Expansive actions on uniform spaces and surjunctive maps

具有 specification 的 sofic 群膨胀作用、强拓扑马尔可夫性质与满射性

Expansive actions with specification of sofic groups, strong topological Markov property, and surjunctivity

本文证明了对于 sofic 群作用的膨胀动力系统，在非负 sofic 拓扑熵且满足弱 specification 与强拓扑马尔可夫性质的条件下，系统必为满射的。

On Dual surjunctivity and applications

We explore the dual version of Gottschalk's conjecture recently introduced by Capobianco, Kari, and Taati, and the notion of dual surjunctivity in general. We show that dual surjunctive groups satisfy Kaplansky's direct finiteness conjecture for all fields of positive characteristic. By quantifying the notions of injectivity and post-surjectivity for cellular automata, we show that the image of the full topological shift under an injective cellular automaton is a subshift of finite type in a quantitative way. Moreover we show that dual surjunctive groups are closed under ultraproducts, under e…

关于非均匀细胞自动机的戈茨克兰猜想

On Gottschalk's surjunctivity conjecture for non-uniform cellular automata

本文研究了非均匀细胞自动机在surjunctive群宇宙中的surjunctivity性质，证明了可逆的非均匀细胞自动机必须可逆，并扩展了Capobianco等人关于细胞自动机的对偶surjunctivity定理。

Expansive actions of countable amenable groups, homoclinic pairs, and the Myhill property

Let $X$ be a compact metrizable space equipped with a continuous action of a countable amenable group $G$. Suppose that the dynamical system $(X,G)$ is expansive and is the quotient by a uniformly bounded-to-one factor map of a strongly irreducible subshift. Let $τ\colon X \to X$ be a continuous map commuting with the action of $G$. We prove that if there is no pair of distinct $G$-homoclinic points in $X$ having the same image under $τ$, then $τ$ is surjective.

Commensurated subgroups and micro-supported actions

Let $Γ$ be a finitely generated group and $X$ be a minimal compact $Γ$-space. We assume that the $Γ$-action is micro-supported, i.e. for every non-empty open subset $U \subseteq X$, there is an element of $Γ$ acting non-trivially on $U$ and trivially on the complement $X \setminus U$. We show that, under suitable assumptions, the existence of certain commensurated subgroups in $Γ$ yields strong restrictions on the dynamics of the $Γ$-action: the space $X$ has compressible open subsets, and it is an almost $Γ$-boundary. Those properties yield in turn restrictions on the structure of $Γ$: $Γ$ is…