Fixed point indices of planar continuous maps

Luis Hernández-Corbato; Francisco R. Ruiz del Portal

doi:10.3934/dcds.2015.35.2979

Article Contents

2015, Volume 35, Issue 7: 2979-2995. Doi: 10.3934/dcds.2015.35.2979

This issue Previous Article Symmetries of the periodic Toda lattice, with an application to normal forms and perturbations of the lattice with Dirichlet boundary conditions Next Article Steplength thresholds for invariance preserving of discretization methods of dynamical systems on a polyhedron

Fixed point indices of planar continuous maps

Luis Hernández-Corbato^1, and
Francisco R. Ruiz del Portal^2,

1.
IMPA, Estrada dona Castorina 110, Rio de Janeiro
2.
Facultad de Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, Madrid

Received: March 31, 2014

Revised: November 30, 2014

Published: May 2017

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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 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.

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Mathematics Subject Classification: 37C25, 54H25.

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