Let f: S¹ × [0,1] → S¹ × [0,1] be a real-analytic diffeomorphism which is homotopic to the identity map and preserves an area form. Assume that for some lift f̃: ℝ × [0,1] → ℝ × [0,1] we have Fix(f̃) = ℝ × 0 and that f̃ positively translates points in ℝ × 1. Let be the perturbation of f̃ by the rigid horizontal translation (x,y) ↦ (x+ϵ,y). We show that for all ϵ > 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm227-1-1,
author = {Salvador Addas-Zanata and Pedro A. S. Salom\~ao},
title = {Persistence of fixed points under rigid perturbations of maps},
journal = {Fundamenta Mathematicae},
volume = {227},
year = {2014},
pages = {1-19},
zbl = {1329.37025},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm227-1-1}
}
Salvador Addas-Zanata; Pedro A. S. Salomão. Persistence of fixed points under rigid perturbations of maps. Fundamenta Mathematicae, Tome 227 (2014) pp. 1-19. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm227-1-1/