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July  2015, 35(7): 2979-2995. doi: 10.3934/dcds.2015.35.2979

Fixed point indices of planar continuous maps

1. 

IMPA, Estrada dona Castorina 110, Rio de Janeiro, Brazil

2. 

Facultad de Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, Madrid, Spain

Received  April 2014 Revised  December 2014 Published  January 2015

We characterize the sequences of fixed point indices $\{i(f^n, p)\}_{n\ge 1}$ of fixed points that are isolated as an invariant set for a continuous map $f$ in the plane. In particular, we prove that the sequence is periodic and $i(f^n, p) \le 1$ for every $n \ge 0$. This characterization allows us to compute effectively the Lefschetz zeta functions for a wide class of continuous maps in the \(2\)-sphere, to obtain new results of existence of infinite periodic orbits inspired on previous articles of J. Franks and to give a partial answer to a problem of M. Shub about the growth of the number of periodic orbits of degree--\(d\) maps in the 2-sphere.
Citation: Luis Hernández-Corbato, Francisco R. Ruiz del Portal. Fixed point indices of planar continuous maps. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2979-2995. doi: 10.3934/dcds.2015.35.2979
References:
[1]

I. K. Babenko and S. A. Bogatyi, The behavior of the index of periodic points under iterations of a mapping,, Math. USSR Izvestiya, 38 (1992), 1.   Google Scholar

[2]

M. Brown, On the fixed point of iterates of planar homeomorphisms,, Proc. Amer. Math. Soc., 108 (1990), 1109.  doi: 10.1090/S0002-9939-1990-0994772-9.  Google Scholar

[3]

A. Dold, Fixed point indices of iterated maps,, Invent. Math., 74 (1983), 419.  doi: 10.1007/BF01394243.  Google Scholar

[4]

R. Easton, Isolating blocks and epsilon chains for maps,, Physica D, 39 (1989), 95.  doi: 10.1016/0167-2789(89)90041-9.  Google Scholar

[5]

J. Franks, Some Smooth Maps with Infinitely Many Hyperbolic Peridoic Points,, Trans. Am. Math. Soc., 226 (1977), 175.   Google Scholar

[6]

J. Franks, Homology and Dynamical Systems,, CBMS Regional Conf. Ser. in Math., (1982).   Google Scholar

[7]

J. Franks, The Conley index and non-existence of minimal homeomorphims,, Illinois J. Math. Soc., 43 (1999), 457.   Google Scholar

[8]

J. Franks and D. Richeson, Shift equivalence and the Conley index,, Trans. Amer. Math. Soc., 352 (2000), 3305.  doi: 10.1090/S0002-9947-00-02488-0.  Google Scholar

[9]

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

[10]

G. Graff, P. Nowak-Przygodzki and F. R. Ruiz del Portal, Local fixed point indices of iterations of planar maps,, J. Dynam. Differ. Equat., 23 (2011), 213.  doi: 10.1007/s10884-011-9204-7.  Google Scholar

[11]

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.  doi: 10.2140/gt.2013.17.2977.  Google Scholar

[12]

J. Iglesias, A. Portela, A. Rovella and J. Xavier, Periodic points for annulus endomorphisms,, preprint, (2014).   Google Scholar

[13]

J. Jezierski and W. Marzantowicz, Homotopy Methods in Topological Fixed and Periodic Points Theory,, Topological Fixed Point Theory and Its Applications, (2006).   Google Scholar

[14]

P. Le Calvez, Dynamique des homéomorphismes du plan au voisinage d'un point fixe,, Ann. Scient. Éc. Norm. Sup., 36 (2003), 139.  doi: 10.1016/S0012-9593(03)00005-3.  Google Scholar

[15]

P. Le Calvez, F. R. Ruiz del Portal and J. M. Salazar, Fixed point indices of the iterates of $\mathbbR^3$-homeomorphisms at fixed points which are isolated invariant sets,, J. London Math. Soc., 82 (2010), 683.  doi: 10.1112/jlms/jdq050.  Google Scholar

[16]

P. Le Calvez and J. C. Yoccoz, Un theoréme d'indice pour les homéomorphismes du plan au voisinage d'un point fixe,, Annals of Math., 146 (1997), 241.  doi: 10.2307/2952463.  Google Scholar

[17]

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., ().   Google Scholar

[18]

K. Mischaikow and M. Mrozek, Conley index,, in Handbook of Dynamical Systems, (2002), 393.  doi: 10.1016/S1874-575X(02)80030-3.  Google Scholar

[19]

C. Pugh and M. Shub, Periodic points on the 2-sphere,, Discrete Contin. Dyn. Syst., 34 (2014), 1171.  doi: 10.3934/dcds.2014.34.1171.  Google Scholar

[20]

D. Richeson and J. Wiseman, A fixed point theorem for bounded dynamical systems,, Illinois Journal of Mathematics, 46 (2002), 491.   Google Scholar

[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), 1199.  doi: 10.1016/S0040-9383(01)00035-0.  Google Scholar

[22]

M. Shub, All, most, some differentiable dynamical systems,, in Proceedings of the International Congress of Mathematicians, (2006), 99.   Google Scholar

[23]

M. Shub and D. Sullivan, A remark on the Lefschetz fixed point formula for differentiable maps,, Topology, 13 (1974), 189.  doi: 10.1016/0040-9383(74)90009-3.  Google Scholar

show all references

References:
[1]

I. K. Babenko and S. A. Bogatyi, The behavior of the index of periodic points under iterations of a mapping,, Math. USSR Izvestiya, 38 (1992), 1.   Google Scholar

[2]

M. Brown, On the fixed point of iterates of planar homeomorphisms,, Proc. Amer. Math. Soc., 108 (1990), 1109.  doi: 10.1090/S0002-9939-1990-0994772-9.  Google Scholar

[3]

A. Dold, Fixed point indices of iterated maps,, Invent. Math., 74 (1983), 419.  doi: 10.1007/BF01394243.  Google Scholar

[4]

R. Easton, Isolating blocks and epsilon chains for maps,, Physica D, 39 (1989), 95.  doi: 10.1016/0167-2789(89)90041-9.  Google Scholar

[5]

J. Franks, Some Smooth Maps with Infinitely Many Hyperbolic Peridoic Points,, Trans. Am. Math. Soc., 226 (1977), 175.   Google Scholar

[6]

J. Franks, Homology and Dynamical Systems,, CBMS Regional Conf. Ser. in Math., (1982).   Google Scholar

[7]

J. Franks, The Conley index and non-existence of minimal homeomorphims,, Illinois J. Math. Soc., 43 (1999), 457.   Google Scholar

[8]

J. Franks and D. Richeson, Shift equivalence and the Conley index,, Trans. Amer. Math. Soc., 352 (2000), 3305.  doi: 10.1090/S0002-9947-00-02488-0.  Google Scholar

[9]

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

[10]

G. Graff, P. Nowak-Przygodzki and F. R. Ruiz del Portal, Local fixed point indices of iterations of planar maps,, J. Dynam. Differ. Equat., 23 (2011), 213.  doi: 10.1007/s10884-011-9204-7.  Google Scholar

[11]

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.  doi: 10.2140/gt.2013.17.2977.  Google Scholar

[12]

J. Iglesias, A. Portela, A. Rovella and J. Xavier, Periodic points for annulus endomorphisms,, preprint, (2014).   Google Scholar

[13]

J. Jezierski and W. Marzantowicz, Homotopy Methods in Topological Fixed and Periodic Points Theory,, Topological Fixed Point Theory and Its Applications, (2006).   Google Scholar

[14]

P. Le Calvez, Dynamique des homéomorphismes du plan au voisinage d'un point fixe,, Ann. Scient. Éc. Norm. Sup., 36 (2003), 139.  doi: 10.1016/S0012-9593(03)00005-3.  Google Scholar

[15]

P. Le Calvez, F. R. Ruiz del Portal and J. M. Salazar, Fixed point indices of the iterates of $\mathbbR^3$-homeomorphisms at fixed points which are isolated invariant sets,, J. London Math. Soc., 82 (2010), 683.  doi: 10.1112/jlms/jdq050.  Google Scholar

[16]

P. Le Calvez and J. C. Yoccoz, Un theoréme d'indice pour les homéomorphismes du plan au voisinage d'un point fixe,, Annals of Math., 146 (1997), 241.  doi: 10.2307/2952463.  Google Scholar

[17]

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., ().   Google Scholar

[18]

K. Mischaikow and M. Mrozek, Conley index,, in Handbook of Dynamical Systems, (2002), 393.  doi: 10.1016/S1874-575X(02)80030-3.  Google Scholar

[19]

C. Pugh and M. Shub, Periodic points on the 2-sphere,, Discrete Contin. Dyn. Syst., 34 (2014), 1171.  doi: 10.3934/dcds.2014.34.1171.  Google Scholar

[20]

D. Richeson and J. Wiseman, A fixed point theorem for bounded dynamical systems,, Illinois Journal of Mathematics, 46 (2002), 491.   Google Scholar

[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), 1199.  doi: 10.1016/S0040-9383(01)00035-0.  Google Scholar

[22]

M. Shub, All, most, some differentiable dynamical systems,, in Proceedings of the International Congress of Mathematicians, (2006), 99.   Google Scholar

[23]

M. Shub and D. Sullivan, A remark on the Lefschetz fixed point formula for differentiable maps,, Topology, 13 (1974), 189.  doi: 10.1016/0040-9383(74)90009-3.  Google Scholar

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