Mayer-Vietoris property of the fixed point index

Héctor Barge, Klaudiusz Wójcik



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.


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

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M. Arkowitz and R.F. Brown, The Lefschetz–Hopf theorem and axioms for the Lefschetz number, Fixed Point Theory Appl. 1 (2004), 1–11.

C. Bonatti and J. Villadelprat, The index of stable critical points, Topology Appl. 126 (2002), 263–271.

M. Bonino, Lefschetz index for orientation reversing planar homeomorphisms, Proc. Amer. Math. Soc. 130 (2002), no. 7, 2173–2177.

M. Brown, On the fixed point index of iterates of planar homeomorphisms, Proc. Amer. Math. Soc. 108 (1990) no. 4, 1109–1114.

R.F. Brown, R.E. Greene and H. Schirmer, Fixed points of map extension, In: Topological Fixed Point Theory and Aplications (Boju Jiang, ed.), Lect. Notes in Math., vol. 1411, pp. 24–45, 1989.

A. Capietto and B.M. Garay, Saturated invariant sets and the boundary behaviour of differential systems, J. Math. Anal. Appl. 176 (1993), 166–181.

E.N. Dancer and R. Ortega, The index of Lyapunov stable fixed points, J. Dynam. Differential Equations 6 (1994), 631–637.

A. Dold, Fixed point index and fixed point theorem for Euclidean neighbourhood retracts, Topology 4 (1965), 1–8.

A. Dold, Fixed point indices of iterated maps, Invent. Math. 74 (1983), 419–435.

F. Dumortier, J. Llibre and J.C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Springer, Berlin, 2006.

B.M. Garay and J. Hofbauer, Robust permanence for ecological differential equations, minimax, and discretizations, SIAM. J. Math. Anal. 34 (2003), 1007–1039.

G. Graff and J. Jezierski, Minimal number of periodic points of smooth boundarypreserving self-maps of simply-connected manifolds, J. Geom Dedicata (2016), DOI: 10.1007/s10711-016-0199-4.

G. Graff and P. Nowak-Przygodzki, Fixed point indices of iterations of planar homeomorphisms, Topol. Methods Nonlinear Anal. 22 (2003), 159–166.

L. Hernandez-Corbato and F.R. Ruiz del Portal, Fixed point indices of planar continuous maps, Discrete Contin. Dyn. Syst. 35 (2015), 2979–2995.

M.W. Hirsch, Differential Topology, Graduate Texts in Mathematics, vol. 33, Springer, New York, 1994.

J. Hofbauer, Saturated equilibria, permanence, and stability for ecological systems, Mathematical Ecology, Proc. Trieste (L.J. Gross, T.G. Hallman and S.A. Levin, eds.), World Scientific, pp. 625–642, 1986.

M. Izydorek, S. Rybicki and S. Szafraniec, A note on the Poincaré–Bendixson index theorem, Kodai Math. J. 19, (1996) no. 2, 145–156.

J. Jezierski and W. Marzantowicz, Homotopy Methods in Topological Fixed and Periodic Points Theory, Topological Fixed Point Theory and Applications, vol. 3, Springer, Berlin, 2005.

B. Jiang, Lectures on Nielsen Fixed Point Theory, Contemporary Mathematics, vol. 14, American Mathematical Society, Providence, 1983.

N. Khamsemanan, R.F. Brown, C. Lee and S. Dhompongsa, Interior fixed points of unit-sphere-preserving Euclidean maps, Fixed Point Theory Appl., 2012/1/183.

N. Khamsemanan, R.F. Brown, C. Lee and S. Dhompongsa, A fixed point theorem for smooth extension maps, Fixed Point Theory Appl., 2014/1/197.

P. Le Calvez and J.-C. Yoccoz, Un théoreme d’indice pour les homéomorphisms du plan au voisinage d’un point fixe, Ann. Math. 146 (1997), 241–293.

J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, vol. 218, Springer, New York, 2003.

J.W. Milnor, Topology from the Differentiable Viewpoint, Princeton University Press, Princeton, 1997.

P. Pelikan and E.E. Slaminka, A bound for the fixed point index of an area-preserving homeomorphism of two-manifolds, Ergodic Theory Dynam. Systems 7 (1987), 463–479.

A. Ruiz-Herrera, Permanence of two species and fixed point index, Nonlinear Anal. 74 (2011), 146–153.

F.R. Ruiz del Portal, Planar isolated and stable fixed points have index = 1, J. Differential Equations 199 (2004), 179–188.

F.R. Ruiz del Portal and J.M. Salazar, Fixed point index o,f iterations of local homeomorphism of the plane: a Conley index approach, Topology 41 (2002), 1199–1212.

F.R. Ruiz del Portal and J.M. Salazar, Indices of the iterates of R3 -homeomorphisms at Lyapunov stable fixed points, I. Differential Equations 244 (2008), 1141–1156.

R. Srzednicki, Generalized Lefschetz theorem and fixed point index formula, Topol. Appl. 81 (1997), 207–224.

A. Szymczak, K. Wójcik and P. Zgliczyński, On the discrete Conley index in the invariant subspace, Topol. Appl. 87 (1998), 105–115.

E.H. Spanier, Algebraic Topology, McGraw–Hill Book, 1966.

K. Wójcik, Conley index and permanence in dynamical systems, Topol. Methods Nonlinear Anal. 12 (1998), 153–158.

K. Wójcik, An attraction result and an index theorem for a continuous flows on Rn[0, +∞), Ann. Polon. Math. 65 (1997), 203–211.


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