A NOTE ON INVARIANT MEASURES. Piotr Niemiec

Opuscula Mathematica Vol. 31 No. 3 2011 A NOTE ON INVARIANT MEASURES Piotr Niemiec Abstract. The aim of the paper is to show that if F is a family of cotiuous trasformatios of a oempty compact Hausdorff space Ω, the there is o F-ivariat probabilistic regular Borel measures o Ω iff there are ϕ 1,...,ϕ p F (for some p 2) ad a cotiuous fuctio u: lim if 1 R such that P σ S p u(x σ(1),...,x σ(p) ) = 0 ad P 1 (u Φk )(x 1,...,x p) 1 for each x 1,...,x p Ω, where Φ: (x 1,...,x p) (ϕ 1(x 1),...,ϕ p(x p)) ad Φ k is the k-th iterate of Φ. A modified versio of this result i case the family F geerates a equicotiuous semigroup is proved. Keywords: ivariat measures, equicotiuous semigroups, compact spaces. Mathematics Subject Classificatio: 28C10, 54H15. 1. INTRODUCTION Ivariat measures are preset i may parts of mathematics, icludig harmoic aalysis, ergodic theory ad topological dyamics. Ergodic theory deals with a sigle measurable trasformatio which preserves a fixed measure ad it focuses o properties of the measure-theoretic discrete dyamical system obtaied i this way. The reader iterested i this subject is referred to stadard textbooks such as [6] or [4]. Aother approach to the aspect of ivariat measures, treated i the recet paper, is, i a sese, related to (commo) fixed poit theory ad it cocetrates o the problem of the existece of a measure preserved by all trasformatios of a fixed family. The most classical result i this topic is the Haar measure theorem which states that o every locally compact topological group there is a uique, up to a costat factor, positive regular Borel measure ivariat uder the left shifts of the group (see e.g. [5, 13] or [12]; for a much more geeral result see [9, 10, 19]). This meaigful result plays a importat role i abstract harmoic aalysis ad group represetatio theory ad gave foudatios to this ew brach of mathematics which is still widely ivestigated. This icludes ivariat measures for both the groups as well as the semigroups of cotiuous or measurable trasformatios actig o metric spaces, compact spaces or totally arbitrary topological spaces. There is a huge rage of literature cocerig 425

426 Piotr Niemiec this subject ad we metio here oly a part: [1,3,7,8,11,14 17,20] or a survey article [21] ad refereces therei. The recet paper deals with a arbitrary semigroup F of cotiuous trasformatios of a compact Hausdorff space. Our aim is to give a equivalet coditio for the existece of a Borel regular probabilistic measure ivariat uder each member of F. The coditio reduces the problem of the existece of the measure to the more friedly issue of the oexistece of a cotiuous fuctio satisfyig certai explicitly stated coditios. 2. PRELIMINARIES I this paper Ω is a oempty compact Hausdorff space ad S stads for the group of all permutatios of {1,...,}. Forσ S, let σ :Ω Ω be a fuctio defied by σ(x 1,...,x )=(x σ(1),...,x σ() ). Wheever Φ is a trasformatio of some set, Φ k deotes the k-th iterate of Φ. The algebra of all the cotiuous real-valued fuctios o Ω is deoted by C(Ω), B(Ω) deotes the σ-algebra of all the Borel subsets of Ω ad C(Ω, Ω) stads for the family of cotiuous trasformatios of Ω; M(Ω) is the vector space of all the (siged) real-valued regular Borel measures o Ω ad Prob(Ω) is its subset of probabilistic measures. The space M(Ω) is equipped with the stadard weak-* topology iduced by liear fuctioals of the form M(Ω) μ f dμ R, where f C(Ω). For Ω a cotiuous trasformatio ϕ: Ω Ω, let ˆϕ: M(Ω) M(Ω) be a trasformatio give by the formula ˆϕ(μ)(A) =μ(ϕ 1 (A)) (μ M(Ω), A B(Ω)). (Thus ˆϕ(μ) is the trasport of the measure μ uder the trasformatio ϕ.) The set Prob(Ω) is compact ad the trasformatio ˆϕ is cotiuous i the weak-* topology (for cotiuous ϕ). (For the proof of the secod statemet see e.g. [14].) For measures μ 1,...,μ M(Ω) ad trasformatios ϕ 1,...,ϕ :Ω Ω, we deote by μ 1... μ ( M(Ω )) ad ϕ 1... ϕ the product of μ 1,...,μ ad of ϕ 1,...,ϕ, respectively. Thus ϕ 1... ϕ :Ω Ω ad (ϕ 1... ϕ )(x 1,...,x )=(ϕ 1 (x 1 ),...,ϕ (x )). If F is a family of cotiuous trasformatios of Ω, we say that F is equicotiuous if ad oly if the closure of F i the compact-ope topology of C(Ω, Ω) is compact (i that topology). Equivaletly, F is equicotiuous if for ay poits x, y Ω ad every ope eighbourhood V Ω of the poit y there exist ope subsets U ad W of Ω such that x U, y W ad for each ϕ F, ϕ(u) V provided ϕ(x) W.IfF is equicotiuous ad h C(Ω), the the closure of the set h F = {h ϕ: ϕ F}i the orm topology of C(Ω) is compact. For proofs, details ad more iformatio o the compact-ope topology the reader is referred to [2]. For a family F C(Ω, Ω), let Iv(F) Prob(Ω) be the set of all the F-ivariat measures, i.e. a measure μ Prob(Ω) belogs to Iv(F) if ad oly if ˆϕ(μ) =μ for each ϕ F. Note, for example, that Iv( ) = Prob(Ω). IfF = {ϕ 1,...,ϕ }, we shall write Iv(ϕ 1,...,ϕ ) istead of Iv({ϕ 1,...,ϕ }). The set Iv(F) is always compact (i the weak-* topology) ad the followig result has etered folklore i ergodic theory:

A ote o ivariat measures 427 Theorem 2.1. If ϕ C(Ω, Ω) ad μ Prob(Ω), the every limit poit of the sequece ( 1 1 (ˆϕ)k (μ)) =1 belogs to Iv(ϕ). A variatio of the above result is the crucial key i the proof of Markov s-kakutai s fixed poit theorem see e.g. [18]. Theorem 2.1 is i fact a special case of this variatio. For simplicity, let A p (Ω) (where p 2) deote the family of all cotiuous fuctios u: R such that u σ 0. (2.1) σ S p The followig result is well kow ad easy to prove. Lemma 2.2. For a family F C(Ω, Ω), the followig coditios are equivalet: (i) the set Iv(F) is empty, (ii) there are a atural umber N 2 ad ϕ 1,...,ϕ N F such that Iv(ϕ 1,...,ϕ N )=. 3. MAIN RESULTS Lemma 2.2 says that we may restrict our ivestigatios to fiite sets of trasformatios, which shall be doe i the sequel. For simplicity, we fix the situatio. Let ϕ 1,...,ϕ p (p 2) be members of C(Ω, Ω) ad Φ=ϕ 1... ϕ p : (ote that Iv(Φ) M( )). Additioally, let Iv(S p ) be the collectio of all siged real-valued regular Borel measures o, ivariat uder all permutatios of variables. The followig simple result may be iterestig i itself. Lemma 3.1. Iv(ϕ 1,...,ϕ p ) is oempty iff Iv(Φ) Iv(S p ). Proof. It is easy to check that if μ Iv(ϕ 1,...,ϕ p ), the λ Iv(Φ) Iv(S p ) for λ = μ... μ M( ). Coversely, if λ belogs to both the sets Iv(Φ) ad Iv(S p ), the it is easily verified that a measure μ M(Ω) defied by μ(a) =λ(a 1 )(A B(Ω)) is ϕ j -ivariat for j =1,...,p. So, if we wat to kow whe Iv(ϕ 1,...,ϕ p )=, it is eough to verify whe the sets Iv(Φ) ad Iv(S p ) are disjoit. Sice the first of them is covex ad compact ad the latter is a closed (i the weak-* topology) liear subspace of M( ),thus by the separatio theorem they are disjoit if ad oly if there is u C( ) such that u dμ =0for μ Iv(S p ), but for some positive t we have u dλ t (3.1) for ay λ Iv(Φ).

428 Piotr Niemiec The proof of the followig fact is immediate. Lemma 3.2. Let u C( ). The u dμ =0for each μ Iv(S p ) if ad oly if u A p (Ω). The property (3.1) of the fuctio u which separates the sets Iv(S p ) ad Iv(Φ) ca be reformulated as follows: Lemma 3.3. For a fuctio u C( ) ad a umber t R the followig coditios are equivalet: (i) the iequality (3.1) holds for every λ Iv(Φ), 1 1 (ii) lim if (u Φk )(z) t for each z. Proof. The implicatio (ii) = (i) follows from Fatou s lemma. Ideed, take m R such that u(z) m for each z. The for λ Iv(Φ) we obtai: ( 1 1 lim if ) u Φ k m dλ ( 1 1 lim if (t m) dλ = t m, ) u Φ k m dλ (3.2) but ( 1 1 ) u Φ k m dλ = 1 1 1 u Φ k 1 dλ m = u dˆφ k (λ) m = = 1 1 u dλ m = u dλ m. For the coverse implicatio, fix z ad take a subsequece (s k ) k of the sequece s = 1 1 u(φj (z)) such that lim s k = lim if s. k Let δ be the Dirac measure o with the atom at z. Sice Prob( ) is compact, the sequece μ k = 1 k 1 k ˆΦ j (δ) has a limit poit i Prob( ), say λ. By Theorem 2.1, λ Iv(Φ) ad hece u dλ t. Fially, sice the fuctio M( ) ν u dν R is cotiuous, therefore the itegral u dλ is a limit poit of the sequece u dμ k. But u dμ k = s k lim if s (k + ), which fiishes the proof.

A ote o ivariat measures 429 Now puttig together the above facts, we obtai the mai result of the paper. Theorem 3.4. The followig coditios are equivalet: (i) Iv(ϕ 1,...,ϕ p )=, (ii) there is u A p (Ω) such that for each z, lim if 1 1 (u Φ k )(z) 1. (3.3) Before we stregthe coditio (ii) of the foregoig theorem i case the family {ϕ 1,...,ϕ p } geerates a equicotiuous semigroup, we shall prove the followig Lemma 3.5. Let v A p (Ω). The followig coditios are equivalet: (i) there are c>0 ad a umber 0 1 such that 1 c(v Φk )(z) >for each z ad 0, (ii) there is m 0 such that the fuctio m v Φk has oly positive values. Each of the above coditios implies that Iv(ϕ 1,...,ϕ p )=. Proof. Thaks to Theorem 3.4, it is eough to prove the equivalece of (i) ad (ii). m To see that (ii) implies (i), put ε = if z (v Φk )(z). By (ii) ad the compactess of Ω, ε>0. For simplicity, put l = m +1 ( 1) ad c = 2l ε > 0. Let t = if { k (v Φj )(z): k {0,...,l 1}, z } (observe that t 0, because v is either costatly equal 0 or is ot oegative). Fially, take 0 1 such that t ε > 1 c (3.4) for every 0. Let be a arbitrary atural umber o less tha 0. Express i the form = sl + r, where s 1 ad 0 r<l. From the defiitio of ε it follows that l 1 (v Φj )(Φ ql+r (z)) ε for every z ad q =0,...,s 1. Furthermore, r 1 (v Φj )(z) t (this is true also for r =0, uder the agreemet that =0). Hece, by (3.4): 1 r 1 s 1 l 1 c(v Φ j )(z) =c (v Φ j )(z)+c (v Φ ql+r+j )(z) ( c(t + sε) =c ( 2 >c c ) =, c q=0 t + r l ) ( ε >c t + l l ) ( ε ) ε = c l + t ε > which fiishes the proof of the implicatio (ii) = (i). The coverse implicatio is immediate.

430 Piotr Niemiec The aouced stregtheig of Theorem 3.4 has the followig form: Theorem 3.6. If the family {ϕ 1,...,ϕ p } geerates a equicotiuous semigroup (with the actio of compositio), the the followig coditios are equivalet: (i) Iv(ϕ 1,...,ϕ p )=, (ii) there are v A p (Ω) ad a umber 0 1 such that 1 (v Φk )(z) >for each 0 ad z. Proof. It suffices to prove that (i) implies (ii). Suppose that the set Iv(ϕ 1,...,ϕ p ) is empty ad let u A p (Ω) be as i the statemet of the coditio (ii) of Theorem 3.4. Put v =2u A p (Ω). We shall show that v is the fuctio which we are lookig for. Suppose, to the cotrary, there exist a icreasig sequece ( k ) k of atural umbers ad a sequece (z k ) k of elemets of such that (v Φ j )(z k ) k (k 1). (3.5) k 1 Let v k = 1 k 1 k v Φj. Sice the semigroup geerated by F is equicotiuous, hece so is the semigroup {Φ j : j 0}. This implies that the uiform closure of V = {v Φ j : j 0} is compact ad therefore the closed covex hull of V i C( ) is compact as well. So, replacig evetually the sequece (v k ) k by a suitable subsequece, we may assume that (v k ) k is uiformly coverget to some v 0 C( ).Letz 0 be a limit poit of the sequece (z k ) k. From the uiform covergece of (v k ) k to v 0 we ifer that v 0 (z 0 ) is a limit poit of the sequece (v k (z k )) k ad thus, by (3.5), v 0 (z 0 ) 1. But o the other had, by (3.3): 1 v 0 (z 0 ) = lim v 1 k(z 0 ) lim if k which is a cotradictio. (v Φ j )(z 0 ) = 2 lim if 1 1 (u Φ j )(z 0 ) 2, Other coditios for the oemptiess of Iv(F) i case the family F geerates a equicotiuous semigroup ca be foud i [14]. REFERENCES [1] R.B. Chuaqui, Measures ivariat uder a group of trasformatios, Pacific J. Math. 68 (1977), 313 329. [2] R. Egelkig, Geeral Topology, PWN Polish Scietific Publishers, Warszawa, 1977. [3] P. Erdös, R.D. Mauldi, The oexistece of certai ivariat measures, Proc. Amer. Math. Soc. 59 (1976), 321 322. [4] N.A. Friedma, Itroductio to Ergodic Theory, Va Nostrad Reihold Compay, 1970.

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