Meagre functions and asymptotic behaviour of dynamical systems

Abstract

A measurable function x : J ⊂R → X (X a metric space) is said to be C-meagre if C ⊂X is non-empty and, for every closed set K ⊂X with K ∩ C = ∅; x−1(K) has 5nite Lebesgue measure. This concept of meagreness is shown to provide a unifying framework which facilitates a variety of characterizations, extensions or generalizations of diverse facts pertaining to asymptotic behaviour of dynamical systems. By way of motivation, consider the initial-value problem ẋ = f(x); x(0) = ∈ RN , with f : RN → RN locally Lipschitz, and denote its unique solution by x : t → (t; ) de5ned on its maximal interval of existence I = [0; ). Let g : RN → R be continuous. Assume that x is a bounded solution of the initial-value problem (in which case, I = [0;∞)), then g ◦ x is uniformly continuous. If, in addition, g ◦ x ∈ L1, then we may conclude (by an elementary observation frequently referred to as Barb> alat’s lemma [5] (see, also, [18, Section 21, Lemma 1])) that g(x(t)) → 0 as t → 0. Since, by boundedness, x(t) approaches, as t → ∞, its non-empty, compact, invariant !-limit set B, we also conclude that B⊂ g−1(0). It follows that x(t) tends, as t → ∞, to the largest invariant subset of g−1(0). This is a short proof of Theorem 1:2 of [6] (the proof in [6] is based on arguments involving properties of the Dow generated by the diEerential