- The class of concentrated periodic diffeomorphisms g: M -M is introduced. A diffeomorphism is called concentrated if, roughly speaking, its normal eigenvalues range in a small (with respect to the period of g and the dimension of M) arc on the circle. In many ways, the cyclic action generated by such a g behaves on the one hand as a circle action and on the other hand as a generic prime power order cyclic action. For example, as for circle actions, Sign(g, M) = Sign(Mg), provided that the left-hand side is an integer; as for prime power order actions, g cannot have a single fixed point if M is closed. A variety of integrality results, relating to the usual signatures of certain characteristic submanifolds of the regular neighbourhood of Mg in M to Sign(g, M) via the normal g-representations, is established.