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Fixed point indices of planar continuous maps

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  • We characterize the sequences of fixed point indices $\{i(f^n, p)\}_{n\ge 1}$ of fixed points that are isolated as an invariant set for a continuous map $f$ in the plane. In particular, we prove that the sequence is periodic and $i(f^n, p) \le 1$ for every $n \ge 0$. This characterization allows us to compute effectively the Lefschetz zeta functions for a wide class of continuous maps in the \(2\)-sphere, to obtain new results of existence of infinite periodic orbits inspired on previous articles of J. Franks and to give a partial answer to a problem of M. Shub about the growth of the number of periodic orbits of degree--\(d\) maps in the 2-sphere.
    Mathematics Subject Classification: 37C25, 54H25.


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