Omega limit sets of nonexpansive maps: finiteness and cardinality estimates
Roger D. Nussbaum · Differential and Integral Equations · 1990
Introduction.If (M, d) is a complete metric space, C is a closed subset of M and T: C--> C a map, then for x E Cone can define the omega limit set w(x; T):(1)In equation (1), cl(A) denotes the closure of a set A. Alternatively, w(x; T) is the set of y E M such that there exists a sequence of integers k; --> oo such thatIf w = w(x; T) is nonempty and Tis nonexpansive it is known (see [5]) that w(y; T) = w(x; T) for ally E w(x; T) and that the restriction ofT to w(x; T) is an isometry of w(x; T) onto itself.In particular, for each y, z E w(x; T) there exists a sequence of integers k; --> oo such that lim Tk;(y) = z.(...... 00If w(x; T) is also compact, then by using the Ascoli-Arzela theorem and equation (3), one easily shows (see Lemma 1 in [7]) that for each y and z in w(x; T) there exists an isometry Sy,z : w(x; T) --> w(x; T) of w(x; T) onto itself such that Sy,z(Y) = z and such that Sy,zSu,v = Su,vSy,z for ally, Z, u, v E w(x; T).(4)If M = Rn, C is compact, the metric d arises from the £1-norm n llxll1 = L ix;i,