New proofs of some results of Nielsen.

*(English)*Zbl 0584.57007The paper contains an excellently written short proof of results (close to those) of J. Nielsen on the types of homeomorphisms of surfaces. The authors prove the following theorem which is in this form due to R. T. Miller [ibid. 45, 189-212 (1982; Zbl 0496.57003)]. Theorem. Let \(\tau\) : \(M^ 2\to M^ 2\) be a homeomorphism of a compact orientable surface with negative Euler characteristic. Then \(\tau\) is isotopic to a homeomorphism \(\phi\) with one of the following properties: i) \(\phi^ n\) is isotopic to the identity; ii) \(\phi\) preserves a pair of transversal geodesic laminations (a configuration close to a foliation) which intersect every closed geodesic in M and have leaves only being dense in the lamination; iii) there is a finite collection \(\gamma\) of simple closed curves such that \(\phi\) permutes the components of an open regular neighbourhood of \(\gamma\). The powers of \(\phi\) preserving some component of the complement of the regular neighbourhood define there a mapping of type i) or ii). The mappings of type ii) are similar to the so-called pseudo-Anosov homeomorphisms.

Reviewer: H.Zieschang

##### MSC:

57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |

57R50 | Differential topological aspects of diffeomorphisms |

37-XX | Dynamical systems and ergodic theory |

##### Keywords:

homeomorphisms of surfaces; geodesic laminations; simple closed curves; pseudo-Anosov homeomorphisms
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\textit{M. Handel} and \textit{W. P. Thurston}, Adv. Math. 56, 173--191 (1985; Zbl 0584.57007)

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##### References:

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