Abstract:
Let (X,σ) be a topological dynamical system, where X is a compact metric space and σ:X → X is a homeomorphism. Its Ellis semigroup is the compactification of the group action generated by σ in the topology of pointwise convergence on the space XX. The Ellis semigroup is, typically, a huge beast, and its computation has been restricted mainly to systems (X,σ) which are metrically equicontinuous; such systems are called tame.
In this talk we give a complete description of the Ellis semigroup for the family of bijective substitution shifts (X,σ). These systems are not tame. (X,σ) admits an equicontinuous factor π: (X,σ)→(Y,δ), and so the Ellis semigroup E(X) is an extension of Y by its subsemigroup Efib(X) of elements which preserve the fibres of π; this includes all idempotents. We give a complete description of Efib(X), expressing it as an uncountable product of the finite group G, defined to be the normal closure of the group generated by the idempotents, with a semigroup Σ. We illustrate with examples all the possibilities that can occur. This is joint work with Johannes Kellendonk.