Stability of compact actions and a result on divided differences

2025/06/17 by Carlos Gustavo Moreira, Moreira, Carlos Gustavo, Nicolau C. Saldanha +1 · 1 citation
#math.DS

paper · pdf · doi:10.48550/arxiv.2506.14446

Abstract

We study smooth locally free actions of \mathbb Rⁿ on manifolds M of dimension n+1. We are interested in compact orbits and in compact actions: actions with all orbits compact. Given a compact orbit in a neighborhood of compact orbits, we give necessary and sufficient conditions for the existence of a C^k perturbation with noncompact orbits in the given neighborhood. We prove that if such a perturbation exists it can be assumed to differ from the original action only in a smaller neighborhood of the initial orbit. As an application, for each k, we give examples of compact actions which admit C^k-1-perturbations with noncompact orbits but such that all C^k-perturbations are compact. The main result generalizes for k > 1 a previous result for the case C¹. A critical auxiliary result is an estimate on divided differences.

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