On the Ellis semigroup of a cascade on a compact metric countable space
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On the Ellis semigroup of a cascade on a compact metric countable space Andres Quintero1 · Carlos Uzcátegui1 Received: 6 October 2019 / Accepted: 16 January 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Let X be a compact metric countable space, let f ∶ X → X be a homeomorphism and let E(X, f) be its Ellis semigroup. Among other results we show that the following statements are equivalent: (1) (X, f) is equicontinuous, (2) (X, f) is distal and (3) every point is periodic. We use this result to give a direct proof of a theorem of Ellis saying that (X, f) is distal if, and only if, E(X, f) is a group. Keywords Discrete dynamical system · Ellis semigroup · Compact metric countable space · Equicontinuity · Distality
1 Introduction Let X be a compact metric space and f ∶ X → X a homeomorphism. The dynamical system (X, f) is usually called a cascade. Ellis introduced [5] the enveloping semigroup (also called the Ellis semigroup) E(X, f) of the dynamical system (X, f) as the closure of {f n ∶ n ∈ ℤ} inside X X with the product topology. E(X, f) is a compact semigroup with composition as the algebraic operation. There is an extensive literature about the enveloping semigroup (see, for instance, [1, 6, 12]). A homeomorphism f ∶ X → X is distal if
inf{d(f n (x), f n (y)) ∶ n ∈ ℤ} > 0 for every pair of different points x and y of X. A well known characterization of distality is as follows (see [1, p. 69]). The system (X, f) is distal if, and only if, every Communicated by Anthony To-Ming Lau. * Carlos Uzcátegui [email protected] Andres Quintero [email protected] 1
Escuela de Matemáticas, Facultad de Ciencias, Universidad Industrial de Santander, Ciudad Universitaria, Carrera 27 Calle 9, Santander, A.A. 678, Bucaramanga, Colombia
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point of (X × X, f ) is almost periodic and also if, and only if, E(X, f) is a group. Distality is a weakening of another very important notion. A system (X, f) is equicontinuous, if {f n ∶ n ∈ ℤ} is an equicontinuous family. It is easy to verify that every equicontinuous system is distal. The converse is not true in general. When (X, f) is equicontinuous, every function in E(X, f) is obviously continuous and such systems are called weakly almost periodic (WAP). It is also well known that a system (X, f) is equicontinuous if, and only if, (X, f) is distal and WAP (see [1, p. 69]). Continuing the work initiated in [8, 9], we are interested on dynamical systems on compact metric countable spaces. Our main result is that, for a compact metric countable space X and a homeomorphism f ∶ X → X , the following statements are equivalent: 1. (X, f) is equicontinuous. 2. (X, f) is distal. 3. Every point of X is periodic. 4. There is g ∈ E(X, f )⧵{f n ∶ n ∈ ℤ} which is a homeomorphism. 5. For all 𝜀 > 0 there is l such that d(x, f nl (x)) < 𝜀 for all x ∈ X and all n ∈ ℕ (where d is the metric on X). 6. E(X, f ) = {f n ∶ n ∈ ℕ}. Even though our main interest is on countable spaces, when possible, we
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