The Ellis semigroup of bijective substitutions

Reem Yassawi (The Open University)

Thursday 21st November, 2019 16:00-17:00 Maths 311B

Abstract

Let (X,T) be a topological dynamical system, where X is a compact metric space and T : X → X is a homeomorphism. Its Ellis semigroup is the compactification of the group action generated by T in the topology of pointwise convergence on the space X^X. The Ellis semigroup is, typically, a huge beast, and its computation has been restricted mainly to systems for which E(X) is a group, or when it has a single minimal left ideal. For topological dynamical systems (X,T,σ) which admit an equicontinuous factor π :(X,T,σ) → (Y,T,δ), the Ellis semigroup E(X) is an extension of Y by its subsemigroup E^fib(X) of elements which preserve the fibres of π. We establish methods to compute E^fib(X), and apply them to 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 of (X,T). This is joint work with Johannes Kellendonk.

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