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Abstract. We study different conditions which turn out to be equivalent to equicontinuity for a transitive compact Hausdorff flow with a general group action. Among them are a notion of "regional" equicontinuity, also known as "Furstenberg" condition, and the condition that every point of the phase space is almost automorphic. Then we study relations on the phase space arising from dynamical properties, among them the regionally proximal relation and two relations introduced by Veech. We generalize Veech's results for minimal actions of non-Abelian groups preserving a probability measure with respect to the regionally proximal relation. We provide proofs in the framework of dynamical systems rather than harmonic analysis as given by Veech.Throughout this note let X be a compact Hausdorff space and T a topological group continuously acting on the left of X, so that (X, T ) is a compact Hausdorff flow. There exists a unique uniform structure on X, defined by a set U of entourages coincident with the set of neighborhoods of the diagonal ∆ = {(x, x) : x ∈ X} in the product topology on X × X. Given an entourage α ∈ U and a point x ∈ X we define the α-neighborhood of x by {y ∈ X : (y, x) ∈ α}, and using these sets as a neighborhood base at x the uniform structure defines the topology on X. Given entourages α, β ∈ U we define the product by αβ = {(x, z) ∈ X × X : there exists y ∈ X s. th. (x, y) ∈ α and (y, z) ∈ β} and the inverse by α −1 = {(y, x) ∈ X × X : (x, y) ∈ α}. If d is a metric on X compatible with the topology, then a set α ⊂ X × X is an element of U if and only if there exists an ε > 0 so that {(x, y) ∈ X × X : d(x, y) < ε} ⊂ α. For a general (non-metrizable) uniform structure, there is always a family D = {d i : i ∈ I} of pseudometrics on X so that α ⊂ X × X is an element of U if and only if there exists an element d i ∈ D and ε > 0 with {(x, y) ∈ X × X : d i (x, y) < ε} ⊂ α. We call a point x ∈ X an equicontinuity point, if for every entourage α ∈ U there exists a neighborhood U x of x so that for every y ∈ U x and every t ∈ T it holds that (tx, ty) ∈ α. If every point in X is an equicontinuity point, then by a finite covering argument for every entourage α ∈ U there exists a entourage β ∈ U so that (x, y) ∈ β implies (tx, ty) ∈ α for all t ∈ T . Such a flow (X, T ) is called equicontinuous. We denote the orbit closure of a point x ∈ X byŌ T (x), hence a flow is transitive ifŌ T (x) = X for some x ∈ X and minimal ifŌ T (x) = X for all x ∈ X. The closure of T in the product space X X has a natural semigroup operation continuous in the left argument, the enveloping semigroup E(X, T ) of the flow (X, T ).2000 Mathematics Subject Classification. Primary: 37B05, 54H20.

We describe various strengthenings of the concept of topological transitivity. Especially when one departs from the family of invertible systems, a number of interesting properties arise. We present the architecture of implications among ten reasonable notions of transitivity.

In this paper we wish to relate the dynamics of the base map to the dynamics of the induced map. In the process, we obtain conditions on the endowed hyperspace topology under which the chaotic behaviour of the map on the base space is inherited by the induced map on the hyperspace. Several of the known results come up as corollaries to our results. We also discuss some metric related dynamical properties on the hyperspace that cannot be deduced for the base dynamics.