Show that if f in Diff^r(M), r >=1, is structurally stable then all the fixed points off are hyperbolic.

Question

i think since f is structurally stable so there exists an open nbd u containig of g then f and f are topoligy equivalent.i think since hyperbolic fixed pints dence and open there exists neighberhood v is small enough of contaning h such that all fixed point h are hyperbolic so all fixed point f are hyperbolic

Answer 1

The famous $C^r$-stability conjecture (it is already proved for $r=1$) says that structural stability is equivalent to Axiom A (nonwandering set is a finite number of disjoint invariant hyperbolic sets -- basic sets) + strong transversality condition (stable and unstable manifolds of points from the basic sets intersect transversally). In particular it gives hyperbolicity of periodic points.

The idea of the proof of your particular statement -- if the periodic point is not hyperbolic one can select a perturbation that would locally act like a translation in some direction (the nonhyperbolic one) which would prevent bounded distance between the conjugacy and the identity map.