# 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 & Continuous Dynamical Systems - A, 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.  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, 51 (1980), 137.   Google Scholar [3] Michal Misiurewicz and Feliks Przytycki, Topological entropy and degree of smooth mappings,, Bull. Acad. Pol., 25 (1977), 573.   Google Scholar [4] Michael Shub, All, most, dome differentiable dynamical systems,, Proceedings of the International Congress of Mathematicians, (2006), 99.   Google Scholar [5] Michael Shub and Dennis Sullivan, A remark on the lefschetz fixed point formula for differentiable maps,, Topology, 13 (1974), 189.  doi: 10.1016/0040-9383(74)90009-3.  Google Scholar [6] Michael Shub, Alexander cocycles and dynamics,, Asterisque, (1978), 395.   Google Scholar

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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.  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, 51 (1980), 137.   Google Scholar [3] Michal Misiurewicz and Feliks Przytycki, Topological entropy and degree of smooth mappings,, Bull. Acad. Pol., 25 (1977), 573.   Google Scholar [4] Michael Shub, All, most, dome differentiable dynamical systems,, Proceedings of the International Congress of Mathematicians, (2006), 99.   Google Scholar [5] Michael Shub and Dennis Sullivan, A remark on the lefschetz fixed point formula for differentiable maps,, Topology, 13 (1974), 189.  doi: 10.1016/0040-9383(74)90009-3.  Google Scholar [6] Michael Shub, Alexander cocycles and dynamics,, Asterisque, (1978), 395.   Google Scholar
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