Abstract. The purpose of this paper is to demonstrate that the families of Poisson stable flows and recurrent flows coincide whenever the phase space is locally compact.1. Introduction. The concepts of recurrent motions, nonwandering points, Poisson stable points, and minimal sets are classical notions central to the qualitative theory of motions for dynamical systems. Birkhoff introduced recurrent motions and nonwandering points (see [6]) and established important interrelationships for recurrent motions, minimal sets, nonwandering points, and Poisson stability for certain «-manifold regions. Most of these results have since been extended to Hausdorff or locally compact Hausdorff phase spaces. Our principal task in this paper is to demonstrate that the notions of recurrence and Poisson stability coincide in certain flows. Moreover, we give examples to show that the results are sharp.Throughout the paper we shall assume that there is a given flow (X, w) on a Hausdorff phase space X. We denote the orbit, orbit closure, limit set, prolongation, and prolongational limit set relations on X, respectively, by C, K, L, D, and J. The unilateral versions of these relations carry the appropriate + or -superscript.

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