On the fixed point index of iterates of planar homeomorphisms

Brown, Morton

doi:10.1090/S0002-9939-1990-0994772-9

On the fixed point index of iterates of planar homeomorphisms
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Proc. Amer. Math. Soc. 108 (1990), 1109-1114 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 ${\text {index(}}{f^n}{\text {,0) = }}p$.

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