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Indices of the iterates of \(\mathbb R^3\)-homeomorphisms at Lyapunov stable fixed points. (English) Zbl 1132.37012
Summary: Given any positive sequence \(\{c_n\}_{n\in\mathbb N}\), we construct orientation preserving homeomorphisms \(f:\mathbb R^3\to \mathbb R^3\) such that \(\text{Fix}(f)=\text{Per}(f)=\{0\}\), 0 is Lyapunov stable and \(\lim\sup\frac{|i(f^m,0)|}{c_m}=\infty\). We will use our results to discuss and to point out some strong differences with respect to the computation and behavior of the sequences of the indices of planar homeomorphisms.

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