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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:

[1] C. Conley, Iso ated Invariant Sets and the Morse index,C.B.M.S. Regional Conference Series in Math.,38 (1978), Amer. Math. Soc. Providence, R.I. · Zbl 0397.34056
[2] J. Franks, Recurrence and Fixed Points of Surface Homeomorphisms,Ergodic Theory and Dyn. Systems,8* (1988), 99–107. · Zbl 0634.58023
[3] J. Franks, A Variation on the Poincaré-Birkhoff Theorem, ”Hamiltonian Dynamics”, Contemporary Math.,Amer. Math. Soc.,81, 111–116.
[4] W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces,Bull. Amer. Math. Soc.,19 (1988), 417–431. · Zbl 0674.57008
[5] W. Wilson, Smoothing derivatives of functions and applications,Trans. Amer. Math. Soc.,139 (1969), 416–428 · Zbl 0175.20203
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