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

TL;DR

This paper demonstrates that in high-dimensional dynamical systems with certain intersections, it is generically possible to find periodic points with real, simple spectra, highlighting the typical spectral properties within these systems.

Contribution

It proves the density of diffeomorphisms with periodic points having real and simple spectra near those with transverse homoclinic intersections, extending understanding of spectral properties in complex dynamics.

Findings

Density of such periodic points in systems with horseshoes

Generic obstruction to real simple spectra is Morse-Smale with non-real eigenvalues

Results apply to a broad class of high-dimensional diffeomorphisms

Abstract

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.