On the density of pre-orbits under linear toral endomorphisms

Authors

  • Saeed Azimi
  • Khosro Tajbakhsh

DOI:

https://doi.org/10.21638/spbu01.2020.301

Abstract

It is well known for non-injective endomorphisms that if for every point the set of preimages is dense in the manifold then the endomorphism is transitive (i. e. there exists a point that its orbit is dense in the manifold). But it has not yet been completely investigated that if the pre-orbit of points are dense under Anosov endomorphisms or what are the necessary conditions that make the pre-orbits of each point dense. By making a great use of the integral lattice properties, we construct our proof on the pre-image sets of points under the iterations of the linear dynamical system. We introduce a class of hyperbolic linear endomorphism that is called absolutely hyperbolic and show that if A : Tm → Tm is an absolutely hyperbolic linear endomorphism of degree more than 1 then the pre-orbit of each point is dense in Tm.

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References

Литература

1. Lizana C., Pinheiro V., Varandas P. Contribution to the ergodic theory of robustly transitive maps // Disc. & Cont. Dynam. Sys. 2015. Vol. 35, no. 1. P. 353–365. https://doi.org/10.3934/dcds.2015.35.353

2. Franks J. Anosov Diffeomorphisms // Proc. Sympos. Pure Math., 1–26 July 1968, Berkeley, California. AMS, Providence, R. I., 1970. Vol. 14. Global Analysis. P. 61–93.

3. Aoki N., Hiraide K. Topological Theory of Dynamical Systems. 1994. (Vol. 52 of North Holland Mathematical library.)

4. Przytycki F. Anosov Endomorphisms // Studia Mathematica. 1976. P. 249–285.

5. Anderson M., Correa J. Transitivity of Codimension One Conservative Skew-products Endomorphisms. 2017. arXiv:1612.09337v2 [math.DS]

6. Shub M. Global Stability of Dynamical Systems. Springer-Verlag, 1987.

7. Hatcher A. Algebraic Topology // Cambridge University Press, 2002.

8. Hilbert D., Cohn-Vossen S. Geometry and the Imagination. 2nd ed. AMS Chelsey Publishing, 1999.

9. Hall B. C. Lie groups, Lie algebras, and representations: An elementary introduction. 2nd ed. Springer, 2015. (Vol. 222 of Graduate Texts in Mathematics.)

References

1. Lizana C., Pinheiro V., Varandas P., “Contribution to the Ergodic Theory of Robustly Transitive Maps”, Disc. & Cont. Dynam. Sys. 35 (1), 353–365 (2015). https://doi.org/10.3934/dcds.2015.35.353

2. Franks J., “Anosov Diffeomorphisms”, Proc. Sympos. Pure Math., 1–26 July 1968, Berkeley, California, 61–93 (Vol. 14. Global Analysis, AMS, Providence, R. I., 1970).

3. Aoki N., Hiraide K., Topological Theory of Dynamical Systems (1994, vol. 52 of North Holland Mathematical library).

4. Przytycki F., “Anosov Endomorphisms”, Studia Mathematica, 249–285 (1976).

5. Anderson M., Correa J., “Transitivity of Codimension One Conservative Skew-products Endomorphisms” (2017). arXiv:1612.09337v2 [math.DS]

6. Shub M., Global Stability of Dynamical Systems (Springer-Verlag, 1987).

7. Hatcher A., Algebraic Topology (Cambridge University Press, 2002).

8. Hilbert D., Cohn-Vossen S., Geometry and the Imagination (2nd ed., AMS Chelsey Publishing, 1999).

9. Hall B. C., Lie groups, Lie algebras, and representations: An elementary introduction (2nd ed., Springer, 2015, vol. 222 of Graduate Texts in Mathematics).

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Published

2020-09-04

How to Cite

Azimi, S., & Tajbakhsh, K. (2020). On the density of pre-orbits under linear toral endomorphisms. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 7(3), 369–376. https://doi.org/10.21638/spbu01.2020.301

Issue

Vol. 7 No. 3 (2020)

Section

Mathematics

License

Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.