Nilpotent Endomorphisms of Expansive Group Actions

TL;DR

This paper investigates conditions under which expansive group actions with certain properties are nilpotent, showing that for many groups, asymptotically nilpotent endomorphisms are actually nilpotent, with implications for cellular automata and symbolic dynamics.

Contribution

It establishes nil-rigidity for expansive actions of broad classes of groups, including residually finite solvable groups and groups of polynomial growth, under natural dynamical properties.

Findings

Nil-rigidity holds for all expansive actions with dense homoclinic points and a specification-like property.

For $ Z$-actions, weak gluing suffices for nil-rigidity.

Block-gluing property ensures nil-rigidity for $ Z^2$-subshifts of finite type.

Abstract

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 $\mathbb{Z}$, we show that a very weak gluing property suffices for nil-rigidity. For $\mathbb{Z}^2$-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.