Persistence of fixed points under rigid perturbations of maps

Salvador Addas-Zanata, and Pedro A. S. Salomão. "Persistence of fixed points under rigid perturbations of maps." Fundamenta Mathematicae 227.1 (2014): 1-19. <http://eudml.org/doc/282661>.

@article{SalvadorAddas2014,
abstract = {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 $f̃_ϵ$ be the perturbation of f̃ by the rigid horizontal translation (x,y) ↦ (x+ϵ,y). We show that $Fix(f̃_ϵ) = ∅$ 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.},
author = {Salvador Addas-Zanata, Pedro A. S. Salomão},
journal = {Fundamenta Mathematicae},
keywords = {topological dynamics; Brouwer theory; generating functions},
language = {eng},
number = {1},
pages = {1-19},
title = {Persistence of fixed points under rigid perturbations of maps},
url = {http://eudml.org/doc/282661},
volume = {227},
year = {2014},
}

TY - JOUR
AU - Salvador Addas-Zanata
AU - Pedro A. S. Salomão
TI - Persistence of fixed points under rigid perturbations of maps
JO - Fundamenta Mathematicae
PY - 2014
VL - 227
IS - 1
SP - 1
EP - 19
AB - 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 $f̃_ϵ$ be the perturbation of f̃ by the rigid horizontal translation (x,y) ↦ (x+ϵ,y). We show that $Fix(f̃_ϵ) = ∅$ 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.
LA - eng
KW - topological dynamics; Brouwer theory; generating functions
UR - http://eudml.org/doc/282661
ER -