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

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 & Continuous Dynamical Systems - A, 2016, 36 (10) : 5347-5368. doi: 10.3934/dcds.2016035

References:

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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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