  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.
