On the fixed point index of iterates of planar homeomorphisms
HTML articles powered by AMS MathViewer
- by Morton Brown
- Proc. Amer. Math. Soc. 108 (1990), 1109-1114
- DOI: https://doi.org/10.1090/S0002-9939-1990-0994772-9
- PDF | Request permission
Abstract:
If $f$ is an orientation preserving homeomorphism of the plane with an isolated fixed point at the origin 0 and ${\text {index(}}f,0{\text {) = }}p$, then ${\text {index(}}{f^n}{\text {,0)}}$ is always well defined provided that $p \ne 1$. In this case, for each $n \ne 0$, ${\text {index(}}{f^n}{\text {,0) = index(}}f,o{\text {) = }}p$. If ${\text {index(}}f,0{\text {) = 1}}$, then there is an integer $p$ (possibly $p = 1$) such that for those values of $n$ for which ${\text {index(}}{f^n}{\text {,0)}}$ is defined (i.e 0 is an isolated fixed point of ${f^n}$), ${\text {index(}}{f^n}{\text {,0) = 1}}$ or ${\text {index(}}{f^n}{\text {,0) = }}p$.References
- Morton Brown, A new proof of Brouwer’s lemma on translation arcs, Houston J. Math. 10 (1984), no. 1, 35–41. MR 736573
- M. Brown, Homeomorphisms of two-dimensional manifolds, Houston J. Math. 11 (1985), no. 4, 455–469. MR 837985
- M. Brown and W. D. Neumann, Proof of the Poincaré-Birkhoff fixed point theorem, Michigan Math. J. 24 (1977), no. 1, 21–31. MR 448339
- 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). MR 546793, DOI 10.1016/0040-9383(79)90006-5
- M. Shub and D. Sullivan, A remark on the Lefschetz fixed point formula for differentiable maps, Topology 13 (1974), 189–191. MR 350782, DOI 10.1016/0040-9383(74)90009-3
Bibliographic Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 108 (1990), 1109-1114
- MSC: Primary 54H25; Secondary 55M20, 55M25, 57N05
- DOI: https://doi.org/10.1090/S0002-9939-1990-0994772-9
- MathSciNet review: 994772