Diffeotopically trivial periodic diffeomorphisms (original) (raw)
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Proceedings of the American Mathematical Society, 1983
This note was inspired by some results of P. A. Smith [S], One proves that for any periodic map of a manifold M and any codimension two invariant submanifold P of M containing part of the stationary point set, connected invariant subsets of the complement of P must carry nontrivial one-dimensional rational cycles, provided that M satisfies some simple homological conditions (Theorem A). This fact has interesting consequences in transformation group theory. 0. Introduction. If m > 2 is a positive integer let Zm-Zm be the cyclic group of order m and if m = oo let Z", resp. Zm be the infinite cyclic group, respectively, 5' = {z G C||z| = 1}. We also denote by Gm ", m, n-2,3,4,..., bo, a semidirect product Z", X rZ" for t: Z"-» AutZm. Such a semidirect product has the inclusion zm ■* Gm,"> tne projection Gmn-Z" and the section s: Z"-» Gmn (tt ° s = id) as part of the data. Clearly if n-oo, Gmn = Zm X Sl. The groups Gm " are regarded as compact Lie groups. Given a compact Lie group G, p: G X M-> M a topological action, N an invariant submanifold, and x E M, then the action /x: G X (M, N)-> (M, N) is called locally smooth at x if there exists a smooth action ß: G X V-» V, an invariant submanifold W E V, and an invariant neighborhood % of x G M together with an equivariant homeomorphism ^(%, % D JV)-* (F, W7). The main result of this note is the following: Theorem A. Let p: Zm X M"-* M" fte a smooth (topological) orientation preserving action, pn~2 c M" an invariant smooth (locally flat) closed oriented submanifold whose orientation is also invariant. Assume that HX(M) is torsion prime to m,H2(M;Q) = 0,and (i) MZm ¥= 02 and P f) Mz-# 0 is a union of connected components of Mz™, (ii) if p is a topological action then there exists y E P n MZm so that p: Zm X (M, P)-> (M, P) is locally smooth aty.
We consider finitely generated groups of real-analytic circle diffeomorphisms. We show that if such a group admits an exceptional minimal set (i.e., a minimal invariant Cantor set), then its Lebesgue measure is zero; moreover, there are only finitely many orbits of connected components of its complement. For the case of minimal actions, we show that if the underlying group is (algebraically) free, then the action is ergodic with respect to the Lebesgue measure. This provides first answers to questions due toÉ. Ghys, G. Hector and D. Sullivan.