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MATHEMATICS
On the density of pre-orbits under linear toral endomorphisms
S. Azimi, Kh. Tajbakhsh University Tarbiat Modares, P. O. Box: 14115-111, Tehran, Iran
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 $A$ : $T^m \to T^m$ is an absolutely hyperbolic linear endomorphism of degree more than $1$ then the pre-orbit of each point is dense in $T^m$.
Keywords:
linear endomorphism, hyperbolicity, transitivity, dynamical systems, Anosov maps.
Received: 22.12.2019 Revised: 20.01.2020 Accepted: 19.03.2020
Citation:
S. Azimi, Kh. Tajbakhsh, “On the density of pre-orbits under linear toral endomorphisms”, Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 7:3 (2020), 369–376; Vestn. St. Petersbg. Univ., Math., 7:3 (2020), 243–247
Linking options:
https://www.mathnet.ru/eng/vspua162 https://www.mathnet.ru/eng/vspua/v7/i3/p369
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