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 : T^m → T^m is an absolutely hyperbolic endomorphism, the pre-orbit of any point is dense in T^m.

中文翻译：

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线性环内同态下的预轨道密度

摘要

众所周知，如果非内射同形的每个预轨道是密集的，则同形是可传递的（即，存在密集的轨道）。但是，仍然不清楚Anosov映射的预轨道是否密集，并且所有预轨道都密集的必要条件也是未知的。利用积分格的性质，我们通过考虑线性同态的预轨道来构造我们的证明。我们引入一类具有绝对双曲性的特征的双曲线性内胚性，以证明如果A：T ^m → T ^m是绝对双曲内胚性，则任何点的前轨道在T ^m中都是密集的。