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Periodic points of Hamiltonian surface diffeomorphisms. (English) Zbl 1034.37028
The main result is that every nontrivial diffeomorphism \(F\) of a closed oriented surface of genus at least one has periodic points of arbitrarily high periods. The same is true for \(S^2\) provided the diffeomorphism has at least three fixed points. Furthermore as a consequence the authors prove that there exist \(n>0\) and a finite set \(P \subset \text{Fix}(F)\) such that \(F^n\) is not isotopic to the identity relative to \(P\). And also, that there exist \(n>0\) and \(p>0\) such that \(F^n\) has \(k\) periodic points for every \(k\geq p\). A second result of the paper, which depends on a result of Thurston says that every orientation-preserving diffeomorphism of a closed orientable surface has a normal form up to isotopy. The main technical work is an analysis of the dynamics of an element \(f \in \mathcal P(S)\), the set of diffeomorphisms of the surface \(S\) that are isotopic to the identity relative to the set of periodic points of \(f\). From this study they construct examples and they prove that the identity is the only area-preserving element of \(\mathcal P(S)\) that fixies at least three points.

37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
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