Periodic points and rotation numbers for area preserving diffeomorphisms of the plane. (English) Zbl 0721.58031

Let f: \({\mathbb{R}}^ 2\to {\mathbb{R}}^ 2\) be an orientation and area preserving diffeomorphism which has two fixed points \(z_ 0\) and \(z_ 1\). The author proves that there are intervals I such that for any non- zero rational number p/q in I, there exists a periodic point x with the property that p/q is the difference of the rotation numbers of x around \(z_ 1\) and \(z_ 0.\)
The notion of a simple heteroclinic cycle for a diffeomorphism f of \({\mathbb{R}}^ 2\) leaving invariant a measure is introduced. If z is a fixed point of such an f, for a suitable rational number p/q there exists a periodic point x whose total rotation number around z is equal to p/q. The rotation number at infinity is defined and studied.


37A99 Ergodic theory
58C25 Differentiable maps on manifolds
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