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By Bonino Marc (f-paris-gl


A dynamical property for planar homeomorphisms and an application to the problem of canonical position around an isolated fixed point. (English summary) Topology 40 (2001), no. 6, 1241–1257. The author first proves a new kind of fixed point theorem, namely: Let h be a homeomorphism of the plane and let D be a closed disc. Assume we can write the boundary of D as the union of two arcs α and β meeting only in their endpoints such that D ⋂ h −1 (β) = ∅ and D ∩ h(α) = ∅. Then, if ⋂ n≥0 h−n (D) = ∅, there exists m ∈ D such that h(m) ∈ D and h 2 (m) = m. If h preserves orientation, we can choose m such that h(m) = m. The strategy of the proof is to find a component of ⋂ n≥0 h−n (D) to which applies the Cartwright-Littlewood-Bell fixed point theorem [H. Bell, Fund. Math. 100 (1978), no. 2, 119–128; MR0500879 (58 #18386)]. This theorem is used to give a corrected and completed proof of Schmitt’s result saying that, in the space of orientationpreserving homeomorphisms of the plane with 0 as unique fixed point, those of fixed index form a path connected subspace for the compact-open topology [B. V. Schmitt, Topology 18 (1979), no. 3, 235–240; MR0546793 (81a:57036)]. Reviewed by L. Guillou (Saint-Martin-d’Hères

Year: 2013
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