Elementary proof for the existence of periodic points with real and simple spectrum for diffeomorphisms in any dimension

Abstract

Using only elementary techniques we prove that for any $ C^r $ diffeomorphism, $ f $, of a compact manifold of dimension $ d>2 $, $ 1\leq r\leq \infty $, admitting a transverse homoclinic intersection, we can find a $ C^1 $-open neighborhood of $ f $ containing a $ C^1 $-open and $ C^r $-dense set of $ C^r $ diffeomorphisms which have a periodic point with real and simple spectrum. We use this fact to prove that $ C^r $-generically among $ C^r $ diffeomorphisms with horseshoes, we have density of periodic points with real and simple spectrum inside the horseshoe. As a corollary, we obtain that generically in the $ C^1 $-topology the unique obstruction to the existence of periodic points with real and simple spectrum are the Morse-Smale diffeomorphisms with all the periodic points admitting non-real eigenvalues.