In this article, we calculate the Ellis semigroup of a certain class of constant length substitutions. This generalises a result of Haddad and Johnson [HJ97] from the binary case to substitutions over arbitrarily large finite alphabets. Moreover, we provide a class of counter-examples to one of the propositions in their paper, which is central to the proof of their main theorem. We give an alternative approach to their result, which centers on the properties of the Ellis semigroup. To do this, we also show a new way to construct an AI tower to the maximal equicontinuous factor of these systems.Definition 1 (dynamical system, cascade, minimal system). By a dynamical system (X, T ), we mean a compact Hausdorff space X together with a homeomorphism T : X → X. The set {T n } n∈Z (alternatively, {T n } n∈N ) forms an action of the group Z (or semigroup N) on the space X. Sometimes, a dynamical system over N or Z is called an N-(respectively, Z-) cascade for short. A dynamical system is minimal if and only if it has no closed set which is invariant under T .Definition 2 (positively/negatively asymptotic points). Let (X, T ) be a dynamical system, and x, y ∈ X. We say x and y are positively (resp, negatively) asymp-
We present results on the existence of long arithmetic progressions in the Thue-Morse word and in a class of generalised Thue-Morse words. Our arguments are inspired by van der Waerden's proof for the existence of arbitrary long monochromatic arithmetic progressions in any finite colouring of the (positive) integers.
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