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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.

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