Enveloping semigroups, distal flows and groups of Heisenberg type

Abstract. We give a concrete representation of the Ellis groups of some dis-tal flows we were lead to when considering distal functions on Heisenberg groups. For example, let H be a locally compact abelian group with dual group • and let G t--- T x [-I x H be the Heisenberg group with multiplica-tion, (Wl,•l,Sl)(W,•,s) : (WlW•l(S),•l 2r- •,Sl 2r- s). Then, denoting by H AP the almost periodic compactification of H, we show that (G t, T x HAP) is a distal flow in a natural way, and its Ellis group is just T x (//d) AP x H AP with multiplication (w•, h•, x•)(•:, h, x) = (w•h•(x), h • + •, • + •). (Here //d is/- / with the discrete topology.) Many examples are discussed. I. Enveloping semigroups. A flow is a pair (G, X), where G is a locally compact group that acts continuously on the compact Hausdorff space X. The enveloping semigroup of (G,X) is the closure of the (perhaps only homomorphic) mage of G in X x and is designated G-. In the relative topology from X x, G- is a compact right topological semigroup [1] with G C A(G-): = {p • G- I v- • tw: G-- • G- is continuous}. The flow (G, X) is called distal if the equality lira s•x • = lira s•x2 for a pair x•, x2 • X and net {s•} C G implies that Xl = x2. A remarkable theorem of Ellis [2, Proposition 5.3] asserts that (G,X) is distal if and only if G- is a group of transformations of X; in this case, we call G- the Ellis group of (o,x). If G • and G • are locally compact groups and o is a homomorphism of G • into the set of automorphisms of G•, then the semidirect product G • ©G• of G • and G • is the topological space G • xG • with product (s•, t•)(s•,t•): (s•o(tl)S2,t•t2); we assume the function (s,t)- • o(t)s: Gx x G2- • G • is continuous (so that G1 x G2 will be a topological group). We quote fro