Existence of periodic points with real and simple spectrum for diffeomorphisms in any dimension

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 result 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.