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An index theorem for the homeomorphisms of the plane near a fixed point. (Un théorème d’indice pour les homéomorphismes du plan au voisinage d’un point fixe.) (French) Zbl 0895.58032
This investigation was motivated by the following question: does the infinite annulus, $$\mathbb R\times (\mathbb R/\mathbb Z)$$, admit a minimal homeomorphism? The authors compute the index, $$i(f^k,z)$$, of a sequence of $$k$$-th interates, $$k=1,2,\dots$$, of a local homeomorphism of $$\mathbb R^2$$ at a fixed point $$z$$ which forms a locally maximal invariant set and is neither a sink nor a source. For suitably chosen integers $$q,r\geq 1$$ it satisfies the following condition: $i(f^k,z)=\begin{cases} 1-rq, & \text{if $$k$$ is a multiple of $$q$$,} \\ 1, & \text{otherwise.} \end{cases}$ Cyclically ordered sets play an important technical role in the proofs.
From the above result the authors deduce that the two-sphere $$\mathbb S^2$$, punctured finitely many times, admits no minimal homeomorphism. This important and definitive result includes, as particular cases, old results on the non-existence of minimal homeomorphisms on $$\mathbb S^2$$ and on $$\mathbb R^2$$, as well as the recent result by M. Handel [Ergodic Theory Dyn. Syst. 12, No. 1, 75-83 (1992; Zbl 0769.58037)], stating that $$\mathbb S^2$$ with $$\geq 3$$ punctures admits no minimal homeomorphism. Thus, the question at the beginning of this review is answered negatively.

##### MSC:
 37B99 Topological dynamics 37C80 Symmetries, equivariant dynamical systems (MSC2010) 54H20 Topological dynamics (MSC2010)
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