Stretched exponential estimates on growth of the number of periodic points for prevalent diffeomorphisms. I
type: Item
instance of: scholarly article
title: Stretched exponential estimates on growth of the number of periodic points for prevalent diffeomorphisms. I
zbMATH Open document ID: 1132.37011
author: Brian R. Hunt
author: Vadim Kaloshin
published in: Annals of Mathematics. Second Series
publication date: 2007-05-29T00:00:00Z
review text: Let \(M\) be a smooth compact finite-dimensional manifold, and let \(f\) be a \(C^{r}\)-diffeo\-mor\-phism on \(M\) for some \(r\in [0,\infty ]\). Denote by \(P_{n}(f)\) the number of periodic points of \(f\) having period \(n\). The growth rate of \(P_{n}(f)\) gives an information on the ``chaoticity'' (or ``complexity'') of the map \(f\). According to a result obtained in \textit{M. Artin} and \textit{B. Mazur} [Ann. Math. 81, 82--99 (1965; Zbl 0127.13401)], diffeomorphisms satisfying \(P_{n}(f)\leq\exp (Cn)\) for some \(C\) are dense in the set of all \(C^{r}\)-diffeomorphisms endowed with the uniform \(C^{r}\)-topology. On the other hand for \(r\geq 2\) it has been proved in [the first author, Commun. Math. Phys. 211, No.~1, 253--271 (2000; Zbl 0956.37017)] that there is a Baire generic set of \(C^{r}\)-diffeomorphisms having arbitrarily large growth rate of \(P_{n}(f)\). The question addressed in this paper is how does the growth rate of \(P_{n}(f)\) behave for the ``most'' diffeomorphisms from a measure theoretic point of view. To this end the notion of prevalence is used in order to describe the meaning of ``a \(C^{r}\)-diffeomorphism satisfies a certain property with probability one''. The manifold \(M\) is embedded in the interior of the closed unit ball \(B^{N}\subset{\mathbb R}^{N}\) for a sufficiently large \(N\). Then \(f\) is extended to an element of \(C^{r}(B^{N},{\mathbb R}^{N})\) in a suitable way. For a ``small'' function \(g\in C^{r}(B^{N},{\mathbb R}^{N})\) the map \(f+g\) corresponds in a natural way to a \(C^{r}\)-diffeomorphism \(h\) on \(M\). A property is called prevalent, if it holds on a Borel set \(S\subset C^{r}(B^{N},{\mathbb R}^{N})\) and there exists a Borel probability measure \(\mu\) on \(C^{r}(B^{N},{\mathbb R}^{N})\) with compact support such that for all \(f\in C^{r}(B^{N},{\mathbb R}^{N})\) there exists a Borel set \(S_{f}\subset C^{r}(B^{N},{\mathbb R}^{N})\) with \(\mu (S_{f})=1\) and \(f+g\in S\) for all \(g\in S_{f}\). For …
reviewed by: Peter Raith
Mathematics Subject Classification ID: 37C05
Mathematics Subject Classification ID: 37E05
Mathematics Subject Classification ID: 37C25
Mathematics Subject Classification ID: 37D05
Mathematics Subject Classification ID: 37C35
zbMATH DE Number: 5157392
zbMATH Keywords: compact manifold
zbMATH Keywords: periodic points
zbMATH Keywords: growth rate
zbMATH Keywords: smooth interval map
MaRDI profile type: Publication
full work available at URL: annals.2007.165.89
OpenAlex ID: W2100299994
DOI: 10.4007/ANNALS.2007.165.89
description: scientific article; zbMATH DE number 5157392