2000/12/21 by Vadim Yu. Kaloshin, Kaloshin, Vadim Yu., Brian Hunt +2
Mathematics · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #math.DS
paper · pdf · doi:10.48550/arxiv.math/0012224
19 pages, 2 figures
arxiv created 2000/12/21 · openalex publication_date 2000/12/21 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M be a compact manifold of dimension at least 2, Diffr(M) be the space of Cr diffeomorphisms of M. Define for any diffeomorphism f in Diffr(M) number of isolated periodic points of period n by Pn(f)=# isolated x in M: fn(x)=x. Artin--Mazur proved that for a dense set of diffeomorphisms the number of periodic points Pn(f) growth at most exponentially fast in n. The author proved that there is an open set N ⊂ Diffr(M) such that for a Baire generic set of diffeomorphisms the number of periodic points Pn(f) growth arbitrarily fast. Arnold posed a problem: Prove that diffeomorphisms with at most exponential growth of the number of periodic points in period have probability one. In this paper we annonce and exhibit key ingredients for a partial solution to Arnold's problem: we prove that for any epsilon>0 with probability one the number of periodic points is bounded by a streched exponential estimate Pn(f)