Equicontinuity on semi-locally connected spaces

We prove that a homeomorphism of a semi-locally connected compact metric space is equicontinuous if and only if it satisfies the following property (CW): the distance between the iterates of a given point and a given subcontinuum not containing that point is bounded away from zero. This equivalence is false for general compact metric spaces. We also prove that a homeomorphism with the property (CW) satisfies that the set of automorphic points contains those points where the space is not semi-locally connected.