×

On the fixed point index of iterates of planar homeomorphisms. (English) Zbl 0686.58028

Summary: If f is an orientation preserving homeomorphism of the plane with an isolated fixed point at the origin 0 and \(index(f,0)=p\), then \(index(f^ n,0)\) is always well defined provided that \(p\neq 1\). In this case, for each \(n\neq 0\), \(index(f^ n,0)=index(f,0)=p\). If \(index(f,0)=1\), then there is an integer p (possibly \(p=1)\) such that for those values of n for which \(index(f^ n,0)\) is defined (i.e. 0 is an isolated fixed point of \(f^ n)\), \(index(f^ n,0)=1\) or \(index(f^ n,0)=p\).

MSC:

37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
54H25 Fixed-point and coincidence theorems (topological aspects)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Morton Brown, A new proof of Brouwer’s lemma on translation arcs, Houston J. Math. 10 (1984), no. 1, 35 – 41. · Zbl 0551.57005
[2] M. Brown, Homeomorphisms of two-dimensional manifolds, Houston J. Math. 11 (1985), no. 4, 455 – 469. · Zbl 0605.57005
[3] M. Brown and W. D. Neumann, Proof of the Poincaré-Birkhoff fixed point theorem, Michigan Math. J. 24 (1977), no. 1, 21 – 31. · Zbl 0402.55001
[4] Bruno V. Schmitt, L’espace des homéomorphismes du plan qui admettent un seul point fixe d’indice donné est connexe par arcs, Topology 18 (1979), no. 3, 235 – 240 (French). · Zbl 0436.58022
[5] M. Shub and D. Sullivan, A remark on the Lefschetz fixed point formula for differentiable maps, Topology 13 (1974), 189 – 191. · Zbl 0291.58014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.