Persistence of fixed points under rigid perturbations of maps

2014/04/04 by Salvador Addas-Zanata, Addas-Zanata, Salvador, Pedro A. S. Salomão +1
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS

paper · pdf · doi:10.48550/arxiv.1404.1302

arxiv created 2014/04/04 · arxiv updated 2014/04/07

Abstract

Let f:S¹× [0,1]→ S¹× [0,1] be a real-analytic annulus diffeomorphism which is homotopic to the identity map and preserves an area form. Assume that for some lift f:ℝ× [0,1]→ ℝ× [0,1] we have \rm Fix(f)=ℝ× \0\ and that f positively translates points in ℝ× \1\. Let f_ε be the perturbation of f by the rigid horizontal translation (x,y)↦ (x+ε,y). We show that for all ε>0 sufficiently small we have \rm Fix (f_ε)=∅ . 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 satisfying all the properties above, except that f is not real-analytic but only smooth, so that the above conclusion is false. Such a map is constructed via generating functions.

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