# American Institute of Mathematical Sciences

November  2016, 36(11): 6187-6199. doi: 10.3934/dcds.2016070

## Periodic points of latitudinal maps of the $m$-dimensional sphere

 1 Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, Narutowicza 11/12, 80-233 Gdańsk, Poland 2 Department of Mathematical Sciences, Indiana University - Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, IN 46202 3 Sopot, Poland

Received  October 2015 Revised  June 2016 Published  August 2016

Let $f$ be a smooth self-map of the $m$-dimensional sphere $S^m$. Under the assumption that $f$ preserves latitudinal foliations with the fibres $S^1$, we estimate from below the number of fixed points of the iterates of $f$. The paper generalizes the results obtained by Pugh and Shub and by Misiurewicz.
Citation: Grzegorz Graff, Michał Misiurewicz, Piotr Nowak-Przygodzki. Periodic points of latitudinal maps of the $m$-dimensional sphere. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6187-6199. doi: 10.3934/dcds.2016070
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##### References:
 [1] Grzegorz Graff, Jerzy Jezierski. Minimization of the number of periodic points for smooth self-maps of closed simply-connected 4-manifolds. Conference Publications, 2011, 2011 (Special) : 523-532. doi: 10.3934/proc.2011.2011.523 [2] John Erik Fornæss. Periodic points of holomorphic twist maps. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1047-1056. doi: 10.3934/dcds.2005.13.1047 [3] Wolfgang Krieger, Kengo Matsumoto. Markov-Dyck shifts, neutral periodic points and topological conjugacy. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 1-18. doi: 10.3934/dcds.2019001 [4] Daniel Wilczak, Piotr Zgliczyński. Topological method for symmetric periodic orbits for maps with a reversing symmetry. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 629-652. doi: 10.3934/dcds.2007.17.629 [5] Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547 [6] Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 545-556. doi: 10.3934/dcds.2011.31.545 [7] Lluís Alsedà, Sylvie Ruette. On the set of periods of sigma maps of degree 1. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4683-4734. doi: 10.3934/dcds.2015.35.4683 [8] Boris Hasselblatt, Zbigniew Nitecki, James Propp. Topological entropy for nonuniformly continuous maps. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 201-213. doi: 10.3934/dcds.2008.22.201 [9] K. H. Kim, F. W. Roush and J. B. Wagoner. Inert actions on periodic points. Electronic Research Announcements, 1997, 3: 55-62. [10] Charles Pugh, Michael Shub. Periodic points on the $2$-sphere. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1171-1182. doi: 10.3934/dcds.2014.34.1171 [11] Marc Henrard. Homoclinic and multibump solutions for perturbed second order systems using topological degree. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 765-782. doi: 10.3934/dcds.1999.5.765 [12] Anna Capietto, Walter Dambrosio. A topological degree approach to sublinear systems of second order differential equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 861-874. doi: 10.3934/dcds.2000.6.861 [13] Jian Lu, Huaiyu Jian. Topological degree method for the rotationally symmetric $L_p$-Minkowski problem. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 971-980. doi: 10.3934/dcds.2016.36.971 [14] Eric Bedford, Kyounghee Kim. Degree growth of matrix inversion: Birational maps of symmetric, cyclic matrices. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 977-1013. doi: 10.3934/dcds.2008.21.977 [15] Anna Go??biewska, S?awomir Rybicki. Equivariant Conley index versus degree for equivariant gradient maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 985-997. doi: 10.3934/dcdss.2013.6.985 [16] Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461 [17] Richard Miles, Thomas Ward. Directional uniformities, periodic points, and entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3525-3545. doi: 10.3934/dcdsb.2015.20.3525 [18] Anna Gierzkiewicz, Klaudiusz Wójcik. Lefschetz sequences and detecting periodic points. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 81-100. doi: 10.3934/dcds.2012.32.81 [19] Viviane Baladi, Daniel Smania. Smooth deformations of piecewise expanding unimodal maps. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 685-703. doi: 10.3934/dcds.2009.23.685 [20] Yong Fang. On smooth conjugacy of expanding maps in higher dimensions. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 687-697. doi: 10.3934/dcds.2011.30.687

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