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.
Lindelöf spaces are studied in any basic Topology course. However, there are other interesting covering properties with similar behaviour, such as almost Lindelöf, weakly Lindelöf, and quasi-Lindelöf, that have been considered in various research papers. Here we present a comparison between the standard results on Lindelöf spaces and analogous results for weakly and almost Lindelöf spaces. Some theorems, similar to the published ones, will be proved. We also consider counterexamples, most of which have not been included in the standard Topological textbooks, that show the interrelations between those properties and various basic topological notions, such as separability, separation axioms, first countability, and others. Some new features of those examples will be noted in view of the present comparison. We also pose several open questions. Historical Overview and MotivationOne of the basic theorems in Real Analysis, the Heine-Borel Theorem, states (in modern terminology) that every closed interval on the real line is compact. Later it was discovered that a similar property holds in more general metric spaces: every closed and bounded subset turns out to be compact and conversely, every compact subset is closed and bounded. It turned out that compactness is in fact a covering property: the modern description of compactness via open covers emerged from the work of P. S. Alexandrov and P. S. Urysohn in their famous "Memoire sur les espaces topologiques compacts" [AU29]. Compact spaces in many ways resemble finite sets. For example, the fact that in Hausdorff topological spaces two different points can be separated by disjoint open sets easily generalizes to the fact that in such spaces two disjoint compact sets can also be separated by disjoint open sets. Any finite set is compact in any topology and the fact that compact spaces should be accompanied by some kind of separation axiom comes from the fact that any set with the co-finite topology is compact (but fails to be Hausdorff).However, our favourite "real" objects such as the real line, real plane etc. fail to be compact. Ernst Lindelöf was able to identify the first compactness-like covering property (which was later given his name): the property that from every open cover, one can choose a countable subcover. The Lindelöf theorem, stating that every second countable space is Lindelöf, was proved by him for Euclidean spaces as early as 1903 in [Lin03]. Many facts that held for compact spaces, such as that every closed subspace of a compact space is also compact, remain true in Lindelöf spaces. In metric spaces the Lindelöf property was proved to be equivalent to separability, the existence of a countable basis and the countable chain condition -all of which hold on the real line. But many other properties, such as preservation under products, fail even in the finite case for Lindelöf spaces.Another generalization that is much closer to compactness is the notion of an H-closed space: this is a Hausdorff space in which from every open cover we can choose a finit...
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