Indices of fixed points not accumulated by periodic points

Luis Hernández-Corbato

DOI: http://dx.doi.org/10.12775/TMNA.2016.021

Abstract


We prove that for every integer sequence $I$ satisfying Dold relations there exists a map $f \colon \mathbb{R}^d\!\to\! \mathbb{R}^d$, $d\! \ge\! 2$, such that $\rom{Per}(f)\! =\! \rom{Fix}(f) \!=\! \{o\}$, where $o$ denotes the origin, and $(i(f^n, o))_n = I$.

Keywords


Fixed point index; Dold relations

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References


A. Aberkane and J. Currie, There exist binary circular 5/2+ power free words of every length, Electron. J. Combin. 11 (2004), no. 1.

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 of iterates of planar homeomorphisms, Proc. Amer. Math. Soc. 108 (1990), no. 4, 1109–1114.

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

G. Graff, J. Jezierski and P. Nowak-Przygodzki, Fixed point indices of iterated smooth maps in arbitrary dimension, J. Differential Equations 251 (2011), no. 6, 1526–1548.

G. Graff and P. Nowak–Przygodzki, Sequences of fixed point indices of iterations in dimension 2, Univ. Iagel. Acta Math. No. 41 (2003), 135–140.

G. Graff, P. Nowak-Przygodzki and F.R. Ruiz del Portal, Local fixed point indices of iterations of planar maps, J. Dynam. Differential Equations 23 (2011), no. 1, 213–223.

L. Hernández-Corbato, P. Le Calvez and F.R. Ruiz del Portal, About the homological Conley index of invariant acyclic continua, Geom. Topol. 17 (2013), 2977–3026.

L. Hernández-Corbato and F. R. Ruiz del Portal, Fixed point indices of planar continuous maps, Discrete Contin. Dyn. Syst. (to appear).

J. Jezierski and W. Marzantowicz, Homotopy Methods in Topological Fixed and Periodic Points Theory, Topological Fixed Point Theory and Its Applications, Vol. 3, Springer, Dordrecht, 2006)

P. Le Calvez, Dynamique des homé omorphismes du plan au voisinage d’un point fixe, Ann. Sci. Éc. Norm. Suppér. (4) 36 (2003), 139–171.

P. Le Calvez, F.R. Ruiz del Portal and J.M. Salazar, Fixed point indices of the iterates of R3 -homeomorphisms at fixed points which are isolated invariant sets, J. London Math. Soc. 82 (2010), no. 2, 683–696.

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

P. Le Calvez and J.C. Yoccoz, Suite des indices de Lefschetz des itérés pour un domaine de Jordan qui est un bloc isolant, (unpublished).

F. Le Roux, Dynamique des homéomorphismes de surfaces, Versions topologiques des théorémes de la fleur de Leau–Fatou et de la variété stable, Astérisque 292 (2004).

M. Morse, Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc. 22 (1921), 84–100.

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

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

F.R. Ruiz del Portal and J.M. Salazar, A Poincaré formula for the fixed point indices of the iterations of planar homeomorphisms, Fixed Point Theory Appl. ID233069 (2010), 1–31.

M. Shub and D. Sullivan, A remark on the Lefschetz fixed point formula for differentiable maps, Topology 13 (1974), 189–191.

A. Thue, Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, Norske Vid. Selsk. Skr., I. Mat.–Nat. Kl., Christiania 1 (1912), 1–67.


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