Abstract
In this paper several results concerning the periodic points of 1-norm nonexpansive
maps will be presented. In particular, we will examine the set R(n),
which consists of integers p ?? 1 such that there exist a 1-norm nonexpansive map
f: Rn ¿ Rn and a periodic point of f of minimal period p. The principal problem is
to find a characterization of the set R(n) in terms of arithmetical and combinatorial
constraints. This problem was posed in [12, section 4]. We shall present here a significant
step towards such a characterization. In fact, we shall introduce for each n ¿ N
a set T(n) that is determined by arithmetical and combinatorial constraints only,
and prove that R(n) ¿ T(n) for all n ¿ N. Moreover, we will see that R(n) = T(n) for
n = 1, 2, 3, 4, 6, 7, and 10, but it remains an open problem whether the sets R(n) and
T(n) are equal for all n ¿ N.
Original language | English |
---|---|
Pages (from-to) | 165-180 |
Journal | Mathematical Proceedings of the Cambridge Philosophical Society |
Volume | 135 |
Issue number | 1 |
DOIs | |
Publication status | Published - 2003 |