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October  2016, 36(10): 5347-5368. doi: 10.3934/dcds.2016035

Periodic and eventually periodic points of affine infra-nilmanifold endomorphisms

 1 KU Leuven Kulak, E. Sabbelaan 53, 8500 Kortrijk, Belgium

Received  November 2015 Revised  January 2016 Published  July 2016

In this paper, we study the periodic and eventually periodic points of affine infra-nilmanifold endomorphisms. On the one hand, we give a sufficient condition for a point of the infra-nilmanifold to be (eventually) periodic. In this way we show that if an affine infra-nilmanifold endomorphism has a periodic point, then its set of periodic points forms a dense subset of the manifold. On the other hand, we deduce a necessary condition for eventually periodic points from which a full description of the set of eventually periodic points follows for an arbitrary affine infra-nilmanifold endomorphism.
Citation: Jonas Deré. Periodic and eventually periodic points of affine infra-nilmanifold endomorphisms. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5347-5368. doi: 10.3934/dcds.2016035
References:
 [1] D. V. Anosov, Geodesic flow on closed Riemannian manifolds with negative curvature, Trudy Mat. Inst. Steklov., 90 (1967), 209pp. [2] K. Dekimpe, Almost-Bieberbach Groups: Affine and Polynomial Structures, Lect. Notes in Math., 1639, Springer-Verlag, 1996. [3] K. Dekimpe, What an infra-nilmanifold endomorphism really should be..., Topological Methods in Nonlinear Analysis, 40 (2012), 111-136. [4] K. Dekimpe and J. Deré, Expanding maps and non-trivial self-covers on infra-nilmanifolds, Topological Methods in Nonlinear Analysis, 47 (2016), 347-368. [5] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley Studies in Nonlinearity $2^{nd}$ edition, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989, Studies in Nonlinearity. Westview Press, Boulder, CO, 2003. [6] M. Gromov, Groups of polynomial growth and expanding maps, Institut des Hautes Études Scientifiques, 53 (1981), 53-73. [7] K. Y. Ha, H. J. Kim and J. B. Lee, Eventually periodic points of infra-nil endomorphisms, Fixed Point Theory Appl., (2010), Art. ID 721736, 15pp. [8] K. B. Lee, Maps on infra-nilmanifolds, Pacific J. Math., 168 (1995), 157-166. doi: 10.2140/pjm.1995.168.157. [9] A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math., 96 (1974), 422-429. doi: 10.2307/2373551. [10] J. R. Munkres, Topology: A First Course, $2^{nd}$ edition, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975. [11] S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math, 92 (1970), 761-770. doi: 10.2307/2373372. [12] D. Segal, Polycyclic Groups, Cambridge University Press, 1983. doi: 10.1017/CBO9780511565953. [13] M. Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math, 91 (1969), 175-199. doi: 10.2307/2373276. [14] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1.

show all references

References:
 [1] D. V. Anosov, Geodesic flow on closed Riemannian manifolds with negative curvature, Trudy Mat. Inst. Steklov., 90 (1967), 209pp. [2] K. Dekimpe, Almost-Bieberbach Groups: Affine and Polynomial Structures, Lect. Notes in Math., 1639, Springer-Verlag, 1996. [3] K. Dekimpe, What an infra-nilmanifold endomorphism really should be..., Topological Methods in Nonlinear Analysis, 40 (2012), 111-136. [4] K. Dekimpe and J. Deré, Expanding maps and non-trivial self-covers on infra-nilmanifolds, Topological Methods in Nonlinear Analysis, 47 (2016), 347-368. [5] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley Studies in Nonlinearity $2^{nd}$ edition, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989, Studies in Nonlinearity. Westview Press, Boulder, CO, 2003. [6] M. Gromov, Groups of polynomial growth and expanding maps, Institut des Hautes Études Scientifiques, 53 (1981), 53-73. [7] K. Y. Ha, H. J. Kim and J. B. Lee, Eventually periodic points of infra-nil endomorphisms, Fixed Point Theory Appl., (2010), Art. ID 721736, 15pp. [8] K. B. Lee, Maps on infra-nilmanifolds, Pacific J. Math., 168 (1995), 157-166. doi: 10.2140/pjm.1995.168.157. [9] A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math., 96 (1974), 422-429. doi: 10.2307/2373551. [10] J. R. Munkres, Topology: A First Course, $2^{nd}$ edition, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975. [11] S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math, 92 (1970), 761-770. doi: 10.2307/2373372. [12] D. Segal, Polycyclic Groups, Cambridge University Press, 1983. doi: 10.1017/CBO9780511565953. [13] M. Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math, 91 (1969), 175-199. doi: 10.2307/2373276. [14] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1.
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