Equicontinuous System Research Articles

The Systems with Almost Banach-Mean Equicontinuity for Abelian Group Actions

In this paper, we present the concept of Banach-mean equicontinuity and prove that the Banach-, Weyl- and Besicovitch-mean equicontinuities of a dynamic system of Abelian group action are equivalent. Furthermore, we obtain that the topological entropy of a transitive, almost Banach-mean equicontinuous dynamical system of Abelian group action is zero. As an application of our main result, we show that the topological entropy of the Banach-mean equicontinuous system under the action of an Abelian groups is zero.

Horseshoes for Anosov Systems on Fibers Driven by an Equicontinuous System

Horseshoes for Anosov Systems on Fibers Driven by an Equicontinuous System

Mean equicontinuity, almost automorphy and regularity

The aim of this article is to obtain a better understanding and classification of strictly ergodic topological dynamical systems with (measurable) discrete spectrum. To that end, we first determine when an isomorphic maximal equicontinuous factor map of a minimal topological dynamical system has trivial (one point) fibers. In other words, we characterize when minimal mean equicontinuous systems are almost automorphic. Furthermore, we investigate another natural subclass of mean equicontinuous systems, so-called diam-mean equicontinuous systems, and show that a minimal system is diam-mean equicontinuous if and only if the maximal equicontinuous factor is regular (the points with trivial fibers have full Haar measure). Combined with previous results in the field, this provides a natural characterization for every step of a natural hierarchy for strictly ergodic topological models of ergodic systems with discrete spectrum. We also construct an example of a transitive almost diam-mean equicontinuous system with positive topological entropy, and we give a partial answer to a question of Furstenberg related to multiple recurrence.

The Proximal Relation, Regionally Proximal Relation and Banach Proximal Relation for Amenable Group Actions

In this paper, we study the proximal relation, regionally proximal relation and Banach proximal relation of a topological dynamical system for amenable group actions. A useful tool is the support of a topological dynamical system which is used to study the structure of the Banach proximal relation, and we prove that above three relations all coincide on a Banach mean equicontinuous system generated by an amenable group action.

On tameness of almost automorphic dynamical systems for general groups

Let ( X , G ) be a minimal equicontinuous dynamical system, where X is a compact metric space and G some topological group acting on X. Under very mild assumptions, we show that the class of regular almost automorphic extensions of ( X , G ) contains examples of tame but non-null systems as well as non-tame ones. To do that, we first study the representation of almost automorphic systems by means of semicocycles for general groups. Based on this representation, we obtain examples of the above kind in well-studied families of group actions. These include Toeplitz flows over G-odometers where G is countable and residually finite as well as symbolic extensions of irrational rotations.

Varieties of mixing

We consider extensions of the notion of topological transitivity for a dynamical system ( X , f ) (X,f) . In addition to chain transitivity, we define strong chain transitivity and vague transitivity. Associated with each, there is a notion of mixing, defined by transitivity of the product system ( X × X , f × f ) (X \times X, f \times f) . These extend the concept of weak mixing which is associated with topological transitivity. Using the barrier functions of Fathi and Pageault, we obtain for each of these extended notions a dichotomy result in which a transitive system of each type either satisfies the corresponding mixing condition or else factors into an appropriate type of equicontinuous minimal system. The classical dichotomy result for minimal systems follows when it is shown that a minimal system is weak mixing if and only if it is vague mixing.

How chaotic is an almost mean equicontinuous system?

The question how chaotic is an almost mean equicontinuous system is addressed. It is shown that every topological dynamical system can be embedded into an almost mean equicontinuous system with the same entropy which is an almost one-to-one extension of some mean equicontinuous system. Besides, there is an almost mean equicontinous system that is topologically K and Devaney chaotic, and as this consequence we know that every ergodic measure of such a topologically K system does not have full support.

Auslander-Yorke dichotomy theorem for indecomposable systems

The Auslander-Yorke dichotomy theorem says that every transitive system is either sensitive or almost equicontinuous. A topological dynamical system $(X,f)$ is said to be indecomposable if every two $f$- invariant closed subsets having non-empty interiors must have common interior points. This notion is strictly weaker than transitivity. In this paper, we show that every indecomposable system with $f$ being semi-open is either almost equicontinuous or sensitive, which complements the Auslander-Yorke dichotomy theorem. Moreover, we prove that in an almost equicontinuous indecomposable system all equicontinuous points have the same $\omega$- limited set.

Limit Sets, Attractors and Chaos

The aim of this paper is a study on relations between omega -chaos and the structure of omega -limit sets. We propose a definition of hat{omega }-chaos which requires stronger relations between limit sets of points from tuples. We present example with relatively simple dynamics (almost equicontinuous system) which is omega -chaotic and propose further restrictions on the conditions in the definition.

Mean equicontinuity and mean sensitivity

Answering an open question affirmatively it is shown that every ergodic invariant measure of a mean equicontinuous (i.e. mean-L-stable) system has discrete spectrum. Dichotomy results related to mean equicontinuity and mean sensitivity are obtained when a dynamical system is transitive or minimal. Localizing the notion of mean equicontinuity, notions of almost mean equicontinuity and almost Banach mean equicontinuity are introduced. It turns out that a system with the former property may have positive entropy and meanwhile a system with the latter property must have zero entropy.

Generalized solution of non‐autonomous photon transport problem

AbstractThe purpose of this paper is to show the existence of a generalized solution of a non‐autonomous transport problem. By means of the theory of equicontinuous evolution system on a sequentially complete locally convex topological vector space, we show that the perturbed abstract non‐autonomous Cauchy problem has a unique solution when the perturbation operator and forcing term function satisfy certain conditions. A consequence of the abstract result is that it can be directly applied to obtain a generalized solution of the non‐autonomous photon transport problem. Copyright © 2009 John Wiley & Sons, Ltd.

The structure of tame minimal dynamical systems

AbstractA dynamical version of the Bourgain–Fremlin–Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of $\beta \mathbb {N}$, or it is a ‘tame’ topological space whose topology is determined by the convergence of sequences. In the latter case, the dynamical system is said to be tame. We use the structure theory of minimal dynamical systems to show that, when the acting group is Abelian, a tame metric minimal dynamical system (i) is almost automorphic (i.e. it is an almost one-to-one extension of an equicontinuous system), and (ii) admits a unique invariant probability measure such that the corresponding measure-preserving system is measure-theoretically isomorphic to the Haar measure system on the maximal equicontinuous factor.

Residual properties and almost equicontinuity

A propertyP of a compact dynamical system (X,f) is called a residual property if it is inherited by factors, almost one-to-one lifts and surjective inverse limits. Many transitivity properties are residual. Weak disjointness from all propertyP systems is a residual property providedP is a residual property stronger than transitivity. Here two systems are weakly disjoint when their product is transitive. Our main result says that for an almost equicontinuous system (X, f) with associated monothetic group Λ, (X, f) is weakly disjoint from allP systems iff the onlyP systems upon which Λ acts are trivial. We use this to prove that every monothetic group has an action which is weak mixing and topologically ergodic.

On the multiplicative ergodic theorem for uniquely ergodic systems

We consider the question of uniform convergence in the multiplicative ergodic theorem lim n→∞ 1 n ·log‖A(T n−1x)…A(x)‖=⋀(A) for continuous function A : X → GL d ( R ), where ( X, T) is a uniquely ergodic system. We show that the inequality lim sup n→∞ n −1 · log ∥ A( T n−1 x) ⋯ A( x) ∥ ≤ Λ( A) holds uniformly on X, but it may happen that for some exceptional zero measure set E ⊂ X of the second Baire category: lim inf n→∞ n −1 · log ∥ A( T n−1 x) ⋯ A( x)∥ < Λ( A). We call such A a non-uniform function. We give sufficient conditions for A to be uniform, which turn out to be necessary in the two-dimensional case. More precisely, A is uniform iff either it has trivial Lyapunov exponents, or A is continuously cohomologous to a diagonal function. For equicontinuous system ( X, T), such as irrational rotations, we identify the collection of non-uniform matrix functions as the set of discontinuity of the functional A on the space C( X, GL 2( R )), thereby proving, that the set of all uniform matrix functions forms a dense G δ -set in C( X, GL 2( R )). It follows, that M. Herman's construction of a non-uniform matrix function on an irrational rotation, gives an example of discontinuity of A on C( X, GL 2( R )).