Mayer-Vietoris property of the fixed point index

Héctor Barge, Klaudiusz Wójcik

DOI: http://dx.doi.org/10.12775/TMNA.2017.034

Abstract


We study a Mayer-Vietoris kind formula for the fixed point index of maps of ENR triplets $f\colon (X;X_1,X_2)\to (X;X_1,X_2) $ having compact fixed point set. We prove it under some suitable conditions. For instance when $(X;X_1,X_2)=(E^n;E^n_+,E^n_-)$. We use these results to generalize the Poincaré-Bendixson index formula for vector fields to continuous maps having a \emph{sectorial decomposition}, to study the fixed point index $i(f,0)$ of orientation preserving homeomorphisms of $E^2_+$ and $(E^3;E^3_+,E^3_-)$ and the fixed point index in the invariant subspace.

Keywords


Fixed point index; Brouwer degree; sectorial decomposition; proper pair; isolated invariant set

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