Index 1 fixed points of orientation reversing planar homeomorphisms

Abstract

Let U subset of R-2 be an open subset, f : U -> f (U) subset of R-2 be an orientation reversing homeomorphism and let 0 is an element of U be an isolated, as a periodic orbit, fixed point. The main theorem of this paper says that if the fixed point indices i(R2)(f, 0) = i(R2)(f(2), 0) = 1 then there exists an orientation preserving dissipative homeomorphism phi: R-2 -> R-2 such that f(2) = phi in a small neighbourhood of 0 and {0} is a global attractor for phi. As a corollary we have that for orientation reversing planar homeomorphisms a fixed point, which is an isolated fixed point for f(2), is asymptotically stable if and only if it is stable. We also present an application to periodic differential equations with symmetries where orientation reversing homeomorphisms appear naturally