Characterizations of distality via weak equicontinuity

For an infinite discrete group $ G acting on a compact metric space X $, we introduce several weak versions of equicontinuity along subsets and show that if minimal system (X, G) admits invariant measure then is distal only it pairwise IP$ ^* $-equicontinuous; the product (X\times X, has dense set points, $-equicontinuous central$ with being abelian, order \infty FIP$ $-equicontinuous.