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

2021/07/16 by Jamerson Bezerra, Bezerra, Jamerson, Carlos Gustavo Moreira +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Analytic and geometric function theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2107.07969

openalex publication_date 2021/07/16 · openalex created_date 2021/08/02 · openalex updated_date 2026/07/28

Abstract

We prove that for any C^r diffeomorphism, f, of a compact manifold of dimension d>2, 1≤ r≤ ∞, admitting a transverse homoclinic intersection, we can find a C¹-open neighborhood of f containing a C¹-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 C¹-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.

Citations

• Lyapunov exponents with multiplicity 1 for deterministic products of matrices

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