Vol. 80, No. 1, 1979

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Semigroups of continuous transformations and generating inverse limit sequences

George Edgar Parker

Vol. 80 (1979), No. 1, 227–235

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 {δn}n=1 and {αn}n=1 of positive numbers converging to 0, the inverse limit spaces generated by {T(δn)(C),T(δn δn+1)}n=1 and {T(αn)(C),T(αn αn+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 {fn}n=1 so that (i) fn : C C, (ii) if x is in C, limn→∞fn(x) = x, and (iii) fn+1 fn+1 = fn, 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
Received: 4 November 1977
Revised: 11 April 1978
Published: 1 January 1979
George Edgar Parker