Criteria for regionally recurrent flows

Abstract

Regionally recurrent and P-regionally recurrent flows are characterized in purely prolongational notions. An example is given to show that the condition z E J(z) is not, as many authors asserted, equivalent to regional recurrence. Introduction. Let (X, T) be a flow, where X is a topological space and T is a topological group. Then a point x E X is said to be T-regionally recurrent or simply regionally recurrent if for each neighbourhood U of x there exists an extensive subset A of T such that U n Ua $ 0 for all a E A [6]. In [3] the author showed that for a flow (X, T), a point x E X is regionally recurrent if and only if x E JP(x) for all replete semigroups P in T. An example is provided (Example 14) to show that the condition x E J(x) is not equivalent to regional recurrence as it was stated in [7] and [8]. However, in the setting of continuous flows, where the phase group is the additive group of real numbers R, a necessary and a sufficient condition for a point x E X to be regionally recurrent is either one of the following (1) x E J+(x) or, (2) x E J-(x) or (3) x E J(x). This is due to the simple structure of replete semigroups in R. Every replete semigroup in R contains either a ray in R+ or a ray in R-. Thus if x E J+(x), then x E JP(x) for all replete semigroups P in R+. Furthermore, x E J+(x) implies that x E J-(x). Consequently, x E JQ(x) for all replete semigroups Q in Rand hence x is regionally recurrent. Similar analysis is available for the cases x E J-(x) and x E J(x). The situation is far more complicated in general flows as is shown in Example 14. In this example we have x E J(x) for each x E X but x ? JQ(x) for some replete semigroups Q in the phase group T. In [3] it was shown that x E X is P-regionally recurrent iff it is P-nonwandering. The same conclusion holds for T-regionally recurrent and T-nonwandering. Hence, the work here is a natural generalization of the work in [1] on nonwandering continuous flows. However, our techniques are different and much simpler. In addition to the above-mentioned, one purpose of this note is to introduce prolongational techniques which are widely used in Dynamical Systems Theory but surprisingly unknown in the theory of Topological Dynamics. As far as I know it was Hajek [7] who suggested the use of prolongational techniques in Topological Dynamics. R. Knight [8, 9] and Elaydi [2-5], with Kaul followed suit. But the power of these techniques has yet to be shown. Our main references for notations and terminology are [7] and [3]. Received by the editors April 6, 1981 and, in revised form, September 20, 1981. 1980 Mathematics Subject Classification. Primary 54H20.