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

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

Grzegorz Graff 1, , Michał Misiurewicz 2, and  Piotr Nowak-Przygodzki 3,

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.
Keywords: topological degree, Periodic points, smooth maps..
Mathematics Subject Classification: Primary: 37C25, 37E30, 55M2.
Citation: Grzegorz Graff, Michał Misiurewicz, Piotr Nowak-Przygodzki. Periodic points of latitudinal maps of the $m$-dimensional sphere. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

B. J. Jiang, Lectures on the Nielsen Fixed Point Theory, Contemp. Math. 14, Amer. Math. Soc., Providence, 1983.  Google Scholar

[5]

V. Kaloshin, Generic diffeomorphisms with superexponential growth of number of periodic orbits, Comm. Math. Phys., 211 (2000), 253-271. doi: 10.1007/s002200050811.  Google Scholar

[6]

N. G. Lloyd, Degree Theory, Cambridge Tracts in Mathematics, 73, Cambridge University Press, Cambridge-New York-Melbourne, 1978.  Google Scholar

[7]

M. Misiurewicz, Periodic points of latitudinal maps, J. Fixed Point Theory Appl., 16 (2014), 149-158. doi: 10.1007/s11784-014-0195-y.  Google Scholar

[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.  Google Scholar

[9]

M. Shub, Alexander cocycles and dynamics, Asterisque, 51, Societé Math. de France, (1978), 395-413.  Google Scholar

[10]

M. Shub, All, most, some differentiable dynamical systems, Proceedings of the International Congress of Mathematicians, Madrid, Spain, (2006), European Math. Society, 99-120.  Google Scholar

[11]

M. Shub, Dynamical systems, filtration and entropy, Bull. Amer. Math. Soc., 80 (1974), 27-41. doi: 10.1090/S0002-9904-1974-13344-6.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

B. J. Jiang, Lectures on the Nielsen Fixed Point Theory, Contemp. Math. 14, Amer. Math. Soc., Providence, 1983.  Google Scholar

[5]

V. Kaloshin, Generic diffeomorphisms with superexponential growth of number of periodic orbits, Comm. Math. Phys., 211 (2000), 253-271. doi: 10.1007/s002200050811.  Google Scholar

[6]

N. G. Lloyd, Degree Theory, Cambridge Tracts in Mathematics, 73, Cambridge University Press, Cambridge-New York-Melbourne, 1978.  Google Scholar

[7]

M. Misiurewicz, Periodic points of latitudinal maps, J. Fixed Point Theory Appl., 16 (2014), 149-158. doi: 10.1007/s11784-014-0195-y.  Google Scholar

[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.  Google Scholar

[9]

M. Shub, Alexander cocycles and dynamics, Asterisque, 51, Societé Math. de France, (1978), 395-413.  Google Scholar

[10]

M. Shub, All, most, some differentiable dynamical systems, Proceedings of the International Congress of Mathematicians, Madrid, Spain, (2006), European Math. Society, 99-120.  Google Scholar

[11]

M. Shub, Dynamical systems, filtration and entropy, Bull. Amer. Math. Soc., 80 (1974), 27-41. doi: 10.1090/S0002-9904-1974-13344-6.  Google Scholar

[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.  Google Scholar

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