Abstract
Let \({f: U\rightarrow {\mathbb R}^2}\) be a continuous map, where U is an open subset of \({{\mathbb R}^2}\). We consider a fixed point p of f which is neither a sink nor a source and such that {p} is an isolated invariant set. Under these assumption we prove, using Conley index methods and Nielsen theory, that the sequence of fixed point indices of iterations \({\{{\rm ind}(f^n,p)\}_{n=1}^\infty}\) is periodic, bounded from above by 1, and has infinitely many non-positive terms, which is a generalization of Le Calvez and Yoccoz theorem (Annals of Math., 146, 241–293 (1997)) onto the class of non-injective maps. We apply our result to study the dynamics of continuous maps on 2-dimensional sphere.
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Graff, G., Nowak-Przygodzki, P. & Ruiz del Portal, F.R. Local Fixed Point Indices of Iterations of Planar Maps. J Dyn Diff Equat 23, 213–223 (2011). https://doi.org/10.1007/s10884-011-9204-7
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DOI: https://doi.org/10.1007/s10884-011-9204-7