We consider expansive group actions on a compact metric space containing a special fixed point denoted by 0, and endomorphisms of such systems whose forward trajectories are attracted toward 0. Such endomorphisms are called asymptotically nilpotent, and we study the conditions in which they are nilpotent, that is, map the entire space to 0 in a finite number of iterations. We show that for a large class of discrete groups, this property of nil-rigidity holds for all expansive actions that satisfy a natural specification-like property and have dense homoclinic points. Our main result in particular shows that the class includes all residually finite solvable groups and all groups of polynomial growth. For expansive actions of the group ℤ, we show that a very weak gluing property suffices for nil-rigidity. For ℤ2-subshifts of finite type, we show that the block-gluing property suffices. The study of nil-rigidity is motivated by two aspects of the theory of cellular automata and symbolic dynamics: It can be seen as a finiteness property for groups, which is representative of the theory of cellular automata on groups. Nilpotency also plays a prominent role in the theory of cellular automata as dynamical systems. As a technical tool of possible independent interest, the proof involves the construction of tiered dynamical systems where several groups act on nested subsets of the original space.

中文翻译：

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扩展群作用的幂零内同态

我们考虑在包含一个特殊固定点的紧凑度量空间上的扩展组动作，表示为0，以及这些系统的内同态，其前向轨迹被吸引到0. 这种内同态被称为渐近幂零, 我们研究它们所处的条件幂零的，即将整个空间映射到0在有限次数的迭代中。我们证明，对于一大类离散群，无刚性适用于满足自然规范性质并具有密集同宿点的所有扩展动作。我们的主要结果特别表明，该类包括所有剩余有限可解群和所有多项式增长群。对于集团的扩张行动ℤ，我们证明了非常弱的粘合特性足以满足零刚性。为了ℤ2-有限类型的子移位，我们证明块粘合特性就足够了。零刚性的研究受到元胞自动机理论和符号动力学两个方面的推动：它可以看作是群的有限性，它代表了群上的元胞自动机理论。幂零性在元胞自动机作为动力系统的理论中也起着重要作用。作为可能具有独立利益的技术工具，证明涉及构造分层动力系统其中几个组作用于原始空间的嵌套子集。