The Ellis semigroup of bijective substitutions

Kellendonk, Johannes and Yassawi, Reem (2020). The Ellis semigroup of bijective substitutions. Groups, Geometry and Dynamics (Early access).

URL: https://arxiv.org/abs/1908.05690

Abstract

For topological dynamical systems (<i>X</i>,<i>T<i>,<i>σ</i>) which admit an equicontinuous factor (<i>Y</i>, <i>T</i>, <i>σ</i>) ➜ (<i>Y</i>, <i>T</i>, <i>δ</i>), the Ellis semigroup E(X) is an extension of <i>Y</i> by its subsemigroup <i>E</i><i><sup>fib</i></sub>(<i>X</i>) of elements which preserve the fibres of π. We establish methods to compute <i>E</i><i><sup>fib</i></sub>(<i>X</i>) 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.

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