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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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