Recurrence and Almost Periodicity in Transformation Groups

Abstract

In [5] Gottschalk comments that an interesting question is to find conditions under which pointwise recurrence in a transformation group (X,T) implies pointwise almost periodicity. In this paper we show that if in a transformation group (X,T), the phase space X is connected locally compact and Hausdorff and T is generative and each point xεX is of P-characteristic 0 for each replete semi-group P in T, then the existence of a single recurrent point implies that (X,T) is pointwise recurrent and each orbit closure is minimal. This implies of course from a general result of Gottschalk and Hedlund that the transformation group is pointwise almost periodic. The condition P-characteristic 0 is weaker than pointwise equicontinuity even when the phase space is compact--see example (4.6) at the end.