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Planar isolated and stable fixed points have index =1. (English) Zbl 1052.37011
Summary: Let \(W\subset\mathbb{R}^2\) be an open subset and \(f:W\to f(W)\subset \mathbb{R}^2\) be an orientation reversing homeomorphism. We prove that if \(p\in W\) is an isolated and stable fixed point of \(f\) then the fixed point index of \(f\) at \(p\), \(i_{\mathbb{R}^2} (f,p)\), is 1. We apply our theorem to the study of the orbital stability of isolated periodic orbits of flows in four-dimensional Riemannian manifolds.

MSC:
37B25 Stability of topological dynamical systems
37B30 Index theory for dynamical systems, Morse-Conley indices
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
54H25 Fixed-point and coincidence theorems (topological aspects)
55M20 Fixed points and coincidences in algebraic topology
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