An index theorem for the homeomorphisms of the plane near a fixed point
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 \textit{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.
scientific article; zbMATH DE number 1088562