P-recurrence in Topological Dynamics1

1. Definitions. The general reference for definitions is [3]. Throughout the paper (X, T, ir) will denote a transformation group for which X is a compact Hausdorff space. Whenever it is stated that X is a uniform space, it will be implicitly assumed that the (Hausdorff) topology of X is the one induced by the uniformity. It is further assumed that T is abelian. P and Q will be used to denote replete semigroups in T, which are distinct from T. Full use of all these hypotheses is not required in every theorem. For x£-X" the P-limit set of x, Px, is defined as in [l, 2.06]. Let EQT, then £ is P-extensive provided EC\pP^0 for each p£P. Let x£X, then x is \P-recurrent] \P-regionally recurrent}3 provided that for each neighborhood, U, of x there exists a P-extensive set £ in T such that r££ implies {xr£t/} { VrC\'Ü9í0\. It is now clear that the restriction P^T is justified, for if we were to allow equality it would follow that each point x£-X" is P-recurrent. This would make P-recurrence too weak a property to be of much interest. It is also clear that a point x£X is recurrent if and only if x is P-recurrent for each replete semigroup P in T. A similar statement holds for regional recurrence. T is said to be \P-regionally recurrent] [regionally recurrent] provided that for each x G X, x is {P-regionally recurrent} {regionally recurrent}.