Local structural stability of actions of R^n on n-manifolds

Carlos Maquera,J L Arraut

doi:10.5269/bspm.v24i1-2.7435

Abstract

Let M^m be a compact m-manifold and  \varphi: R^n × M^m \rightarrow M^m a C^r, r \geq 1, action with infinitesimal generators of class C^r . We introduce the concept of transversally hyperbolic singular orbit for an action \varphi and explore this concept in its relations to stability. Our main result says that if m = n and O_p is a compact singular orbit of \varphi that is transversally hyperbolic, then \varphi is C^1 locally structurally stable at O_p.

Highlights

Theorem 1.1 If Op is a transversally hyperbolic compact singular orbit of φ ∈ Ar(Rn, N ), r ≥ 1, φ is locally C1 structurally stable at Op
We denote by X1, . . . , Xn the infinitesimal generators of φ associated to the canonical base of Rn
Op is transversally hyperbolic if there exist a chart adapted to Op at p such that 0 ∈ Dεm−k is a hyperbolic fixed point of the action φT

Summary

Introduction

Theorem 1.1 If Op is a transversally hyperbolic compact singular orbit of φ ∈ Ar(Rn, N ), r ≥ 1, φ is locally C1 structurally stable at Op. Assume that we already defined hyperbolicity for linear actions of groups H such that rank(H) < k + l. Let p a hyperbolic fixed point of φ ∈ Ar(Rn, M m), r ≥ 1, and : Rn × TpM m → TpM m the induced linear action.

Results

Conclusion