Persistence of fixed points under rigid perturbations of maps

Autor(es)

Addas-Zanata, Salvador ^[1] ; Salomao, Pedro A. S. ^[2]

Afiliação do(s) autor(es)

^[1] Univ Sao Paulo, Dept Matemat Aplicada, Inst Matemat & Estat, Sao Paulo - Brazil

^[2] Univ Sao Paulo, Dept Matemat, Inst Matemat & Estat, Sao Paulo - Brazil

• Tipo de documento: Artigo Científico
• Fonte: FUNDAMENTA MATHEMATICAE; v. 227, n. 1, p. 1-19, 2014.
• Citações Web of Science: 0

Resumo

Let f : S-1 x {[}0, 1] -> S-1 x {[}0, 1] be a real-analytic diffeomorphism which is homotopic to the identity map and preserves an area form. Assume that for some lift (f) over tilde : R x {[}0, 1] -> R x {[}0, 1] we have Fix((f) over tilde) = R x [0] and that (f) over tilde positively translates points in R x [1]. Let (f) over tilde (is an element of) be the perturbation of (f) over tilde by the rigid horizontal translation (x, y) -> (x+is an element of, y). We show that Fix((f) over tilde (is an element of)) = empty set for all is an element of > 0 sufficiently small. The proof follows from Kerekjarto'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 diffeomorphisra 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. (AU)

• Processo FAPESP: 11/16265-8 - Dinâmica em baixas dimensões
• Beneficiário: Edson Vargas
• Modalidade de apoio: Auxílio à Pesquisa - Temático