For topological dynamical systems $(X,T,\sigma)$ with abelian group $T$, which admit an equicontinuous factor $\pi:(X,T,\sigma)\to (Y,T,\delta)$, the Ellis semigroup $E(X)$ is an extension of $Y$ by its subsemigroup $E^{\operatorname{fib}}(X)$ of elements which preserve the fibres of $\pi$. We establish methods to compute $E^{\operatorname{fib}}(X)$ and use them to determine the Ellis semigroup of dynamical systems arising from primitive aperiodic bijective substitutions. As an application, we show that for these substitution shifts, the virtual automorphism group is isomorphic to the classical automorphism group.

中文翻译：

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双射代换的 Ellis 半群

对于具有阿贝尔群 $T$ 的拓扑动力系统 $(X,T,\sigma)$，它承认一个等连续因子 $\pi:(X,T,\sigma)\to (Y,T,\delta)$， Ellis 半群 $E(X)$ 是 $Y$ 通过其子半群 $E^{\operatorname{fib}}(X)$ 的扩展，这些元素保留了 $\pi$ 的纤维。我们建立了计算 $E^{\operatorname{fib}}(X)$ 的方法，并使用它们来确定由原始非周期双射替换产生的动力系统的 Ellis 半群。作为一个应用，我们展示了对于这些替换移位，虚拟自同构群与经典自同构群同构。