LÉVY GROUP ACTION AND INVARIANT MEASURES ON βN

For f ∈ `∞(N) let Tf be defined by Tf(n) = 1 n ∑n i=1 f(i). We investigate permutations g of N, which satisfy Tf(n)−Tfg(n)→ 0 as n→∞ with fg(n) = f(gn) for f ∈ `∞(N) (i.e. g is in the Lévy group G), or for f in the subspace of Cesàro-summable sequences. Our main interest are G-invariant means on `∞(N) or equivalently G-invariant probability measures on βN. We show that the adjoint T ∗ of T maps measures supported in βN \ N onto a weak*-dense subset of the space of G-invariant measures. We investigate the dynamical system (G, βN) and show that the support set of invariant measures on βN is the closure of the set of almost periodic points and the set of nontopologically transitive points in βN \ N. Finally we consider measures which are invariant under T ∗. Introduction For A ⊂ N the density function is given by dA(n) = 1 n |{i ∈ A : i ≤ n}|, and its upper density dA as dA = limn→∞ dA(n). A set A has density α if limn→∞ dA(n) = α. For a function f ∈ `∞(N) (the space of bounded real functions on N) we define the operator T by Tf(n) = 1 n ∑n i=1 f(i). Any element f of `∞(N) has a unique continuous extension to the Stone-Čech compactification βN of N. The continuous extension of the characteristic functions χA of a subset A of N is the characteristic function χA of the closure A of A in βN. For a general f in `∞(N) we will denote the continuous extension of a function f to βN again by f . This isometry between `∞(N) and C(βN) (the space of real continuous functions on βN) allows us to express continuous functions on βN by their values on the integers, i.e. as bounded sequences on N, to interpret T as a continuous operator on C(βN), and to identify the dual spaces of `∞(N) (finitely additive measures on N) and of C(βN) (regular Borel measures on βN). In particular means on `∞(N) correspond to probability measures on βN (see proof of Lemma 1 below). This correspondence between finitely additive measures on N and measures on βN becomes more transparent when we realize βN as the set of ultrafilters on N, with ultrafilters being finitely additive 0-1 valued measures (cf. [11], 20.37 (b)): Then for an ultrafilter q the Dirac measure δq is the 0-1 valued Borel measure on βN corresponding to the finitely additive 0-1 valued measure q on N. For our approach however it will be more convenient to work with the usual topological definition of a filter on a set M as a nonempty collection F of subsets of Received by the editors September 29, 1995. 1991 Mathematics Subject Classification. Primary 54H20. Part of this work was carried out at Macquarie University with financial support from the Australian Research Council. c ©1996 American Mathematical Society 5087 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 5088 MARTIN BLÜMLINGER M satisfying (i) ∅ / ∈ F, (ii) A ∈ F, B ∈ F ⇒ A∩B ∈ F, (iii) A ∈ F, A ⊂ B ⊂M ⇒ B ∈ F. An ultrafilter is a filter with the additional property that it is not properly contained in any other filter (i.e. it is maximal), or equivalently, A ⊂M ⇒ A ∈ F or {A ∈ F. We identify the natural numbers with their neighborhood filters, so N ⊂ βN. A base for the topology on βN is now given by the clopen sets. A subset of βN is clopen iff it is the closure A of a subset A of N, i.e. iff it is the set of all ultrafilters containing A. With this abuse of notation we can see A ⊂ N both as a subset of βN and as an element of an ultrafilter p in A, i.e. we have p ∈ A iff A ∈ p. See [20] for a detailed exposition. A permutation g of N acts naturally as an automorphism on βN: For p ∈ βN let the ultrafilter gp be defined as gp = {gA : A ∈ p}. So gA ∈ gp ⇔ A ∈ p and g1(g2p) = (g1g2)p, i.e. the group of permutations of N acts as a transformation group on βN. Then for any clopen set A, A ⊂ N, we have gA = {gp : p ∈ A} = {gp : A ∈ p} = {gp : gA ∈ gp} = {q : gA ∈ q} = gA. (1) Since the clopen sets are a base for the topology on βN, it follows that g is an automorphism. A permutation g of N also induces an isometry of `∞(N) : f → fg, fg(n) = f(gn) and an isometry of C(βN) : f → fg, fg(p) = f(gp), p ∈ βN, so g acts naturally on the spaceM of (signed real regular Borel) measures on βN by 〈gμ, f〉 = 〈μ, fg〉 or equivalently by gμ(U) = μ(g−1U) for Borel sets U in βN. A measure μ in M is G-invariant (G being some group of permutations) if μ = μg ∀g ∈ G, i.e. if μ is in the annihilator {f − fg : f ∈ C(βN), g ∈ G}⊥ (with respect to the duality 〈M, C(βN)〉). We shall be interested in measures on βN, which are invariant under certain permutations defined via invariance of density functions at infinity. The Lévy group G is the group of permutations g satisfying Tf − Tfg vanishes on N∗ ∀f ∈ C(βN). We construct a weak*-dense subspace of the space of G-invariant measures and show that the bigger group Gδ of permutations satisfying this requirement only for Cesáro-summable bounded sequences f allows no Gδ-invariant measures. We investigate the topological dynamics of the permutation group G acting on βN; similar problems of invariance under right translations (or more generally under motions, i.e. 1-1 maps of N without periodic points) were discussed in [4], [12], [16] and [17]. See [2], II.5 for an overview. Although right translation τ of N is not a permutation of N, there are elements g in G which satisfy gn = τn except for a set of density 0 (cf. proof of Theorem 7). Since sets of density 0 will turn out to be negligible for the questions considered, it is safe to think of translations as being a subsemigroup of the Lévy group. In particular the action of τ on the support set S of measures invariant under the Lévy group coincides with the action of certain elements of the Lévy group on S. While the characterization of the support set Sτ of τ -invariant measures looks like our characterization of S (Theorem 4) with the notion of upper density replaced by that of upper Banach density, the orbit structures of the pertinent dynamical systems differ drastically: While we will show that points with minimal G-orbit closure are dense in S, it is known that there are open sets in Sτ without points with minimal τ -orbit closure. On the other hand every minimal τ -orbit closure is the support set of some τ -invariant measure, whereas there are minimal G-orbit closures which do not support any G-invariant measure. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use LÉVY GROUP ACTION AND INVARIANT MEASURES ON βN 5089 Some of these differences are due to the fact that the Lévy group is not amenable, whereas the semigroup of translations is amenable. An essential difference in the investigations of the dynamical systems (G, βN) and (τ, βN) stems from the fact that translations of N suggest the consideration of N as a semigroup under addition, which then allows some characterizations via algebraic properties of the left topological (right topological in the notation of [2]) semigroup βN. So the set S is a right ideal of the semigroup (βN,+) cf. [5], 7.4 and minimal closed right ideals are just minimal orbit closures under τ cf. [1], 2.1. Finally we characterize the support set of measures which are invariant under the adjoint T ∗ of T . This invariance is a stronger requirement than G-invariance. The support set SI of these measures allows a characterization similar to that of S or of SI , again with an appropriate notion of density. Permutation groups and invariant means on N We first clarify why the existence of positive invariant functionals is equivalent to the existence of nonzero invariant functionals for `∞(N) resp. C(βN). The proof of these facts is included for the reader’s convenience. We recall that a mean on N is a positive normalized finitely additive measure (set function) on N or equivalently a positive normalized functional on `∞(N). Lemma 1. For a group G of permutations of N the following are equivalent: (a) There exists a G-invariant mean on N; (b) There exists a nonzero G-invariant finitely additive measure on βN; (c) There exists a G-invariant probability measure on βN; (d) There exists a nonzero G-invariant (finite Borel) measure on βN. Proof. The restriction map from βN to N induces an isometry Φ between the Banach spaces C(βN) and `∞(N). The adjoint Φ∗ then is an isometry between their dual spaces: Φ∗ : `∞(N)∗ → C(βN)∗. The dual of `∞(N) is the space of finitely additive measures on N (cf. [6], IV 8.16) and the dual of C(βN) is the space M of (finite real regular Borel) measures on βN by the Riesz representation theorem (cf. [6], IV 6.2), both spaces with the total variation norm. We further note that, since for f ∈ C(βN) and a permutation g of N the action of g on f is defined by fg(p) = f(gp)∀p ∈ βN, we have (Φf)g(n) = (Φf)(gn) = f(gn) = fg(n) = Φ(fg)(n), (2) i.e. the action of G commutes with the isometry Φ and we may use the notation fg both for the continuous extension of fg ∈ `∞(N) to βN and for the action of g on the continuous extension of f ∈ `∞(N) to βN. Since the space of G-invariant measures on βN is just the annihilator of the set {f−fg : f ∈ C(βN), g ∈ G} and the space of G-invariant finitely additive measures on N is just the annihilator of the set {f − fg : f ∈ `∞(N), g ∈ G}, it follows that G-invariant measures on βN correspond to G-invariant finitely additive measures on N via the isometry Φ∗, since by (2) Φ is a bijection between the above two sets. Finally Φ is order preserving since the restriction to N of a positive function on C(βN) is positive, so means correspond to probability measures and (a) ⇔ (c) and (b) ⇔ (d) follow. Since (c)⇒ (d) is trivial, it remains to show (d) ⇒ (c): Let μ be a nontrivial Ginvariant measure on βN and let βN = X1∪X2 be the Hahn decomposition (cf. [11]) of βN, i.e. μ is nonnegative when restricted to X1 and nonpositive when restricted License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 5090 MARTIN BLÜMLINGER to X2, wher