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

Periodic points on the $2$-sphere

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 $n$.
Citation: Charles Pugh, Michael Shub. Periodic points on the $2$-sphere. Discrete and 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.

[2]

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

[3]

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

[4]

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

[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.

[6]

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

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.

[2]

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

[3]

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

[4]

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

[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.

[6]

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

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