Normed versus topological groups: Dichotomy and duality

Normed groups 133converging to the identity and divergent sequence. These are critical to the definition ofregular variation [BOst-TRI].Definition. Let ψ n : X → X be self-homeomorphisms.We say that a sequence ψ n in H(X) converges to the identity if‖ψ n ‖ = ˆd(ψ n , id) := sup t∈X d(ψ n (t), t) → 0.Thus, for all t, we have z n (t) := d(ψ n (t), t) ≤ ‖ψ n ‖ and z n (t) → 0. Thus the sequence‖ψ n ‖ is bounded.Illustrative Examples. In R we may consider ψ n (t) = t + z n with z n → 0. In a moregeneral context, we note that a natural example of a convergent sequence of homeomorphismsis provided by a flow parametrized by discrete time (thus also termed a ‘chain’)towards a sink. If ψ : N × X → X is a flow and ψ n (x) = ψ(n, x), then, for each t, theorbit {ψ n (t) : n = 1, 2, ...} is the image of the real null sequence {z n (t) : n = 1, 2, ...}.Proposition 13.1. (i) For a sequence ψ n in H(X), ψ n converges to the identity iff ψ −1nconverges to the identity.(ii) Suppose X has abelian norm. For h ∈ H(X), if ψ n converges to the identity then sodoes h −1 ψ n h.Proof. Only (ii) requires proof, and that follows from ∣ ∣ ∣ ∣h −1 ψ n h ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣hh −1 ψ n∣ ∣∣ ∣ = ||ψ n || ,by the cyclic property.Definitions. 1. Again let ϕ n : X → X be self-homeomorphisms. We say that thesequence ϕ n in G diverges uniformly if for any M > 0 we have, for all large enough n,thatd(ϕ n (t), t) ≥ M, for all t.Equivalently, puttingd ∗ (h, h ′ ) = inf x∈X d(h(x), h ′ (x)),d ∗ (ϕ n , id) → ∞.2. More generally, let A ⊆ H(S) with A a metrizable topological group. We say that α nis a pointwise divergent sequence in A if, for each s ∈ S,d S (α n (s), s) → ∞,equivalently, α n (s) does not contain a bounded subsequence.3. We say that α n is a uniformly divergent sequence in A if‖α n ‖ A := d A (e A , α n ) → ∞,equivalently, α n does not contain a bounded subsequence.