# 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 and Continuous Dynamical Systems, 2016, 36 (11) : 6187-6199. doi: 10.3934/dcds.2016070
##### References:
 [1] I. K. Babenko, S. A. Bogatyi, The behavior of the index of periodic points under iterations of a mapping, Math. USSR Izv., 38 (1992), 1-26. [2] G. Graff and J. Jezierski, On the growth of the number of periodic points for smooth self-maps of a compact manifold, Proc. Amer. Math. Soc., 135 (2007), 3249-3254. doi: 10.1090/S0002-9939-07-08836-3. [3] L. Hernández-Corbato and F. R. Ruiz del Portal, Fixed point indices of planar continuous maps, Discrete Contin. Dyn. Syst., 35 (2015), 2979-2995. doi: 10.3934/dcds.2015.35.2979. [4] B. J. Jiang, Lectures on the Nielsen Fixed Point Theory, Contemp. Math. 14, Amer. Math. Soc., Providence, 1983. [5] V. Kaloshin, Generic diffeomorphisms with superexponential growth of number of periodic orbits, Comm. Math. Phys., 211 (2000), 253-271. doi: 10.1007/s002200050811. [6] N. G. Lloyd, Degree Theory, Cambridge Tracts in Mathematics, 73, Cambridge University Press, Cambridge-New York-Melbourne, 1978. [7] M. Misiurewicz, Periodic points of latitudinal maps, J. Fixed Point Theory Appl., 16 (2014), 149-158. doi: 10.1007/s11784-014-0195-y. [8] C. Pugh and M. Shub, Periodic points on the 2-sphere, Discrete Contin. Dynam. Sys., 34 (2014), 1171-1182. doi: 10.3934/dcds.2014.34.1171. [9] M. Shub, Alexander cocycles and dynamics, Asterisque, 51, Societé Math. de France, (1978), 395-413. [10] M. Shub, All, most, some differentiable dynamical systems, Proceedings of the International Congress of Mathematicians, Madrid, Spain, (2006), European Math. Society, 99-120. [11] M. Shub, Dynamical systems, filtration and entropy, Bull. Amer. Math. Soc., 80 (1974), 27-41. doi: 10.1090/S0002-9904-1974-13344-6. [12] M. Shub and P. Sullivan, A remark on the Lefschetz fixed point formula for differentiable maps, Topology, 13 (1974), 189-191. doi: 10.1016/0040-9383(74)90009-3.

show all references

##### References:
 [1] I. K. Babenko, S. A. Bogatyi, The behavior of the index of periodic points under iterations of a mapping, Math. USSR Izv., 38 (1992), 1-26. [2] G. Graff and J. Jezierski, On the growth of the number of periodic points for smooth self-maps of a compact manifold, Proc. Amer. Math. Soc., 135 (2007), 3249-3254. doi: 10.1090/S0002-9939-07-08836-3. [3] L. Hernández-Corbato and F. R. Ruiz del Portal, Fixed point indices of planar continuous maps, Discrete Contin. Dyn. Syst., 35 (2015), 2979-2995. doi: 10.3934/dcds.2015.35.2979. [4] B. J. Jiang, Lectures on the Nielsen Fixed Point Theory, Contemp. Math. 14, Amer. Math. Soc., Providence, 1983. [5] V. Kaloshin, Generic diffeomorphisms with superexponential growth of number of periodic orbits, Comm. Math. Phys., 211 (2000), 253-271. doi: 10.1007/s002200050811. [6] N. G. Lloyd, Degree Theory, Cambridge Tracts in Mathematics, 73, Cambridge University Press, Cambridge-New York-Melbourne, 1978. [7] M. Misiurewicz, Periodic points of latitudinal maps, J. Fixed Point Theory Appl., 16 (2014), 149-158. doi: 10.1007/s11784-014-0195-y. [8] C. Pugh and M. Shub, Periodic points on the 2-sphere, Discrete Contin. Dynam. Sys., 34 (2014), 1171-1182. doi: 10.3934/dcds.2014.34.1171. [9] M. Shub, Alexander cocycles and dynamics, Asterisque, 51, Societé Math. de France, (1978), 395-413. [10] M. Shub, All, most, some differentiable dynamical systems, Proceedings of the International Congress of Mathematicians, Madrid, Spain, (2006), European Math. Society, 99-120. [11] M. Shub, Dynamical systems, filtration and entropy, Bull. Amer. Math. Soc., 80 (1974), 27-41. doi: 10.1090/S0002-9904-1974-13344-6. [12] M. Shub and P. Sullivan, A remark on the Lefschetz fixed point formula for differentiable maps, Topology, 13 (1974), 189-191. doi: 10.1016/0040-9383(74)90009-3.
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