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Local fixed point indices of iterations of planar maps. (English) Zbl 1230.37029
Let \(f: U \rightarrow \mathbb R^2\) be a continuous map on an open subset \(U\) of \(\mathbb R^2\), and let \(p\) be a fixed point of \(f\) which is neither a source nor a sink. Consider \(S_p: = (\mathrm{ind}(f^n,p))^\infty_{n=1}\), the sequence of fixed point indices of the iterates of \(f\). If \(\{p\}\) is an isolated invariant set for \(f\) then it is proved that \(S_p\) is periodic, bounded above by \(1\), and has infinitely many non-positive terms.
This generalizes a result by P. Le Calvez and J.-C. Yoccoz [Ann. Math. (2) 146, No. 2, 241–293 (1997; Zbl 0895.58032)] which gives a rather precise description of \(S_p\) if \(f\) is an orientation preserving local homeomorphism. The hypothesis that \(\{p\}\) is an isolated invariant set amounts to assuming that there is a compact neighborhood \(N\) of \(p\) such that any two-sided sequence \((\sigma(n))^\infty_{n=-\infty}\) in \(N\) satisfying \(\sigma(n+1)=f(\sigma(n))\) for all \(n \in \mathbb Z\) is the constant sequence with value \(p\). Applications are given that concern the existence of minimal maps on the \(2\)-sphere.
The methods employed involve replacing \(f\) by a self-map on a suitable space allowing to compute the index at \(p\) by the Lefschetz-Hopf theorem. Another ingredient of the proof is a result of B. Jiang from Nielsen theory of surfaces.

MSC:
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37B30 Index theory for dynamical systems, Morse-Conley indices
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