Problem 88

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Index 0 fixed point of diffeo is removable by small perturbation (Hirsch)




Hirsch mentioned this problem in [1].

In [2], we show that the conjecture is true for diffeomorphisms of 2-manifolds, but only in the C1- topology of diffeomorphisms. The problem is especially interesting within the space of symplectic diffeomorphisms, or equivalently (see [4], e.g., for the equivalence) for zeroes of gradient vector fields. In [3] we discuss how this is related to the proof of the second fixed point in the Poincare-Birkhoff Twist Theorem on the annulus. 

In [3] we show that the answer is yes for analytic symplectic transformations of 2-manifolds but only when certain regularity conditions are satisfied. We also show that the answer is no in dimensions > 2, basically because there are more invariants to consider than just the fixed point index. The answer for symplectic maps of M2 is equivalent to the problem of “removing “ index zero intersections of two 2-dimensional Lagrangian submanifolds in R4.


1. Hirsch, Morris. 1973. Stability of Compact Leaves of Foliations. In: Dynamical Systems. (M. Peixoto, editor) NY: Academic Press.

2. Simon, Carl and Titus, Charles. 1973. Removing Index--Zero Singularities with C1-Small Perturbations. In: Dynamical Systems---Warwick 1974. (A. Manning, editor). New York: Springer-Verlag Lecture Notes in Math 468 (1975) 278-286.

3. Simon, Carl and Titus, Charles. 1973. The Fixed Point Index of Symplectic Maps. In: (J-M Souriau, editor). Geometrie Symplectique et Physique Mathematique. Paris: C.N.R.S. (1975) No. 237, 19-28.

4. Simon, Carl. 1974. A Bound for the Fixed-Point Index of an Area-Preserving Map with Applications to Mechanics. Inventiones Math., 26 (1974) 187-200.

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