A characterization of the minimal invariant sets of Alspach's mapping
If \(C\) is a nonempty, weakly compact, convex subset of a Banach space and \(T:C\to C\) is a nonexpansive mapping without a fixed point, it is well known that Zorn's Lemma ensures the existence of a \textit{minimal} nonempty, weakly compact, convex subset \(M\) of \(C\) such that \(T(M)\subset M\). Although the existence of minimal \(T\)-invariant sets is frequently used in proofs that particular classes of weakly compact, convex sets have the fixed point property, until now, no one had given an example of a minimal \(T\)-invariant set that could be explicitly described. In the article under review, the authors consider Alspach's example of a fixed point free, nonexpansive self-map \(T\) of the weakly compact, convex set \(C = \{f\in L^{1}[0,1]: 0\leq f\leq 1, \int f = 1/2\}\) [see \textit{D.\,E.\thinspace Alspach}, Proc.\ Am.\ Math.\ Soc.\ 82, 423--424 (1981; Zbl 0468.47036)]. The authors derive a formula for the \(n\)-th iterate \(T^n\) of Alspach's map and they use it to prove that, for each \(f \in C\), the sequence \((T^nf)\) converges to \(\frac12 \chi_{[0,1]}\). From this, the authors prove that the set \(C\) contains a unique minimal \(T\)-invariant subset and they give an explicit characterization of the minimal set. Finally, the authors provide an alternative proof of their result using ergodic theory.
scientific article; zbMATH DE number 5718404
scientific article