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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

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 & Continuous Dynamical Systems - A, 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).   Google Scholar

[2]

K. Dekimpe, Almost-Bieberbach Groups: Affine and Polynomial Structures,, Lect. Notes in Math., 1639 (1996).   Google Scholar

[3]

K. Dekimpe, What an infra-nilmanifold endomorphism really should be...,, Topological Methods in Nonlinear Analysis, 40 (2012), 111.   Google Scholar

[4]

K. Dekimpe and J. Deré, Expanding maps and non-trivial self-covers on infra-nilmanifolds,, Topological Methods in Nonlinear Analysis, 47 (2016), 347.   Google Scholar

[5]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems,, Addison-Wesley Studies in Nonlinearity $2^{nd}$ edition, (1989).   Google Scholar

[6]

M. Gromov, Groups of polynomial growth and expanding maps,, Institut des Hautes Études Scientifiques, 53 (1981), 53.   Google Scholar

[7]

K. Y. Ha, H. J. Kim and J. B. Lee, Eventually periodic points of infra-nil endomorphisms,, Fixed Point Theory Appl., (2010).   Google Scholar

[8]

K. B. Lee, Maps on infra-nilmanifolds,, Pacific J. Math., 168 (1995), 157.  doi: 10.2140/pjm.1995.168.157.  Google Scholar

[9]

A. Manning, There are no new Anosov diffeomorphisms on tori,, Amer. J. Math., 96 (1974), 422.  doi: 10.2307/2373551.  Google Scholar

[10]

J. R. Munkres, Topology: A First Course,, $2^{nd}$ edition, (1975).   Google Scholar

[11]

S. E. Newhouse, On codimension one Anosov diffeomorphisms,, Amer. J. Math, 92 (1970), 761.  doi: 10.2307/2373372.  Google Scholar

[12]

D. Segal, Polycyclic Groups,, Cambridge University Press, (1983).  doi: 10.1017/CBO9780511565953.  Google Scholar

[13]

M. Shub, Endomorphisms of compact differentiable manifolds,, Amer. J. Math, 91 (1969), 175.  doi: 10.2307/2373276.  Google Scholar

[14]

S. Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747.  doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

show all references

References:
[1]

D. V. Anosov, Geodesic flow on closed Riemannian manifolds with negative curvature,, Trudy Mat. Inst. Steklov., 90 (1967).   Google Scholar

[2]

K. Dekimpe, Almost-Bieberbach Groups: Affine and Polynomial Structures,, Lect. Notes in Math., 1639 (1996).   Google Scholar

[3]

K. Dekimpe, What an infra-nilmanifold endomorphism really should be...,, Topological Methods in Nonlinear Analysis, 40 (2012), 111.   Google Scholar

[4]

K. Dekimpe and J. Deré, Expanding maps and non-trivial self-covers on infra-nilmanifolds,, Topological Methods in Nonlinear Analysis, 47 (2016), 347.   Google Scholar

[5]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems,, Addison-Wesley Studies in Nonlinearity $2^{nd}$ edition, (1989).   Google Scholar

[6]

M. Gromov, Groups of polynomial growth and expanding maps,, Institut des Hautes Études Scientifiques, 53 (1981), 53.   Google Scholar

[7]

K. Y. Ha, H. J. Kim and J. B. Lee, Eventually periodic points of infra-nil endomorphisms,, Fixed Point Theory Appl., (2010).   Google Scholar

[8]

K. B. Lee, Maps on infra-nilmanifolds,, Pacific J. Math., 168 (1995), 157.  doi: 10.2140/pjm.1995.168.157.  Google Scholar

[9]

A. Manning, There are no new Anosov diffeomorphisms on tori,, Amer. J. Math., 96 (1974), 422.  doi: 10.2307/2373551.  Google Scholar

[10]

J. R. Munkres, Topology: A First Course,, $2^{nd}$ edition, (1975).   Google Scholar

[11]

S. E. Newhouse, On codimension one Anosov diffeomorphisms,, Amer. J. Math, 92 (1970), 761.  doi: 10.2307/2373372.  Google Scholar

[12]

D. Segal, Polycyclic Groups,, Cambridge University Press, (1983).  doi: 10.1017/CBO9780511565953.  Google Scholar

[13]

M. Shub, Endomorphisms of compact differentiable manifolds,, Amer. J. Math, 91 (1969), 175.  doi: 10.2307/2373276.  Google Scholar

[14]

S. Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747.  doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

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