Lower and upper bounds for \(\omega\)-limit sets of nonexpansive maps
Let \((X,d)\) be a complete metric space, \(D_f\) a closed subset of \(X\) and let \(f\) map \(D_f\) info itself. For each \(x\in D_f\) the \(\omega\)-limit set, \(\omega(x)=\omega(x;f)\), is defined by: \[ \omega(x)=\Bigl\{y\in D_f: y=\lim_{i\to\infty}f^{k_i}(x) \text{for some sequence of integers } k_i\to\infty \Bigr\}. \] In the paper under review, the authors discuss the procedure to obtain a priori upper bounds on the cardinality of the \(\omega\)-limit of \(l_1\)-norm nonexpansive maps, based on the approach introduced by \textit{M. Misiurewicz} [Math. Gottingensis, Schriftenr. Sonderforschungsbereichs Geom. Anal. 45 (1987; Zbl 0657.46012)]. In the case when the dimension of the space is less than six, they give a sharper upper bound for the cardinality of \(\omega\)-limit sets of \(l_1\)-norm nonexpansive maps than the upper bound obtained by Misiurewicz. The authors also describe two different procedures to construct periodic orbits of \(l_1\)-norm nonexpansive maps. From these constructions it follows, in particular, that a lower bound for the largest possible minimal period of a periodic point of an \(l_1\)-norm nonexpansive map is given by \(3\cdot 2^{n-1}\) for \(n\geq 3\). The paper also contains some interesting examples and open questions.
scientific article; zbMATH DE number 1758184
scientific article