The Center of a Transformation Group

Criteria for regionally recurrent flows

“…Then a point x £ 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 Uf] Ua ^ 0 for all o G A [6]. In [3] the author showed that for a flow (X, T), a point x £ X is regionally recurrent if and only if x £ Jp(x) for all replete semigroups P in T. An example is provided (Example 14) to show that the condition x £ 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 G X to be regionally recurrent is either one of the following (1) x £ J+(x) or, (2) x £ J~(x) or (3) x £ J(x).…”

“…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][3][4][5], with Kaul followed suit. But the power of these techniques has yet to be shown.…”

“…We need only to replace T by Q, where Q is any replete semigroup contained in P, in the proof of [8,1]. REMARK 1.13.…”

“…REMARK 1.13. Although the proof of [8,1] uses wrong equivalences to regionally recurrence and recurrence it can be corrected by a slight modification. We will give an example of a flow in which x G L(x) for each x G X and yet it is neither regionally recurrent nor recurrent.…”