Estimates of the periods of periodic points for nonexpansive operators

Abstract

Suppose thatE is a finite-dimensional Banach space with a polyhedral norm ‖·‖, i.e., a norm such that the unit ball inE is a polyhedron. ℝn with the sup norm or ℝn with thel1-norm are important examples. IfD is a bounded set inE andT:D→D is a map such that ‖T(y)−T(z)‖≤ ‖y−z‖ for ally andz inE, thenT is called nonexpansive with respect to ‖·‖, and it is known that for eachx ∈D there is an integerp=p(x) such that limj→∞Tjp(x) exists. Furthermore, there exists an integerN, depending only on the dimension ofE and the polyhedral norm onE, such thatp(x)≤N: see [1,12,18,19] and the references to the literature there. In [15], Scheutzow has raised a question about the optimal choice ofN whenE=ℝn,D=Kn, the set of nonnegative vectors in ℝn, and the norm is thel1-norm. We provide here a reasonably sharp answer to Scheutzow’s question, and in fact we provide a systematic way to generate examples and use this approach to prove that our estimates are optimal forn≤24. See Theorem 2.1, Table 2.1 and the examples in Section 3. As we show in Corollary 2.3, these results also provide information about the caseD=ℝn, i.e.,T:ℝn→ℝn isl1-nonexpansive. In addition, it is conjectured in [12] thatN=2n whenE=ℝn and the norm is the sup norm, and such a result is optimal, if true. Our theorems here show that a sharper result is true for an important subclass of nonexpansive mapsT:(ℝn,‖ · ‖∞)→(ℝn,‖ · ‖∞).