Publication: Mayer-vietoris property of the fixed point index
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2017
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In this paper we study a Mayer-Vietoris kind of formula for the fixed point index of maps of ENR triplets f : (X;X1,X2) → (X;X1,X2) having compact fixed point set. We prove it under some suitable conditions. For instance when (X;X1,X2) = (En;En+,En −). We use these results to generalize Poincar´e-Bendixson index formula for vector fields to continuos maps having a sectorial decomposition, to study the fixed point index i(f, 0) of orientation preserving homeomorphisms of E2 + and (E3;E3 +,E3 −) and the fixed point index in the invariant subspace.
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1. M. Arkowitz , R. F. Brown, The Lefschetz-Hopf theorem and axioms for the Lefschetz number , Fixed Point
Theory Appl. 1 (2004), 1–11.
2. C. Bonatti, J. Villadelprat, The index of stable critical points, Topology Appl. 126, (2002), 263-271.
3. M. Bonino, Lefschetz index for orientation reversing planar homeomorphisms, Proc. AMS 130 (2002) (7),
2173-2177.
4. M. Brown, On the fixed point index of iterates of planar homeomorphisms, Proc. AMS 108 (1990) (4),
1109-1114.
5. R. F. Brown, R. E. Greene, H. Schirmer, Fixed points of map extension, in Boju Jiang (Ed.), Topological
Fixed Point Theory and Aplications, Lect. Notes in Math. 1411 (1989), pp. 24–45.
6. A. Capietto, B. M. Garay, Saturated invariant sets and the boundary behaviour of differential Systems, J.
Math. Anal. Appl. 176, (1993), 166–181.
7. E. N. Dancer, R. Ortega, The index of Lyapunov stable fixed points, J. Dyn. Diff. Eq. 6 (1994), 631–637.
8. A. Dold, Fixed point index and fixed point theorem for Euclidean neighborhood retracts, Topology 4 (1965),
1–8.
9. A. Dold, Fixed point indices of iterated maps, Invent. Math. 74 (1983), 419–435.
10. F. Dumortier, J. Llibre, J.C. Art´es, Qualitative Theory of Planar Differential Systems, Series: Universitext, Springer-Verlag, Berl´ın, 2006.
11. B. M. Garay, J. Hofbauer, Robust permanence for ecological differential equations, minimax, and discretizations, SIAM. J. Math. Anal. 34 (2003) 1007–1039.
12. G. Graff, P. Nowak-Przygodzki, Fixed point indices of iterations of planar homeomorphisms, Topol. Meth.
Nonl. Anal. 22 (2003), 159-166.
13. G. Graff, J. Jezierski, Minimal number of periodic points of smooth boundary-preserving self-maps of simplyconnected manifolds, J. Geom Dedicata (2016) doi:10.1007/s10711-016-0199-4.
14. L. Hernandez-Corbato, F. R. Ruiz del Portal, Fixed point indices of planar continuous maps, Discrete and
Cont. Dyn. Sys. 35, (2015), 2979–2995.
15. M. W. Hirsch, Differential Topology, Series: Graduate Texts in Mathematics, vol. 33, Springer-Verlag, NewYork 1994.
16. J. Hofbauer, Saturated equilibria, permanence, and stability for ecological systems, in “Mathematical Ecology, Proc. Trieste, 1986” (L. J. Gross, T. G. Hallman, and S. A. Levin, Eds.), pp. 625–642, World Scientific.
17. M. Izydorek, S. Rybicki, Z. Szafraniec, A note on the Poincar´e-Bendixson index theorem, Kodai Math. J. 19,
(1996) (2), 145–156.
18. J. Jezierski, W. Marzantowicz, Homotopy Methods in Topological Fixed and Periodic Points Theory, Series:
Topological Fixed Point Theory and Appl., vol. 3, Springer, 2005.
19. B. Jiang, Lectures on Nielsen Fixed Point Theory, Series: Contemporary Mathematics, vol. 14, Amer. Math.
Soc., Providence, RI, 1983.
20. N. Khamsemanan, R. F. Brown, C. Lee, S. Dhompongsa, Interior fixed points of unit-sphere-preserving
Euclidean maps, Fixed Point Th. Appl., 2012/1/183.
21. N. Khamsemanan, R. F. Brown, C. Lee, S. Dhompongsa, A fixed point theorem for smooth extension maps,
Fixed Point Th. Appl. , 2014/1/197.
22. P. Le Calvez, J.-C. Yoccoz, Un th´eoreme d’indice pour les hom´eomorphisms du plan au voisinage d’un point
fixe, Ann. Math. 146, (1997), 241–293.
23. J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, 218, Springer-Verlag, New
York 2003.
24. J.W. Milnor, Topology from the Differentiable Viewpoint, Princeton University Press , 1997.
25. P. Pelikan, E. E. Slaminka, A bound for the fixed point index of an area-preserving homeomorphism of
two-manifolds, Ergod. Th. Dyn. Syst. 7 (1987), 463–479.
26. A. Ruiz-Herrera, Permanence of two species and fixed point index, Nonlinear Anal. 74 (2011) 146153.
27. F. R. Ruiz del Portal, Planar isolated and stable fixed points have index = 1, J. Diff. Eq. 199, (2004),
179–188.
28. F. R. Ruiz del Portal, J. M. Salazar, Fixed point index o,f iterations of local homeomorphism of the plane:
a Conley index approach, Topology 41 (2002), 1199–1212.
29. F. R. Ruiz del Portal, J. M. Salazar, Indices of the iterates of R 3-homeomorphisms at Lyapunov stable fixed
points, I. Diff. Eq. 244 (2008), 1141–1156.
30. R. Srzednicki, Generalized Lefschetz Theorem and Fixed Point Index Formula, Topol. Appl. 81, (1997),
207-224.
31. A. Szymczak, K. W´ojcik, P. Zgliczy´nski, On the discrete Conley index in the invariant subspace, Topol. Appl. 87, (1998), 105–115.
32. E. H. Spanier, Algebraic topology, McGraw-Hill Book Co., 1966.
33. K. W´ojcik, Conley index and permanence in dynamical systems, Top. Meth. Nonl. Anal. 12 (1998), 153–158.
34. K. W´ojcik, An attraction result and an index theorem for a continuous flows on Rn × [0, +∞), Ann. Polon.
Math. 65 (1997), 203–211.