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Indices of the iterates of \(\mathbb R^3\)-homeomorphisms at Lyapunov stable fixed points. (English) Zbl 1132.37012
Summary: Given any positive sequence \(\{c_n\}_{n\in\mathbb N}\), we construct orientation preserving homeomorphisms \(f:\mathbb R^3\to \mathbb R^3\) such that \(\text{Fix}(f)=\text{Per}(f)=\{0\}\), 0 is Lyapunov stable and \(\lim\sup\frac{|i(f^m,0)|}{c_m}=\infty\). We will use our results to discuss and to point out some strong differences with respect to the computation and behavior of the sequences of the indices of planar homeomorphisms.

37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37B30 Index theory for dynamical systems, Morse-Conley indices
54H25 Fixed-point and coincidence theorems (topological aspects)
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[1] Babenko, I.K.; Bogatyi, S.A., The behavior of the index of periodic points under iterations of a mapping, Math. USSR izvestiya, 38, 1-26, (1992) · Zbl 0742.58027
[2] Brechner, B.L.; Lee, J.S., A three dimensional prime end theory, Topology proc., 20, 15-47, (1995) · Zbl 0873.57012
[3] Bonatti, C.; Villadelprat, J., The index of stable critical points, Topology appl., 126, 1-2, 263-271, (2002) · Zbl 1013.37016
[4] Brown, R.F., The Lefschetz fixed point theorem, (1971), Scott Foresman Co. Glenview, IL · Zbl 0216.19601
[5] Chow, S.N.; Mallet-Paret, J.; Yorke, J.A., A periodic orbit index which is a bifurcation invariant, (), 109-131
[6] Dancer, E.N.; Ortega, R., The index or Lyapunov stable fixed points, J. dynam. differential equations, 6, 631-637, (1994) · Zbl 0811.34018
[7] Dold, A., Fixed point indices of iterated maps, Invent. math., 74, 419-435, (1983) · Zbl 0583.55001
[8] Erle, E., Stable equilibria and vector field index, Topology appl., 49, 231-235, (1993) · Zbl 0777.58032
[9] Franks, J.; Richeson, D., Shift equivalence and the Conley index, Trans. amer. math. soc., 352, 7, 3305-3322, (2000) · Zbl 0956.37010
[10] Graff, G.; Nowak-Przygodzki, P., Fixed point indices of iterations of \(C^1\)-maps in \(\mathbb{R}^3\), Discrete contin. dyn. syst., 4, 843-856, (2006) · Zbl 1185.37043
[11] Hu, S.T., Theory of retracts, (1965), Wayne State University Press · Zbl 0137.01701
[12] Jezierski, J.; Marzantowicz, W., Homotopy methods in topological fixed and periodic points theory, (2005), Springer
[13] Le Calvez, P.; Yoccoz, J.C., Un théoréme d’indice pour LES homéomorphismes du plan au voisinage d’un poin fixe, Ann. of math., 146, 241-293, (1997) · Zbl 0895.58032
[14] P. Le Calvez, J.C. Yoccoz, Suite des indices de Lefschetz des itérés pour un domaine de Jordan qui est un bloc isolant, unpublished
[15] Le Calvez, P., Dynamique des homéomorphismes du plan au voisinage d’un point fixe, Ann. sci. ecole norm. sup. (4), 36, 1, 139-171, (2003) · Zbl 1017.37017
[16] Nussbaum, R.D., The fixed point index and some applications, () · Zbl 0174.45402
[17] Ruiz del Portal, F.R., Planar isolated and stable fixed points have index =1, J. differential equations, 199, 179-188, (2004) · Zbl 1052.37011
[18] Ruiz del Portal, F.R.; Salazar, J.M., Fixed point index of iterations of local homeomorphisms of the plane: A Conley-index approach, Topology, 41, 1199-1212, (2002) · Zbl 1009.54043
[19] F.R. Ruiz del Portal, J.M. Salazar, A Poincaré formula for the fixed point indices of the iterations of arbitrary planar homeomorphisms, preprint · Zbl 1196.54068
[20] F.R. Ruiz del Portal, J.M. Salazar, Fixed point indices of the iterations of \(\mathbb{R}^3\)-homeomorphisms, preprint · Zbl 1009.54043
[21] Shub, M.; Sullivan, D., A remark on the Lefschetz fixed point formula for differentiable maps, Topology, 13, 189-191, (1974) · Zbl 0291.58014
[22] ()
[23] Wilson, F.W., On the minimal sets of non-singular vector fields, Ann. of math., 84, 529-536, (1966) · Zbl 0156.43803
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