Semigroups of continuous transformations and generating inverse limit sequences

George Edgar Parker

Vol. 80 (1979), No. 1, 227–235

DOI: 10.2140/pjm.1979.80.227

Abstract

Suppose that T denotes a strongly continuous semigroup of continuous transformations on a closed subset C of a complete metric space. For arbitrary decreasing sequences {δ_n}_n=1^∞ and {α_n}_n=1^∞ of positive numbers converging to 0, the inverse limit spaces generated by {T(δ_n)(C),T(δ_n − δ_n+1)}_n=1^∞ and {T(α_n)(C),T(α_n − α_n+1)}_n=1^∞ are homeomorphic and contain a dense one-to-one continuous image of C. Conversely, given an inverse limit system with bonding maps {f_n}_n=1^∞ so that (i) f_n : C → C, (ii) if x is in C, lim_n→∞f_n(x) = x, and (iii) f_n+1 ∘f_n+1 = f_n, conditions are given under which a semigroup, and consequently a family of homeomorphic inverse limits, can be recovered.

Examples are given which illustrate analytical applications and topological implications.

Mathematical Subject Classification 2000

Primary: 54H15

Secondary: 54B25

Milestones

Received: 4 November 1977

Revised: 11 April 1978

Published: 1 January 1979