On the Density of Pre-Orbits under Linear Toral Endomorphisms

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.