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In "Geometry and Billiards" by S. Tabachnikov the author proves Poincaré's last geometric theorem:

"An area-preserving transformation of an annulus that moves the boundary circles in opposite directions has at least two distinct fixed points."

He states that the hardest part of the argument is to prove the existence of one fixed point and that the existence of the second point follows from a standard topological argument involving Euler characteristic.

Can anybody tell me what this "standard topological argument" is?

aes
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Klaas
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    The question seems delicate. I quote from a 2013 paper by Kirillov and Starkov: "G. Birkhoff was the first who responded to the appeal of Poincaré. In 1913 he proved the theorem using his ingenious method, but the existence of the second fixed point was not correctly justified. Only in 1977 M. Brown and W. D. Neumann gave explicit proof clarifying the Birkhoff’s one." – Marcos Cossarini Sep 23 '14 at 00:27

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Given an automorphism $f:A\to A$ of the annulus, we can define a vector field on $A$ by $v(x)=f(x)-x$, with fixed points of $f$ corresponding to zeros of $v$. By the Poincare-Hopf theorem, the sum of the indices of the zeros of $v$ must be $\chi(A)=0$, so $v$ cannot have only one zero (since the index of a zero is never $0$).

Alex Becker
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    The index of a zero can be 0. If a vector field $v$ is non-zero at some point $x$, then you can multiply it by a non-negative smooth function $\phi$ that vanishes only at zero, creating a vector field $\phi v$ that has a topologically invisible zero at $x$ (it's index is 0). Regular zeros do have index $\pm 1$, but I don't see how that could help. – Marcos Cossarini Sep 22 '14 at 23:29
  • @MarcosCossarini perhaps the case of an index 0 zero can be avoided by using the area-preserving property of the map? – AnonymousCoward Dec 28 '14 at 02:05
  • This paper claims to do precisely that, but it contains some technical arguments I haven't looked at. http://www.ams.org/journals/tran/1993-340-01/S0002-9947-1993-1145963-5/S0002-9947-1993-1145963-5.pdf – AnonymousCoward Dec 28 '14 at 02:53
  • I'm sorry that I can't read that paper now, but Birkhoff´s original paper (http://www.ams.org/journals/tran/1913-014-01/S0002-9947-1913-1500933-9/S0002-9947-1913-1500933-9.pdf), supplemented by Brown and Neumann's details (https://projecteuclid.org/euclid.mmj/1029001816), are clear, short and enjoyable. Once you proved the existence of the first fixed point using Birkhoff's methods, you can establish the existence of the second one by a more careful analysis of the same type. – Marcos Cossarini Dec 28 '14 at 20:19