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

The paper proves the existence (in any dimension d ≥ 3) of periodic points with real and simple spectrum near a transverse homoclinic intersection via a detailed linear-algebraic mechanism after Sternberg linearization and a key matrix product proposition. The model’s solution reaches the same conclusion by a different route, using a small horseshoe inside a chosen neighborhood and a Franks/Bonatti–Gourmelon–Vivier perturbation along a long periodic orbit to steer the spectrum. Both arguments are sound at a high level; the paper’s proof is self-contained and complete within its framework, and the model’s relies on standard perturbative tools. Minor clarifications are noted for each (openness justification in the paper; C^r-small realization details in the model), but neither affects the main claim.