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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