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• # Artykuł - szczegóły

## Fundamenta Mathematicae

2014 | 227 | 1 | 1-19

## Persistence of fixed points under rigid perturbations of maps

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.

1-19

wydano
2014

### Twórcy

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