American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Well-posedness and optimal control of a hemivariational inequality for nonstationary Stokes fluid flow
  • DCDS Home
  • This Issue
  • Next Article
    Bifurcation and one-sign solutions of the $p$-Laplacian involving a nonlinearity with zeros
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, 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.  Google Scholar

[2]

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

[3]

K. Dekimpe, What an infra-nilmanifold endomorphism really should be..., Topological Methods in Nonlinear Analysis, 40 (2012), 111-136.  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), 347-368. Google Scholar

[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.  Google Scholar

[6]

M. Gromov, Groups of polynomial growth and expanding maps, Institut des Hautes Études Scientifiques, 53 (1981), 53-73.  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), Art. ID 721736, 15pp.  Google Scholar

[8]

K. B. Lee, Maps on infra-nilmanifolds, Pacific J. Math., 168 (1995), 157-166. 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), 422-429. doi: 10.2307/2373551.  Google Scholar

[10]

J. R. Munkres, Topology: A First Course, $2^{nd}$ edition, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975.  Google Scholar

[11]

S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math, 92 (1970), 761-770. 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), 175-199. doi: 10.2307/2373276.  Google Scholar

[14]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. 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), 209pp.  Google Scholar

[2]

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

[3]

K. Dekimpe, What an infra-nilmanifold endomorphism really should be..., Topological Methods in Nonlinear Analysis, 40 (2012), 111-136.  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), 347-368. Google Scholar

[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.  Google Scholar

[6]

M. Gromov, Groups of polynomial growth and expanding maps, Institut des Hautes Études Scientifiques, 53 (1981), 53-73.  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), Art. ID 721736, 15pp.  Google Scholar

[8]

K. B. Lee, Maps on infra-nilmanifolds, Pacific J. Math., 168 (1995), 157-166. 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), 422-429. doi: 10.2307/2373551.  Google Scholar

[10]

J. R. Munkres, Topology: A First Course, $2^{nd}$ edition, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975.  Google Scholar

[11]

S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math, 92 (1970), 761-770. 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), 175-199. doi: 10.2307/2373276.  Google Scholar

[14]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

[1]

André Caldas, Mauro Patrão. Entropy of endomorphisms of Lie groups. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1351-1363. doi: 10.3934/dcds.2013.33.1351
[2]

Velimir Jurdjevic. Affine-quadratic problems on Lie groups. Mathematical Control & Related Fields, 2013, 3 (3) : 347-374. doi: 10.3934/mcrf.2013.3.347
[3]

Kanat Abdukhalikov. On codes over rings invariant under affine groups. Advances in Mathematics of Communications, 2013, 7 (3) : 253-265. doi: 10.3934/amc.2013.7.253
[4]

Luca Capogna. Optimal regularity for quasilinear equations in stratified nilpotent Lie groups and applications. Electronic Research Announcements, 1996, 2: 60-68.

[5]

Gabriela P. Ovando. The geodesic flow on nilpotent Lie groups of steps two and three. Discrete & Continuous Dynamical Systems, 2022, 42 (1) : 327-352. doi: 10.3934/dcds.2021119
[6]

Viorel Niţică. Stable transitivity for extensions of hyperbolic systems by semidirect products of compact and nilpotent Lie groups. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 1197-1204. doi: 10.3934/dcds.2011.29.1197
[7]

Clara Cufí-Cabré, Ernest Fontich. Differentiable invariant manifolds of nilpotent parabolic points. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4667-4704. doi: 10.3934/dcds.2021053
[8]

Ludovic Rifford. Ricci curvatures in Carnot groups. Mathematical Control & Related Fields, 2013, 3 (4) : 467-487. doi: 10.3934/mcrf.2013.3.467
[9]

Sergei V. Ivanov. On aspherical presentations of groups. Electronic Research Announcements, 1998, 4: 109-114.

[10]

Benjamin Weiss. Entropy and actions of sofic groups. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3375-3383. doi: 10.3934/dcdsb.2015.20.3375
[11]

Neal Koblitz, Alfred Menezes. Another look at generic groups. Advances in Mathematics of Communications, 2007, 1 (1) : 13-28. doi: 10.3934/amc.2007.1.13
[12]

Robert McOwen, Peter Topalov. Groups of asymptotic diffeomorphisms. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6331-6377. doi: 10.3934/dcds.2016075
[13]

Steven T. Piantadosi. Symbolic dynamics on free groups. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 725-738. doi: 10.3934/dcds.2008.20.725
[14]

Hans Ulrich Besche, Bettina Eick and E. A. O'Brien. The groups of order at most 2000. Electronic Research Announcements, 2001, 7: 1-4.

[15]

Światosław R. Gal, Jarek Kędra. On distortion in groups of homeomorphisms. Journal of Modern Dynamics, 2011, 5 (3) : 609-622. doi: 10.3934/jmd.2011.5.609
[16]

Marc Peigné. On some exotic Schottky groups. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 559-579. doi: 10.3934/dcds.2011.31.559
[17]

Paul Skerritt, Cornelia Vizman. Dual pairs for matrix groups. Journal of Geometric Mechanics, 2019, 11 (2) : 255-275. doi: 10.3934/jgm.2019014
[18]

Uri Bader, Alex Furman. Boundaries, Weyl groups, and Superrigidity. Electronic Research Announcements, 2012, 19: 41-48. doi: 10.3934/era.2012.19.41
[19]

Javier Pérez Álvarez. Invariant structures on Lie groups. Journal of Geometric Mechanics, 2020, 12 (2) : 141-148. doi: 10.3934/jgm.2020007
[20]

Martin Kassabov. Symmetric groups and expanders. Electronic Research Announcements, 2005, 11: 47-56.

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (170)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy