Persistence of fixed points under rigid perturbations of maps
Volume 227 / 2014
Abstract
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.
