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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.

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