Fixed point indices of planar continuous maps

Abstract

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 and continuous maps in the plane. In particular, we prove that the sequence is periodic and $i(f^n,p)\le 1$ for every $n\ge 1$. 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 Shub about the growth of the number of periodic orbits of degree--$d$ maps in the 2-sphere.

15 pages, 4 figures. Final version published in DCDS