The Ellis semigroup of certain constant-length substitutions

Abstract

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-

Characterization of Quasifactors

“…We look into this study via an example: Example 2.11. We look into a substitution system, the square of the Morse-Thue substitution as considered in [18,21]. This is a continuous substitution Q defined by the rule…”

“…Following the calculations in [18,21], here E(X) has exactly four minimal idempotents u 1 , v 1 , u 2 , v 2 such that they are the identity off the orbits of a, b, ā, b and on the orbits of a, b, ā, b are defined as:…”

Uniform enveloping semigroupoids for groupoid actions

“…In particular the elegant theory of compact, right-topological semigroups has been used to describe and study properties of topological dynamical systems. We refer to [Aus88, Chapters 3 and 6] and [Gla07] for the general theory of the Ellis semigroup and to [KLS15, Chapter 5], [Sta19], or [GGY18, Section 4] for some recent applications.…”