Persistence of fixed points under rigid perturbations of maps

• Autorzy: Salvador Addas-Zanata , Pedro A. S. Salomão
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 37C05: Smooth mappings and diffeomorphisms
• 37C25: Fixed points, periodic points, fixed-point index theory
• 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2014
• Tom: 227
• Numer: 1
• Strony: 1-19

Daty

wydano

2014

Twórcy

autor

Salvador Addas-Zanata

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

autor

Pedro A. S. Salomão

• Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, Brazil

Identyfikatory

DOI

10.4064/fm227-1-1