(f) = λf′, where λ : ℝ → ℝ is continuous, generates a C₀-group on C₀(ℝ) (corresponding to a flow on ℝ) if and only if 1∕λ is not locally integrable on either side of any zero of λ or at ±∞.

If S is a flow on a locally compact, Hausdorff, space X with fixed point set X_S⁰, δ_S is the generator of the induced action on C₀(X), λ : X ∖X_S⁰ → ℝ is continuous, and bounded on sets of low frequency under S, and t → λ(S_tω)⁻¹ is not locally integrable on either side of any zero or at ±∞, then the flows along the orbits of S form a flow on X whose generator acts as λδ_S.

Mathematical Subject Classification 2000

Primary: 46L55

Secondary: 28D10, 47D99

Milestones

Received: 1 August 1985

Revised: 6 February 1986

Published: 1 February 1987