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March  2014, 34(3): 1171-1182. doi: 10.3934/dcds.2014.34.1171

Periodic points on the $2$-sphere

Charles Pugh 1, and  Michael Shub 2,

1. 

Department of Mathematics, University of Chicago, 5734 S. University Ave, Chicago, Illinois 60637, United States
2. 

CONICET, IMAS, Universidad de Buenos Aires, Buenos Aires

Received  October 2012 Revised  March 2013 Published  August 2013

For a $C^{1}$ degree two latitude preserving endomorphism $f$ of the $2$-sphere, we show that for each $n$, $f$ has at least $2^{n}$ periodic points of period $n$.
Keywords: latitude preserving., Periodic points, smoothness, two sphere, degree two.
Mathematics Subject Classification: Primary: 37C2.
Citation: Charles Pugh, Michael Shub. Periodic points on the $2$-sphere. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 1171-1182. doi: 10.3934/dcds.2014.34.1171
References:
[1]

Katrin Gelfert and Christian Wolf, On the distribution of periodic orbits, Discrete and Continuous Dynamical Systems, 36 (2010), 949-966. doi: 10.3934/dcds.2010.26.949.  Google Scholar

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Anatole Katok, Lyapunov Exponents, Entropy, and Periodic Points for Diffeomorphisms, Institute des Hautes Études Scientifiques, Publications Mathématiques, 51 (1980), 137-173.  Google Scholar

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Michal Misiurewicz and Feliks Przytycki, Topological entropy and degree of smooth mappings, Bull. Acad. Pol., 25 (1977), 573-574  Google Scholar

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

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

show all references

References:
[1]

Katrin Gelfert and Christian Wolf, On the distribution of periodic orbits, Discrete and Continuous Dynamical Systems, 36 (2010), 949-966. doi: 10.3934/dcds.2010.26.949.  Google Scholar

[2]

Anatole Katok, Lyapunov Exponents, Entropy, and Periodic Points for Diffeomorphisms, Institute des Hautes Études Scientifiques, Publications Mathématiques, 51 (1980), 137-173.  Google Scholar

[3]

Michal Misiurewicz and Feliks Przytycki, Topological entropy and degree of smooth mappings, Bull. Acad. Pol., 25 (1977), 573-574  Google Scholar

[4]

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

[5]

Michael Shub and Dennis 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

[6]

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

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