On tameness of almost automorphic dynamical systems for general groups

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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On the structure of the Levinson center for monotone non-autonomous dynamical systems with a first integral

Carpathian Journal of Mathematics ◽

10.37193/cjm.2022.01.07 ◽

2021 ◽

Vol 38 (1) ◽

pp. 67-94

Author(s):

DAVID CHEBAN ◽

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Global Attractor ◽

Difference Equations ◽

First Integral ◽

Almost Periodic ◽

Almost Automorphic ◽

Stable Solutions ◽

Autonomous Dynamical Systems

In this paper we give a description of the structure of compact global attractor (Levinson center) for monotone Bohr/Levitan almost periodic dynamical system $x'=f(t,x)$ (*) with the strictly monotone first integral. It is shown that Levinson center of equation (*) consists of the Bohr/Levitan almost periodic (respectively, almost automorphic, recurrent or Poisson stable) solutions. We establish the main results in the framework of general non-autonomous (cocycle) dynamical systems. We also give some applications of theses results to different classes of differential/difference equations.

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Decomposing isometric extensions using group extensions

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700007604 ◽

1993 ◽

Vol 13 (4) ◽

pp. 661-673

Author(s):

A. H. Forrest

Keyword(s):

Dynamical Systems ◽

Group Extension ◽

Natural Projection ◽

Orbit Closure ◽

Isometric Extension ◽

Group Extensions ◽

Almost Automorphic ◽

Distal Point ◽

Group Of Isometries ◽

Topological Dynamical Systems

AbstractThis paper studies the structure of isometric extensions of compact metric topological dynamical systems with ℤ action and gives two decompositions of the general case to a more structured case. Suppose that Y → X is a M-isometric extension. An extension, Z, of Y is constructed which is also a G-isometric extension of X, where G is the group of isometries of M. The first construction shows that, provided that (X, T) is transitive, there are almost-automorphic extensions Y′ → Y and X′ → X, so that Y′ is homeomorphic to X′ × M and the natural projection Y′ → X′ is a group extension. The second shows that, provided that (X, T) is minimal, there is a G-action on Z which commutes with T and which preserves fibres and acts on each of them minimally. Each individual orbit closure, Za, in Z is a G′-isometric extension of X, where G′ is a subgroup of G, and there is a G′-action on Za which commutes with T, preserves fibres and acts minimally on each of them. Two illustrations are presented. Of the first: to reprove a result of Furstenberg; that every distal point is IP*-recurrent. Of the second: to describe the minimal subsets in isometric extensions of minimal topological dynamical systems.

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