The Ellis semigroup of bijective substitutions

Complete regularity of Ellis semigroups of -actions

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.118 ◽

2020 ◽

pp. 1-17

Author(s):

MARCY BARGE ◽

JOHANNES KELLENDONK

Keyword(s):

Complete Regularity ◽

Completely Regular ◽

Ellis Semigroup ◽

Totally Disconnected

Abstract It is shown that the Ellis semigroup of a $\mathbb Z$ -action on a compact totally disconnected space is completely regular if and only if forward proximality coincides with forward asymptoticity and backward proximality coincides with backward asymptoticity. Furthermore, the Ellis semigroup of a $\mathbb Z$ - or $\mathbb R$ -action for which forward proximality and backward proximality are transitive relations is shown to have at most two left minimal ideals. Finally, the notion of near simplicity of the Ellis semigroup is introduced and related to the above.

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About the Ellis semigroup of a simple k-od

Topology and its Applications ◽

10.1016/j.topol.2019.06.001 ◽

2019 ◽

Vol 265 ◽

pp. 106756 ◽

Cited By ~ 1

Author(s):

Ivon Vidal-Escobar ◽

Salvador Garcia-Ferreira

Keyword(s):

Ellis Semigroup

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The Ellis semigroup of certain constant-length substitutions

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2019.75 ◽

2019 ◽

pp. 1-26

Author(s):

PETRA STAYNOVA

Keyword(s):

Dynamical Systems ◽

Constant Length ◽

Almost Automorphic ◽

Recurrent Sequences ◽

Alternative Approach ◽

Binary Case ◽

The One ◽

Ellis Semigroup ◽

Cluster Points

In this article, we calculate the Ellis semigroup of a certain class of constant-length substitutions. This generalizes a result of Haddad and Johnson [IP cluster points, idempotents, and recurrent sequences. Topology Proc.22 (1997) 213–226] from the binary case to substitutions over arbitrarily large finite alphabets. Moreover, we provide a class of counterexamples 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 almost automorphic–isometric tower to the maximal equicontinuous factor of these systems, which gives a more particular approach than the one given by Dekking [The spectrum of dynamical systems arising from substitutions of constant length. Z. Wahrscheinlichkeitstheor. Verw. Geb.41(3) (1977/78) 221–239].

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The Ellis semigroup of a nonautonomous discrete dynamical system

Quaestiones Mathematicae ◽

10.2989/16073606.2017.1313332 ◽

2017 ◽

Vol 40 (6) ◽

pp. 753-767

Author(s):

S. García-Ferreira ◽

M. Sanchis

Keyword(s):

Dynamical System ◽

Discrete Dynamical System ◽

Ellis Semigroup

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Semigroup compactifications by generalized distal functions and a fixed point theorem

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171291000285 ◽

1991 ◽

Vol 14 (2) ◽

pp. 253-260

Author(s):

R. D. Pandian

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Point Theorem ◽

Extensive Study ◽

Point Of View ◽

Almost Periodic ◽

Semigroup Compactification ◽

Semigroup Compactifications ◽

Invariant Means ◽

Ellis Semigroup

The notion of Semigroup compactification which is in a sense, a generalization of the classical Bohr (almost periodic) compactification of the usual additive realsR, has been studied by J. F. Berglund et. al. [2]. Their approach to the theory of semigroup compactification is based on the Gelfand-Naimark theory of commutativeC*algebras, where the spectra of admissibleC*-algebras, are the semigroup compactifications. H. D. Junghenn's extensive study of distal functions is from the point of view of semigroup compactifications [5]. In this paper, extending Junghenn's work, we generalize the notion of distal flows and distal functions on an arbitrary semitopological semigroupS, and show that these function spaces are admissibleC*- subalgebras ofC(S). We then characterize their spectra (semigroup compactifications) in terms of the universal mapping properties these compactifications enjoy. In our work, as it is in Junghenn's, the Ellis semigroup plays an important role. Also, relating the existence of left invariant means on these algebras to the existence of fixed points of certain affine flows, we prove the related fixed point theorem.

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Cardinality of the Ellis semigroup on compact metric countable spaces

Semigroup Forum ◽

10.1007/s00233-017-9888-z ◽

2017 ◽

Vol 97 (1) ◽

pp. 162-176 ◽

Cited By ~ 1

Author(s):

S. García-Ferreira ◽

Y. Rodríguez-López ◽

C. Uzcátegui

Keyword(s):

Ellis Semigroup

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Model theoretic aspects of the ellis semigroup

Israel Journal of Mathematics ◽

10.1007/s11856-011-0202-6 ◽

2011 ◽

Vol 190 (1) ◽

pp. 477-507 ◽

Cited By ~ 12

Author(s):

Ludomir Newelski

Keyword(s):

Ellis Semigroup

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Boundedness and absoluteness of some dynamical invariants in model theory

Journal of Mathematical Logic ◽

10.1142/s0219061319500120 ◽

2019 ◽

Vol 19 (02) ◽

pp. 1950012

Author(s):

Krzysztof Krupiński ◽

Ludomir Newelski ◽

Pierre Simon

Keyword(s):

Left Ideal ◽

Topological Dynamics ◽

Degree Of Saturation ◽

Minimal Left Ideal ◽

Ellis Group ◽

Absolute Property ◽

Dynamical Invariants ◽

Bounded Size ◽

Ellis Semigroup ◽

Theory Formula

Let [Formula: see text] be a monster model of an arbitrary theory [Formula: see text], let [Formula: see text] be any (possibly infinite) tuple of bounded length of elements of [Formula: see text], and let [Formula: see text] be an enumeration of all elements of [Formula: see text] (so a tuple of unbounded length). By [Formula: see text] we denote the compact space of all complete types over [Formula: see text] extending [Formula: see text], and [Formula: see text] is defined analogously. Then [Formula: see text] and [Formula: see text] are naturally [Formula: see text]-flows (even [Formula: see text]-ambits). We show that the Ellis groups of both these flows are of bounded size (i.e. smaller than the degree of saturation of [Formula: see text]), providing an explicit bound on this size. Next, we prove that these Ellis groups do not depend (as groups equipped with the so-called [Formula: see text]-topology) on the choice of the monster model [Formula: see text]; thus, we say that these Ellis groups are absolute. We also study minimal left ideals (equivalently subflows) of the Ellis semigroups of the flows [Formula: see text] and [Formula: see text]. We give an example of a NIP theory in which the minimal left ideals are of unbounded size. Then we show that in each of these two cases, boundedness of a minimal left ideal (equivalently, of all the minimal left ideals) is an absolute property (i.e. it does not depend on the choice of [Formula: see text]) and that whenever such an ideal is bounded, then in some sense its isomorphism type is also absolute. Under the assumption that [Formula: see text] has NIP, we give characterizations (in various terms) of when a minimal left ideal of the Ellis semigroup of [Formula: see text] is bounded. Then we adapt the proof of Theorem 5.7 in Definably amenable NIP groups, J. Amer. Math. Soc. 31 (2018) 609–641 to show that whenever such an ideal is bounded, a certain natural epimorphism (described in [K. Krupiński, A. Pillay and T. Rzepecki, Topological dynamics and the complexity of strong types, Israel J. Math. 228 (2018) 863–932]) from the Ellis group of the flow [Formula: see text] to the Kim–Pillay Galois group [Formula: see text] is an isomorphism (in particular, [Formula: see text] is G-compact). We also obtain some variants of these results, formulate some questions, and explain differences (providing a few counterexamples) which occur when the flow [Formula: see text] is replaced by [Formula: see text].

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