# American Institute of Mathematical Sciences

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.

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##### 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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