Minimal Transformation Groups Research Articles

Disjointness of Some Types of Extensions of Topological Transformation Semigroups

We extend (with some generalizations) Bronshtein's theorems about the disjointness of F-, PE-, V-extensions of minimal topological transformation groups.

On equicontinuous factors of linear extensions of minimal dynamical systems

The concept of the equicontinuous factor of the linear extension of a minimal transformation group is introduced and investigated. It is shown that a subset of motions, bounded and distal with respect to the extension, forms a maximal equicontinuous subsplitting of the linear extension. As a consequence, any distal linear extension has a nontrivial equicontinuous invariant subsplitting. The linear extensions without exponential dichotomy possess similar subsplittings if the Favard condition is satisfied. The same statement holds for linear extensions with the property of recurrent motions additivity provided that at least one nonzero motion of this sort exists.

Weak disjointness and the equicontinuous structure relation

AbstractWe discuss weak disjointness of homomorphisms of minimal transformation groups and use the techniques involved to deepen our knowledge of the equicontinuous structure relation.

Weak mixing and a note on a structure theorem for minimal transformation groups

Weak mixing and a note on a structure theorem for minimal transformation groups

Relativized problems with abelian phase group in topological dynamics

Let (X, T) be the equicontinuous minimal transformation group with X = pi(infinity)Z(2), the Cantor group, and S = [unk](infinity)Z(2) endowed with the discrete topology acting on X by right multiplication. For any countable group T we construct a function F:X x S --> T such that if (Y, T) is a minimal transformation group, then (X x Y, S) is a minimal transformation group with the action defined by (x, y)s = [xs, yF(x, s)]. If (W, T) is a minimal transformation group and varphi:(Y, T) --> (W, T) is a homomorphism, then identity x varphi:(X x Y, S) --> (X x W, S) is a homomorphism and has many of the same properties that varphi has. For this reason, one may assume that the phase group is abelian (or S) without loss of generality for many relativized problems in topological dynamics.

On the connectedness of homomorphisms in topological dynamics

Let (X, T) be a minimal transformation group with compact Hausdorff phase space. We show that if ϕ : ( X , T ) → ( Y , T ) \phi :(X,T) \to (Y,T) is a distal homomorphism and has a structure similar to the structure Furstenberg derived for distal minimal sets, then for T belonging to a class of topological groups T, the homomorphism X → X / S ( ϕ ) X \to X/S(\phi ) has connected fibers, where S ( ϕ ) S(\phi ) is the relativized equicontinuous structure relation. The class T is defined by Sacker and Sell as consisting of all groups T with the property that there is a compact set K ⊆ T K \subseteq T such that T is generated by each open neighborhood of K. They show that for such T, a distal minimal set which is a finite-to-one extension of an almost periodic minimal set is itself an almost periodic minimal set. We provide an example that shows that the restriction on T cannot be dropped. As one of the preliminaries to the above we show that given ϕ : ( X , T ) → ( Y , T ) \phi :(X,T) \to (Y,T) , the relation R c ( ϕ ) Rc(\phi ) induced by the components in the fibers relative to ϕ \phi , i.e., ( x , x ′ ) ∈ R c ( ϕ ) (x,x’) \in Rc(\phi ) if and only if x and x ′ x’ are in the same component of ϕ − 1 ( ϕ ( x ) ) {\phi ^{ - 1}}(\phi (x)) , is a closed invariant equivalence relation. We also consider the question of when a minimal set (X, T) is such that Q ( x ) Q(x) is finite for some x in X, where Q is the regionally proximal relation. This problem was motivated by Veech’s work on almost automorphic minimal sets, i.e., the case in which Q ( x ) Q(x) is a singleton for some x in X.

Proximal-isometric ( [formula omitted]) flows

We develop some new tools for the study of minimal transformation groups (flows) and apply these tools to extend the classification theory of minimal flows to a larger class than has previously been classified. The tools we develop arise from a simplification and unification of the techniques used by Furstenberg, Ellis, and Veech in their proofs of structure theorems about distal and point distal flows. This simplification is made possible largely through the use of recent results of Glasner on the structure of 2 X (whose points consist of closed subsets of the phase space X of a given flow).

Finite Extensions of Minimal Transformation Groups

In this paper we shall study homomorphisms $p:W \to Y$ on minimal transformation groups. We shall prove, in the case that W and Y are metrizable, that W is a finite (N-to-1) extension of Y if and only if W is an N-fold covering space of Y and p is a covering map. This result places no further restrictions on the acting group. We shall then use this characterization to investigate the question of lifting an equicontinuous structure from Y to W. We show that, under very weak restrictions on the acting group, this lifting is always possible when W is a finite extension of Y.

Finite extensions of minimal transformation groups

In this paper we shall study homomorphisms p : W → Y p:W \to Y on minimal transformation groups. We shall prove, in the case that W and Y are metrizable, that W is a finite (N-to-1) extension of Y if and only if W is an N-fold covering space of Y and p is a covering map. This result places no further restrictions on the acting group. We shall then use this characterization to investigate the question of lifting an equicontinuous structure from Y to W. We show that, under very weak restrictions on the acting group, this lifting is always possible when W is a finite extension of Y.

Skew-product flows, finite extensions of minimal transformation groups and almost periodic differential equations

where a is itself a flow on Y and T is a topological group. In other words the natural projection p:X x Y -+ Y is a homomorphism of the transformation group (X x y, T, 7i) onto (y, T, cr). Skew-product flows arise in a natural way in the study of ordinary differential equations x' = g(x, t) (cf. [6] and [7]). In this case the group T would be the real numbers and Y would be a topological function space containing g and closed under time-translations. The flow a would be given by <x( ƒ, T) = ft where /T(x, t) = ƒ (x, T + £). The space X would be the phase space for the differential equation, usually X is the Euclidean space R or perhaps some n-dimensional manifold, and cp(x9 ƒ, t) would represent the solution of x' = ƒ (x, t) passing through x at time t = 0. (We assume that all differential equations in Y give rise to unique solutions, although some of our results are valid without this restriction (cf. [8]).) Now assume that y is a compact minimal set under the flow o and let M a X x y be a compact invariant set of the skew-product flow. Motivated by the above model for differential equations we ask : When can certain structures be lifted from Y to M? For example, if we assume that y is an almost periodic minimal set (that is, the flow G is equicontinuous on Y) under what conditions will M contain an almost periodic minimal set ? We shall say that the flow n has the distal property on M if for any ye Y and x l 5 x 2 e l with x t ^ x2, (xl5y)e M and (x2,y)eM there is an

Disjointness in Transformation Groups

In this paper, we shall be concerned with the question of what conditions on minimal transformation groups will guarantee that they are disjoint. Generalizing a result of I. Bronšteĭn about lifting of minimality through group extensions to associated bitransformation groups, we prove that in a large class of transformation groups, disjointness is equivalent to disjointness of their maximal equicontinuous factors. In the abelian case, this means that disjointness is equivalent to no common factor in the class of flows discussed.

A Characterization of Distal and Point-Distal Minimal Transformation Groups

If G is a locally compact topological group, let $BC(G)$ denote the set of real-valued, bounded, uniformly continuous functions on G with the compact-open topology. Using the fact that the distal (weakly distal) functions are the elements of $BC(G)$ whose orbit closures are compact distal (point-distal) minimal sets, we can characterize compact distal and point-distal minimal transformation groups. Let $(X,G,\phi )$ be a right transformation group where X is compact Hausdorff and minimal under G. Then X is a compact distal (point-distal) minimal set if and only if there is a point $x \in X$ such that for any homomorphism $h:X \to BC(G),h(x)$ is a right distal (weakly distal) function.

Disjointness in transformation groups

In this paper, we shall be concerned with the question of what conditions on minimal transformation groups will guarantee that they are disjoint. Generalizing a result of I. Bronstein about lifting of minimality through group extensions to associated bitransformation groups, we prove that in a large class of transformation groups, disjointness is equivalent to disjointness of their maximal equicontinuous factors. In the abelian case, this means that disjointness is equivalent to no common factor in the class of flows discussed.

A characterization of distal and point-distal minimal transformation groups

If G is a locally compact topological group, let BC(G) denote the set of real-valued, bounded, uniformly continuous functions on G with the compact-open topology. Using the fact that the distal (weakly distal) functions are the elements of BC(G) whose orbit closures are compact distal (point-distal) minimal sets, we can characterize compact distal and point-distal minimal transformation groups. Let (X, G, A) be a right transformation group where X is compact Hausdorff and minimal under G. Then X is a compact distal (point-distal) minimal set if and only if there is a point xCX such that for any homomorphism h: X-*BC(G), h(x) is a right distal (weakly distal) function.

Extension of minimal transformation groups

Extension of minimal transformation groups

A Characterization of the Equicontinuous Structure Relation

The main result in the paper is to show that in a large class of minimal transformation groups (including those with abelian phase groups, and point-distal transformation groups), the equicontinuous structure relation is precisely the regionally proximal relation. The techniques involved enable one to recover and extend the previously known characterizations. Several corollaries are indicated, among which the most important is a new criterion (which is easily applicable) for the existence of a nontrivial equicontinuous image of a given transformation group.

The structure of weakly mixing minimal transformation groups

The structure of weakly mixing minimal transformation groups

Distal extensions of minimal transformation groups

Distal extensions of minimal transformation groups

A characterization of the equicontinuous structure relation

The main result in the paper is to show that in a large class of minimal transformation groups (including those with abelian phase groups, and point-distal transformation groups), the equicontinuous structure relation is precisely the regionally proximal relation. The techniques involved enable one to recover and extend the previously known characterizations. Several corollaries are indicated, among which the most important is a new criterion (which is easily applicable) for the existence of a nontrivial equicontinuous image of a given transformation group.

Proximality in Minimal Transformation Groups

Proximality in Minimal Transformation Groups