Generic family displaying robustly a fast growth of the number of periodic points

2017/01/09 by Pierre Berger, Berger, Pierre · 3 citations
Mathematics · #37C35 37L10 37J40 37E20 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.1701.02393

openalex publication_date 2017/01/09 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

For any 2 ≤ r ≤ ∞, n≥2, we prove the existence of an open set U of C^r-self-mappings of any n-manifold so that a generic map f in U displays a fast growth of the number of periodic points: the number of its n-periodic points grows as fast as asked. This complements the works of Martens-de Melo-van Strien, Kaloshin, Bonatti-D' iaz-Fisher and Turaev, to give a full answer to questions asked by Smale in 1967, Bowen in 1978 and Arnold in 1989, for any manifold of any dimension and for any smoothness. Furthermore for any 1≤ r

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