Fixed point indices of planar continuous maps

TL;DR

This paper characterizes the fixed point index sequences of isolated fixed points for planar continuous maps, showing they are periodic and bounded by 1, enabling effective computation of Lefschetz zeta functions and advancing understanding of periodic orbits.

Contribution

It provides a complete characterization of fixed point index sequences in the plane, facilitating new computations and theoretical insights into periodic orbits and fixed point dynamics.

Findings

Fixed point index sequences are periodic and bounded by 1.

Effective computation of Lefschetz zeta functions for 2-sphere maps.

New results on the existence and growth of periodic orbits.

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.