Abstract
It is known that if each pre-orbit of a non-injective endomorphism is dense, the endomorphism is transitive (i.e., a dense orbit exists). However, it is still unknown whether the pre-orbits of an Anosov map are dense, and the conditions necessary for all pre-orbits to be dense are also unknown. Using the properties of integral lattices, we construct our proof by considering the pre-orbits of linear endomorphisms. We introduce a class of hyperbolic linear endomorphisms characterized by the property of absolute hyperbolicity to prove that if A : Tm → Tm is an absolutely hyperbolic endomorphism, the pre-orbit of any point is dense in Tm.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
C. Lizana, V. Pinheiro, and P. Varandas, “Contribution to the ergodic theory of robustly transitive maps,” Discrete Contin. Dyn. Syst. 35, 353–365 (2015). https://doi.org/10.3934/dcds.2015.35.353
J. Franks, “Anosov diffeomorphisms,” in Global Analysis (Proc. Symp. in Pure Mathematics, Berkeley, CA, 1–26 July 1968) (American Mathematical Society, Providence, RI, 1970), in Ser.: Proceedings of Symposia in Pure Mathematics, Vol. 14, pp. 61–93.
N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems (North-Holland, Amsterdam, 1994), in Ser.: North Holland Mathematical Library, Vol. 52.
F. Przytycki, “Anosov endomorphisms,” Stud. Math. 58, 249–285 (1976).
M. Anderson and J. Correa, “Transitivity of codimension one conservative skew-products endomorphisms” (2017). arXiv 1612.09337v2 [math.DS].
M. Shub, Global Stability of Dynamical Systems (Springer-Verlag, New York, 1987).
A. Hatcher, Algebraic Topology (Cambridge Univ. Press, Cambridge, 2002).
D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, 2nd ed. (AMS Chelsea, Providence, RI, 1999).
B. C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, 2nd ed. (Springer-Verlag, Cham, 2015), in Ser.: Graduate Texts in Mathematics, Vol. 222.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Translated by E. Glushachenkova
Corresponding author.
About this article
Cite this article
Azimi, S., Tajbakhsh, K. On the Density of Pre-Orbits under Linear Toral Endomorphisms. Vestnik St.Petersb. Univ.Math. 53, 243–247 (2020). https://doi.org/10.1134/S1063454120030036
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1063454120030036