Concentrated Cyclic Actions of High Periodicity

Abstract

The class of concentrated periodic diffeomorphisms $g:M \to M$ 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, $\operatorname {Sign} (g,M) = \operatorname {Sign} ({M^g})$, 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 ${M^g}$ in $M$ to $\operatorname {Sign} (g,M)$ via the normal $g$-representations, is established.