Periodic and eventually periodic points of affine infra-nilmanifold endomorphisms

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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

Jonas Deré ^1,

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

Received November 2015 Revised January 2016 Published July 2016

Abstract
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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.

Keywords: periodic points., almost-crystallographic groups, Nilpotent groups, affine infra-nilmanifold endomorphisms.

Mathematics Subject Classification: Primary: 37C25; Secondary: 20F18, 20F34, 22E2.

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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