On the density of pre-orbits under linear toral endomorphisms

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 → T m is an absolutely hyperbolic linear endomorphism of degree more than 1 then the pre-orbit of each point is dense in Tm.