Complutense University Library

Local fixed point indices of iterations of planar maps

Romero Ruiz del Portal, Francisco and Graff, Grzegorz and Nowak-Przygodzki, Piotr (2011) Local fixed point indices of iterations of planar maps. Journal of Dynamics and Differential Equations, 23 (1). pp. 213-223. ISSN 1040-7294

[img]
Preview
PDF
200kB

Official URL: http://www.springerlink.com/openurl.asp?genre=journalissn=1040-7294

View download statistics for this eprint

==>>> Export to other formats

Abstract

Let f : U →R2 be a continuous map, where U is an open
subset of R2. We consider a fixed point p of f which is neither a sink nor
a source and such that p is an isolated invariant set. Under these assumption
we prove, using Conley index methods and Nielsen theory, that the sequence of fixed point indices of iterations ind(fn, p) n=1 is periodic,bounded by 1, and has infinitely many non-positive terms, which is a generalization of Le Calvez and Yoccoz theorem [Annals of Math., 146 (1997), 241-293] onto the class of non-injective maps. We apply our result to study the dynamics of continuous maps on 2-dimensional
sphere.


Item Type:Article
Uncontrolled Keywords:Fixed point index; Conley index, Nielsen number; Periodic points; Iterations
Subjects:Sciences > Mathematics > Topology
ID Code:13994
References:

[1] I. K. Babenko and S. A. Bogatyi, The behavior of the index of periodic points under iterations of a mapping, Math. USSR Izv., 38 (1992), 1-26.

[2] N. Bernardes, On the set of points with a dense orbit, Proc. Amer. Math. Soc. 128 (2000), no. 11, 3421-3423.

[3] S. A. Bogatyi, Local indices of iterations of a holomorphic mapping. (Russian) General topology. Spaces and mappings, 48{61, Moskov. Gos. Univ., Moscow, 1989.

[4] S. N. Chow, J. Mallet-Parret, J. A. Yorke, A periodic point index which is a bifurcation invariant, Geometric dynamics (Rio de Janeiro, 1981), 109-131, Springer Lecture Notes in Math. 1007, Berlin 1983.

[5] A. Dold, Fixed point indices of iterated maps, Invent. Math. 74 (1983), 419-435.

[6] N. Fagella, J. Llibre, Periodic points of holomorphic maps via Lefschetz numbers, Trans. Amer. Math. Soc. 352 (2000), no. 10, 4711-4730.

[7] J. Franks, The Conley index and non-existence of minimal homeomorphisms, Illinois J. Math., 43 (1999), no. 3, 457-464.

[8] J. Franks and D. Richeson, Shift equivalence and the Conley index Trans. Amer. Math. Soc. 352 (2000), no. 7, 3305-3322.

[9] G. Graff and P. Nowak-Przygodzki, Fixed point indices of iterations of C1 maps in R3, Discrete Cont. Dyn. Syst. 16 (2006), no. 4, 843-856.

[10] G. Graff and P. Nowak-Przygodzki, Sequences of fixed point indices of iterations in dimension 2, Univ. Iagel. Acta Math., XLI (2003), 135-140.

[11] J. Jezierski and W. Marzantowicz, Homotopy methods in topological fixed and periodic points theory, Topological Fixed Point Theory and Its Applications, 3. Springer, Dordrecht, 2005.

[12] B. J. Jiang, Bounds for fixed points on surfaces, Math. Ann. 311 (1998), no. 3, 467-479.

[13] B. J. Jiang, Lectures on the Nielsen Fixed Point Theory, Contemp. Math. 14, Amer. Math. Soc., Providence 1983.

[14] P. Le Calvez, Dynamique des homomorphismes du plan au voisinage d'un point fixe, Ann. Sci. École Norm. Sup. (4) 36, (2003), no. 1, 139-171.

[15] P. Le Calvez and J.-C. Yoccoz, Un théoreme d'indice pour les hom_eomorphismes du plan au voisinage d'un point fixe, Annals of Math., 146 (1997), 241-293.

[16] K. Mischaikow, M. Mrozek, Conley index. Handbook of dynamical systems, Vol. 2, 393-460, North-Holland, Amsterdam, 2002.

[17] W. Marzantowicz and P. Przygodzki, Finding periodic points of a map by use of a k-adic expansion, Discrete Contin. Dyn. Syst. 5 (1999), 495-514.

[18] R. Daniel Mauldin (red), The Scottish Book, Birkhäuser, Boston 1981.

[19] F. Ruiz del Portal and J.M. Salazar, A Poincaré formula for the fixed point indices of the iterations of planar homeomorphisms, preprint.

[20] F. Ruiz del Portal, J.M Salazar, A stable/unstable manifold theorem for local homeo-morphisms of the plane, Ergodic Theory Dynam. Systems 25 (2005), no. 1, 301-317.

[21] 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), no. 6, 1199-1212.

[22] F. Ruiz del Portal and J.M. Salazar, Indices of the iterates of R3-homeomorphisms at Lyapunov stable fixed points, J. Diff. Eq. 244 no. 5 (2008), 1141-1156.

[23] M. Shub, P. Sullivan, A remark on the Lefschetz fixed point formula for differentiable maps, Topology 13 (1974), 189-191.

[24] G.Y. Zhang, Fixed point indices and invariant periodic sets of holomorphic systems, Proc. Amer. Math. Soc. 135 (2007), no. 3, 767-776 (electronic).

[25] G.Y. Zhang, Fixed point indices and periodic points of holomorphic mappings, Math. Ann. 337 (2007), no. 2, 401-433.

Deposited On:07 Dec 2011 09:12
Last Modified:06 Feb 2014 09:57

Repository Staff Only: item control page