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We prove that for any $C^r$ diffeomorphism, $f$, of a compact manifold ofdimension $d>2$, $1\leq r\leq \infty$, admitting a transverse homoclinicintersection, 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 periodicpoint with real and simple spectrum. We use this result to prove that$C^r$-generically among $C^r$ diffeomorphisms with horseshoes, we have densityof periodic points with real and simple spectrum inside the horseshoe. As acorollary, we obtain that generically in the $C^1$-topology the uniqueobstruction to the existence of periodic points with real and simple spectrumare the Morse-Smale diffeomorphisms with all the periodic points admittingnon-real eigenvalues.

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