Persistence of fixed points under rigid perturbations of maps

Salvador Addas-Zanata¹ ; Pedro A. S. Salomão²

¹ Departamento de Matemática Aplicada Instituto de Matemática e Estatística Universidade de São Paulo São Paulo, Brazil
² Departamento de Matemática Instituto de Matemática e Estatística Universidade de São Paulo São Paulo, Brazil

Fundamenta Mathematicae, Tome 227 (2014) no. 1, pp. 1-19.

Résumé

Let $f:S^1\times [0,1]\to S^1\times [0,1]$ be a real-analytic diffeomorphism which is homotopic to the identity map and preserves an area form. Assume that for some lift $\tilde {f}:\mathbb{R}\times [0,1]\rightarrow \mathbb{R}\times [0,1]$ we have ${\rm Fix}(\tilde{f})=\mathbb{R}\times \{0\}$ and that $\tilde{f}$ positively translates points in $\mathbb{R}\times \{1\}$. Let $\tilde{f}_\epsilon $ be the perturbation of $\tilde{f}$ by the rigid horizontal translation $(x,y)\mapsto (x+\epsilon ,y)$. We show that ${\rm Fix} (\tilde{f}_\epsilon )=\emptyset $ for all $\epsilon >0$ sufficiently small. The proof follows from Kerékjártó's construction of Brouwer lines for orientation preserving homeomorphisms of the plane with no fixed points. This result turns out to be sharp with respect to the regularity assumption: there exists a diffeomorphism $f$ with all the properties above, except that $f$ is not real-analytic but only smooth, such that the above conclusion is false. Such a map is constructed via generating functions.

Détail

DOI : 10.4064/fm227-1-1

Keywords: times times real analytic diffeomorphism which homotopic identity map preserves area form assume lift tilde mathbb times rightarrow mathbb times have fix tilde mathbb times tilde positively translates points mathbb times tilde epsilon perturbation tilde rigid horizontal translation mapsto epsilon fix tilde epsilon emptyset epsilon sufficiently small proof follows ker construction brouwer lines orientation preserving homeomorphisms plane fixed points result turns out sharp respect regularity assumption there exists diffeomorphism properties above except real analytic only smooth above conclusion false map constructed via generating functions