A Poincaré formula for the fixed point indices of the iterates of arbitrary planar homeomorphisms

Abstract

Let U ⊂ ℝ2 be an open subset and f:U → ℝ2 be an arbitrary local homeomorphism with Fix(f)={p}. We compute the fixed point indices of the iterates of f at p, i ℝ2 (fk,p), and we identify these indices in dynamical terms. Therefore, we obtain a sort of Poincaré index formula without differentiability assumptions. Our techniques apply equally to both orientation preserving and orientation reversing homeomorphisms. We present some new results, especially in the orientation reversing case. Copyright © 2010 F. R. Ruiz del Portal and J. M. Salazar.