Periodic points on the 2-sphere

Previous Article
The Landau--Kolmogorov inequality revisited
DCDS Home
This Issue
Next Article
Discrete gradient methods for preserving a first integral of an ordinary differential equation

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

Abstract
Related Papers

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 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.

[1]	Gabriele Benedetti, Kai Zehmisch. On the existence of periodic orbits for magnetic systems on the two-sphere. Journal of Modern Dynamics, 2015, 9: 141-146. doi: 10.3934/jmd.2015.9.141
[2]	Grzegorz Graff, Michał Misiurewicz, Piotr Nowak-Przygodzki. Periodic points of latitudinal maps of the $m$-dimensional sphere. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6187-6199. doi: 10.3934/dcds.2016070
[3]	Ihsan Topaloglu. On a nonlocal isoperimetric problem on the two-sphere. Communications on Pure and Applied Analysis, 2013, 12 (1) : 597-620. doi: 10.3934/cpaa.2013.12.597
[4]	Jorge Rebaza. Uniformly distributed points on the sphere. Communications on Pure and Applied Analysis, 2005, 4 (2) : 389-403. doi: 10.3934/cpaa.2005.4.389
[5]	Alexander Nabutovsky and Regina Rotman. Lengths of geodesics between two points on a Riemannian manifold. Electronic Research Announcements, 2007, 13: 13-20.
[6]	Juan Vicente Gutiérrez-Santacreu. Two scenarios on a potential smoothness breakdown for the three-dimensional Navier–Stokes equations. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2593-2613. doi: 10.3934/dcds.2020142
[7]	Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic and Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009
[8]	Akhtam Dzhalilov, Isabelle Liousse, Dieter Mayer. Singular measures of piecewise smooth circle homeomorphisms with two break points. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 381-403. doi: 10.3934/dcds.2009.24.381
[9]	P. De Maesschalck. Gevrey normal forms for nilpotent contact points of order two. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 677-688. doi: 10.3934/dcds.2014.34.677
[10]	Duraisamy Balraj, Muthaiah Marudai, Zoran D. Mitrovic, Ozgur Ege, Veeraraghavan Piramanantham. Existence of best proximity points satisfying two constraint inequalities. Electronic Research Archive, 2020, 28 (1) : 549-557. doi: 10.3934/era.2020028
[11]	Jorge Sotomayor, Ronaldo Garcia. Codimension two umbilic points on surfaces immersed in $R^3$. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 293-308. doi: 10.3934/dcds.2007.17.293
[12]	Kazuyuki Yagasaki. Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 387-402. doi: 10.3934/dcds.2011.29.387
[13]	Elena Nozdrinova, Olga Pochinka. Solution of the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1101-1131. doi: 10.3934/dcds.2020311
[14]	Li Li, Yanyan Li, Xukai Yan. Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. Ⅲ. Two singularities. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 7163-7211. doi: 10.3934/dcds.2019300
[15]	Ye Zhang, Bernd Hofmann. Two new non-negativity preserving iterative regularization methods for ill-posed inverse problems. Inverse Problems and Imaging, 2021, 15 (2) : 229-256. doi: 10.3934/ipi.2020062
[16]	Jingwei Hu, Shi Jin, Li Wang. An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: A splitting approach. Kinetic and Related Models, 2015, 8 (4) : 707-723. doi: 10.3934/krm.2015.8.707
[17]	Baiju Zhang, Zhimin Zhang. Polynomial preserving recovery and a posteriori error estimates for the two-dimensional quad-curl problem. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022124
[18]	K. H. Kim, F. W. Roush and J. B. Wagoner. Inert actions on periodic points. Electronic Research Announcements, 1997, 3: 55-62.
[19]	Futoshi Takahashi. On the number of maximum points of least energy solution to a two-dimensional Hénon equation with large exponent. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1237-1241. doi: 10.3934/cpaa.2013.12.1237
[20]	Erik I. Verriest. Generalizations of Naismith's problem: Minimal transit time between two points in a heterogenous terrian. Conference Publications, 2011, 2011 (Special) : 1413-1422. doi: 10.3934/proc.2011.2011.1413

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Charles Pugh Michael Shub
on Google Scholar
Charles Pugh Michael Shub

[Back to Top]