On Galvin’s theorem for compact Hausdorff right-topological semigroups with dense topological centers

We generalize an important theorem of Fred Galvin from the Stone-Cˇech compactification βT of any discrete semigroup T to any compact Hausdorff right-topological semigroup with a dense topological center;and then apply it to Ellis' semigroups to prove that a point is distal if and only if it is IP*-recurrent, for any semiflow(T, X) with arbitrary compact Hausdorff phase space X not necessarily metrizable and with arbitrary phase semigroup T not necessarily cancelable.