Nilpotent Endomorphisms of Expansive Group Actions

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Date(s) - 30/01/2018
11 h 00 min - 12 h 00 min


Consider a cellular automaton with a special quiescent state 0. The automaton is called nilpotent if it sends every initial configuration to the 0-uniform configuration in a bounded number of steps. It is asymptotically nilpotent if the forward orbit of every configuration converges toward the 0-uniform configuration in the product topology. Guillon and Richard showed in 2008 that on a one-dimensional full shift, these notions are equivalent. In 2012, Salo extended the result to multidimensional full shifts.

We further generalize these results to the setting of expansive group actions. More formally, given a continuous action of a group G on a compact metric space X and a single fixed point 0∈X, one can ask whether there exists an endomorphism f:X→X such that fⁿ(x)⟶0 for every x∈X, but the convergence is not uniform. We focus on expansive actions that have dense 0-homoclinic points and a specification-like property that we call 0-gluing. For example, every strongly irreducible G-SFT with a uniform configuration satisfies these conditions. We show that for a large class of groups, containing in particular all residually finite solvable groups, and all such actions on them, all asymptotically nilpotent endomorphisms exhibit uniform convergence. In the course of the proof, we develop a technical tool called a tiered dynamical system, which consists of a nested family of compact subsets of an ambient space, each of which is equipped with a group action. (Joint work with Ville Salo)

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