## 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.

### MSC:

 37A99 Ergodic theory 58C25 Differentiable maps on manifolds
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### References:

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