132 N. H. Bingham <strong>and</strong> A. J. OstaszewskiRemark. Alternatively, working in T r L (X) rather than in H u (X) <strong>and</strong> with d X R againright-invariant, since ξ x (λ y )(z) = λ y λ −1x (z)) = λ yx −1(z), we havepossibly infinite. Indeedsup w d H (ξ x (λ w ), ξ e (λ w )) = sup v d X v (e, x) = ‖x‖ X ∞,sup w d H (ξ x (λ w ), ξ y (λ w )) = sup w sup z d X R (ξ x (λ w )(z), ξ y (λ w )(z))= sup w sup z d X R (wx −1 z, wy −1 z) = sup w d X R (vxx −1 , vxy −1 )= sup v d X R (vy, vx) = sup v d X v (y, x).(Here we have written w = vx.)The refinement metric sup v d X (vy, vx) is left-invariant on the bounded elements (i.e.bounded under the corresponding norm ‖x‖ := sup{‖vxv −1 ‖ : v ∈ X}; cf. Proposition2.12). Of course, if d X were bi-invariant (both right- <strong>and</strong> left-invariant), we would havesup w d H (ξ x (λ w ), ξ y (λ w )) = d X (x, y).13. Divergence in the bounded subgroupIn earlier sections we made on occasion the assumption of a bounded norm. Here we areinterested in norms that are unbounded. For S a space <strong>and</strong> A a subgroup of Auth(S)equipped with the supremum norm, suppose ϕ : A × S → S is a continuous flow (seeLemma 3.8, for an instance). We will write α(s) := ϕ α (s) = ϕ(α, s). This is consistentwith A being a subgroup of Auth(S). As explained at the outset of Section 12, we havein mind two pairs (A, S), as follows.Example 1. Take S = X to be a normed <strong>topological</strong> group <strong>and</strong> A = T ⊆ H(X) to bea subgroup of automorphisms of X such that T is a <strong>topological</strong> group with supremummetricd T (t 1 , t 2 ) = sup x d X (t 1 (x), t 2 (x)),e.g. T = H u (X). Note that here e T = id X .Example 2. (A, S) = (Ξ, T ) = (X, T ). Here X is identified with its second dual Ξ (ofthe preceding section).Given a flow ϕ(t, x) on T × X, with T closed under translation, the action defined byϕ(ξ x , t) := ξ x −1(t)is continuous, hence a flow on Ξ × T, which is identified with X × T . Observe thatt(x) = ξ x −1(t)(e X ), i.e. projection onto the e X coordinate retrieves the T -flow ϕ. Here,for ξ = ξ x −1, writing x(t) for the translate of t, we haveξ(t) := ϕ ξ (t) = ϕ(ξ, t) = x(t),so that ϕ may be regarded as a X-flow on T. We now formalize the notion of a sequence
<strong>Normed</strong> <strong>groups</strong> 133converging to the identity <strong>and</strong> 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 ‖ <strong>and</strong> 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 <strong>and</strong> 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, <strong>and</strong> 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 <strong>topological</strong> 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.
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N. H. BINGHAM and A. J. OSTASZEWSKI
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Normed groups 3ContentsContents . .
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1. IntroductionGroup-norms, which b
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Normed groups 3Topological complete
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Normed groups 5abelian group has se
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Normed groups 74 (Topological permu
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Normed groups 9The following result
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Normed groups 11Corollary 2.4. For
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Normed groups 13More generally, for
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Normed groups 15definitions, our pr
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Normed groups 17so that fg is in th
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Normed groups 19(iii) The ¯d H -to
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Normed groups 21so‖αβ‖ ≤
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Normed groups 23Remark. Note that,
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Normed groups 25shows that [z n , y
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Normed groups 27Denoting this commo
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Normed groups 29Theorem 3.4 (Equiva
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Normed groups 31argument as again p
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Normed groups 33(ii) For α ∈ H u
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Normed groups 35Definition. A group
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Normed groups 37We now give an expl
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Normed groups 39Theorem 3.19 (Abeli
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Normed groups 412. Further recall t
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Normed groups 43Theorem 3.22 (Lipsc
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Normed groups 45Proof. Z γ = G (cf
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Normed groups 47Theorem 3.30. Let G
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Normed groups 49Remark. On the matt
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Normed groups 51As for the conclusi
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Normed groups 53By (C-adm), we may
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Normed groups 55equipped with an in
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Normed groups 57Proof. To apply Th.
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Normed groups 59Definition. A point
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Normed groups 61Proposition 3.46 (M
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Normed groups 63Thus ω δ (s) ≤
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Normed groups 65Remark. In the penu
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Normed groups 67The result confirms
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Normed groups 69Proof. By the Baire
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Normed groups 715. Generic Dichotom
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Normed groups 73Returning to the cr
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Normed groups 75Examples. Here are
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Normed groups 77cf. [Eng, 4.3.23].)
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Normed groups 79Remarks. 1. See [Fo
- Page 85 and 86: Normed groups 81Theorem 6.1 (Catego
- Page 87 and 88: Normed groups 83is continuous at th
- Page 89 and 90: Normed groups 85compact. Evidently,
- Page 91 and 92: Normed groups 87j ∈ ω} which enu
- Page 93 and 94: Normed groups 89The result below ge
- Page 95 and 96: Normed groups 91left-shift, not in
- Page 97 and 98: Normed groups 93As a corollary of t
- Page 99 and 100: Normed groups 953. For X a normed g
- Page 101 and 102: Normed groups 97Proof. Note that‖
- Page 103 and 104: Normed groups 99Taking h(x) := ‖
- Page 105 and 106: Normed groups 1019. The Semigroup T
- Page 107 and 108: Normed groups 103Theorem 9.5 (Semig
- Page 109 and 110: Normed groups 105By the Category Em
- Page 111 and 112: Normed groups 107Proof. Say f is bo
- Page 113 and 114: Normed groups 109Thus G is locally
- Page 115 and 116: Normed groups 111Theorem 10.10 (Bar
- Page 117 and 118: Normed groups 113K-analyticity was
- Page 119 and 120: Normed groups 115Theorem 11.6 (Disc
- Page 121 and 122: Normed groups 117restricted to X\M
- Page 123 and 124: Normed groups 119groups need not be
- Page 125 and 126: Normed groups 121Proof. In the meas
- Page 127 and 128: Normed groups 123Hence, as t i n
- Page 129 and 130: Normed groups 125The corresponding
- Page 131 and 132: Normed groups 127(t, x) ✛✻Φ T
- Page 133 and 134: Normed groups 129Fix s. Since s is
- Page 135: Normed groups 131Hence,‖x‖ −
- Page 139 and 140: Normed groups 135Definition. Let {
- Page 141 and 142: Normed groups 137However, whilst th
- Page 143 and 144: Normed groups 139embeddable, 14enab
- Page 145 and 146: Normed groups 141Bibliography[AL]J.
- Page 147 and 148: Normed groups 143Series 378, 2010.[
- Page 149 and 150: Normed groups 145abelian groups, Ma
- Page 151 and 152: Normed groups 147[Kak] S. Kakutani,
- Page 153 and 154: Normed groups 149fields. I. Basic p
- Page 155: Normed groups 151[So]R. M. Solovay,