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EXPONENTIALLY MIXING, LOCALLY CONSTANT SKEW EXTENSIONS OF SHIFT MAPS FRE´DE´RICNAUD 7 1 0 Abstract. We build examples of locally constant SU2(C)-extensions of the full shift map which are exponentially mixing for the measure of 2 maximal entropy. n a J 1. Introduction and result 0 If T :X X is a uniformly hyperbolic map (for example an Anosov dif- 3 → feomorphism or an expanding map on a smooth manifold), one can produce ] a skew extension by considering the new map T : X G X G defined S × → × D by b T(x,w) := (Tx,τ 1(x)w), . − h at where G is a compact cobnnected Lie group and τ : X → G is a given G- valued function. Such a map is an example of partially hyperbolic system. m Given a mixing invariant probability measure µ for (X,T), the product by [ thenormalized Haar measuremon Gis obviously aT-invariantmeasurefor 1 which we can ask natural questions such as mixing, stable ergodicity etc... v b The qualitative ergodic theory is now fairly well understood, and we refer 9 5 to the works [4, 5, 6]. The paper of Dolgopyat [6] shows that exponential 6 mixing (for regular enough observables) is generic for extensions of smooth 8 expandingmaps,whilerapidmixingisalsogenericforextensionsofsubshifts 0 . of finite type. If the map τ is piecewise constant on X, we call this skew 1 extension locally constant. It is has been observed first by Ruelle [12], that 0 7 locally constant extensions of hyperbolic systems cannot be exponentially 1 mixing when G is a torus. If G = S1 = R/Z, this essentially boils down to : v the fact that when τ takes only finitely many values, Dirichlet box principle Xi shows that one can find sequences of integers Nk such that ar lim sup e2iπNkτ(x) 1 = 0. k x X(cid:12) − (cid:12) →∞ ∈ (cid:12) (cid:12) Foranin-depthstudyoflocallyc(cid:12)onstanttorale(cid:12)xtensions,wereferthereader to [7] where various rates of polynomial mixing are obtained, depending on the diophantine properties of the values of τ. Inthispaper,wewillshowthatsurprisinglyinthenon-commutative case, it is possible to exhibit a large class of locally constant extensions that are exponentially mixing. Let us introduce some notations. Let k 2 be an ≥ integer, and let Σ+ be the one-sided shift space Σ+ = 1,...,2k N, { } Key words and phrases. Symbolic dynamics, Compact Lie group extensions, mixing rates. 1 2 FRE´DE´RICNAUD endowed with the shift map σ :Σ+ Σ+ defined by → (σξ) = ξ , n N. n n+1 ∀ ∈ Given 0 < θ < 1, a standard distance d on Σ+ is defined by setting θ 1 if x =ξ 0 0 d (x,ξ) = 6 θ (cid:26)θN(x,ξ) where N := max n 1 : x = ξ j n otherwise. j j { ≥ ∀ ≤ } Given a continuous observable F : Σ+ G C, which is d -Lipschitz with θ × → respect to the first variable, we will use the norm F θ,G defined by k k 1 F 2 := sup F(x,g)2dm(g)+sup F(x,g) F(ξ,g)2dm(g). k kθ,G x Σ+ZG| | x=ξ d2θ(x,ξ) ZG| − | ∈ 6 We will denote by W2 the completion of the space of continuous functions θ,G F(x,g) on Σ+ G, which are dθ-Lipschitz with respect to x for the . θ,G × kk norm. Pick τ ,...,τ G and set for all ξ Σ+, 1 k ∈ ∈ τ(ξ) = τ if ξ 1,...,k , and τ(ξ) = τ 1 if ξ k+1,...,2k . ξ0 0 ∈ { } ξ−0 k 0 ∈ { } − Our main result is the following. Theorem 1.1. Let µ be the measure of maximal entropy on Σ+. Assume that G= SU (C) and suppose that the group generated by 2 τ ,...,τ ,τ 1,...,τ 1 1 k 1− k− is Zariski dense in G and that τ ,...,τ all have algebraic entries. Then 1 k there exist C > 0, 0 < γ < 1 such that for all F L2(Σ+ G,dµdm), G W2 we have for all n N, ∈ × ∈ θ,G ∈ (cid:12)(cid:12)ZΣ+ G(F ◦σn)Gdµdm−Z FdµdmZ Gdµdm(cid:12)(cid:12) ≤ CγnkFkL2kGkθ,G. I(cid:12)(cid:12)n oth×er wordbs, we have exponential decay of co(cid:12)(cid:12)rrelations for Hölder/L2- observables on Σ+ G, in sharp contrast with the Abelian case. This × surprising behaviour will follow from a deep result of Bourgain-Gamburd [3]. Extensions to more general groups are possible in view of the results obtained more recently [2, 1]. It is also very likely that the result holds for more general equilibrium measures on Σ+, not just the measure of maximal entropy, and more general subshifts of finite type. This work should be pursued elsewhere. A corollary of this exponential rate of mixing is the following Central limittheorem for randomproductsinSU (C), which may beof independent 2 interest. Corollary 1.2. Let F W2 be real valued, and consider the random ∈ θ,G variable on the probability space (Σ+ G,µ m) × × S (F)(ξ,g) n Fdmdµ n Z := − , n √nR R where we have set S (F)(ξ,g) = F(ξ,g)+F(σξ,τ 1g)+...F(σn 1ξ,τ 1 ...τ 1g). n ξ−0 − ξ−n−2 ξ−0 LOCALLY CONSTANT SKEW EXTENSIONS 3 Then there exists σ 0 such that as n , Z converges to a gaussian F n variable N(0,σ ) in la≥w. → ∞ F The validity of this CLT for generic extensions of hyperbolic systems (including locally constant extensions) and smooth enough F with respect to the G-variable is due to Dolgopyat in [6]. The point of our result is that no smoothness is required on F with respect to the G-variable, which is unusual in the CLT literature for hyperbolic systems. In particular, if F(ξ,g) = f(g) is in L2(G), then the CLT holds. This CLT follows directly fromabsolutesummabilityofcorrelation functionsandtheresultofLiverani [11], more details can be found at the end of section 2. § 2. Proofs 2.1. Bourgain-Gamburd’s spectral gap result. As mentioned earlier, the proof is based on a result of Bourgain-Gamburd [3] on the spectral gap of certain Hecke operators acting on L2(G). We start by recalling their result. Given S = τ ,...,τ G, one consider the operator given by 1 k { } ⊂ k 1 T (f)(g) := f(τ g)+f(τ 1g) . S 2k ℓ ℓ− Xℓ=1(cid:0) (cid:1) This operator T is self-adjoint on L2(G) and leaves invariant the one- S dimensional space of constant functions. Let L2(G) be 0 L2(G) := f L2(G) : fdm = 0 . 0 (cid:26) ∈ Z (cid:27) The main result of [3] asserts that if the generators τ ,...,τ have algebraic 1 k entries and the group Γ = τ ,...,τ ,τ 1,...,τ 1 h 1 k 1− k− i is Zariski dense, then T has a spectral gap i.e. S T < 1. k S|L20(G)k By self-adjointness, this is equivalent to say that theL2-spectrumof T con- S sists of the simple eigenvalue 1 while the rest of the spectrum is included { } in a disc of radius ρ < 1. A special case is when Γ is a free group, which is enough for many applications. A consequence of the spectral gap property is the following. If T has a spectral gap then there exists 0 < ρ < 1 such S that for all n 0 and f L2, ≥ ∈ (1) Tn(f) fdm 2ρn f . (cid:13) S −Z (cid:13) ≤ k kL2 (cid:13) (cid:13)L2 (cid:13) (cid:13) This deep result of s(cid:13)pectral gap is re(cid:13)lated to the (now solved) Ruziewicz measure problem on invariant means on the sphere. For more details on the genesis of these problems and explicit examples, we refer the reader to the book of Sarnak [13], chapter 2, and to the paper [8] which was the starting pointof[3]. Thisestimateisthemainingredientoftheproof,combinedwith a decoupling argument and exponential mixing of the measure of maximal 4 FRE´DE´RICNAUD entropy. We recall that given a finite word α 1,...,2k n, the cylinder ∈ { } set [α] of length α = n associated to α is just the set | | [α] := ξ Σ+ : ξ = α ,...,ξ = α . 0 1 n 1 n { ∈ − } We will use the following Lemma. Lemma 2.1. For each α 1,...,2k n, choose a sequence ξ [α]. let F α ∈ { } ∈ be an observable with F θ,G < + . k k ∞ Then we have the bound for all n 1, ≥ 1 (cid:12)(cid:12)(2k)n Z F(ξα,g)dm(g)−Z Z Fdµdm(cid:12)(cid:12) ≤ kFkθ,Gθn. (cid:12) αX=n G (cid:12) (cid:12) | | (cid:12) (cid:12) (cid:12) Proof. T(cid:12)his is a rephrasing of the fact that the measu(cid:12)re of maximal entropy is exponentially mixing for Hölder observables on Σ+. Indeed, recall that in our case, the measure of maximal entropy µ is just the Bernoulli measure such that 1 µ([α]) = (2k)n when α = n. By Schwarz inequality, for all x [α], we have | | ∈ F(x,g)dm(g) F(ξα,g)dm(g) F θ,Gdθ(ξα,x) (cid:12)ZG −ZG (cid:12) ≤ k k (cid:12) (cid:12) (cid:12)(cid:12) F θ,Gθn. (cid:12)(cid:12) ≤ k k Writing Fdµdm = F(x,g)dm(g), Z Z αX=nZ[α]ZG | | we deduce (cid:12) µ([α]) F(ξα,g)dm(g) Fdµdm(cid:12) F θ,Gθn, (cid:12) Z −Z Z (cid:12) ≤ k k (cid:12)αX=n G (cid:12) (cid:12)| | (cid:12) and the(cid:12)proof is done. (cid:3) (cid:12) (cid:12) (cid:12) In the sequel, we will use the following notation: given a finite word α 1,...,2k n and ξ Σ+, we will denote by αξ the concatenation of ∈ { } ∈ the two words i.e. the new sequence αξ Σ+ such that (αξ) = α for j j+1 ∈ j = 0,...,n 1 and σn(αξ) = ξ. We recall that given f,g C0(Σ+), we − ∈ have the transfer operator identity (f σn)gdµ = fLn(g)dµ, ZΣ+ ◦ ZΣ+ where we have 1 Ln(g)(ξ) = g(αξ). (2k)n X α=n | | This identity follows straightforwardly from the σ-invariance of the measure µ and its value on cylinder sets. Notice that the operator L is normalized i.e. satisfies L(1) = 1, LOCALLY CONSTANT SKEW EXTENSIONS 5 which will be used throughout all the computations below. Notice also that using the above notations, we have for all f L2(G), for all x Σ+, ∈ ∈ 2k 1 T (f)(g) = f(τ(ℓx)g), S 2k Xℓ=1 while 1 Tn(f)(g) = f(τ(βx)...τ(β β x)τ(β x)g). S (2k)n n−1 n n βX=n | | The fact that τ(x) depends only on the firstcoordinate of x is critical in the above identities. 2.2. Main proof. We now move on to the proof of the main result. Let F,G C0(Σ+ G), and compute the correlation function: ∈ × (F σn)Gdµdm Z Z ◦ b = F(σnx,τ 1(σn 1x)...τ 1(σx)τ 1(x)g)G(x,g)dµ(x)dm(g). − − − − Z Z By using Fubini and translation invariance of the Haar measure we get (F σn)Gdµdm = F(σnx,g)G(x,τ(n)(x)g)dm(x)dm(g), Z Z ◦ Z Z where b τ(n)(x) = τ(x)τ(σx)...τ(σn 1x). − Again by Fubini and the transfer operator formula we get (F σn)Gdµdm = FLn(G)dµdm, Z Z ◦ Z Z where b b 1 Ln(G)(x,g) = G(αx,τ(n)(αx)g). (2k)n X α=n b | | The main result will follow from the following estimate. Proposition 2.2. There exist C > 0 and 0 <γ < 1 such that for all n 1, ≥ sup Ln(G)(x,g) Gdµdm C G θ,Gγn. x Σ+(cid:13)(cid:13) −Z Z (cid:13)(cid:13)L2(G) ≤ k k ∈ (cid:13)b (cid:13) Indeed, write (cid:13) (cid:13) FLn(G)dµdm Fdµdm Gdµdm Z Z −Z Z Z Z b = F Ln(G) Gdµdm dµdm, Z Z (cid:18) −Z Z (cid:19) and use Schwarz inequalitybcombined with the above estimate to get the conclusion of the main theorem. Let us prove Proposition 2.2. Writing n = n +n , we get 1 2 1 1 Ln(G)(x,g) = G(αβx,τ(n)(αβx)g). (2k)n1 (2k)n2 b |αX|=n1 |βX|=n2 6 FRE´DE´RICNAUD Observe that we have (since τ depends only on the first coordinate) τ(n)(αβx) = τ(αβx)...τ(α βx)τ(βx)...τ(β x). n1 n2 depends only on α depends only on β For all word α 1,...,2|k n1, we{zchoose ξ }| [α], an{zd set } α ∈ { } ∈ G (g) := G(ξ ,τ ...τ g). α α α1 αn1 We now have 1 Ln(G)(x,g) = Tn2(G )(g)+R (x,g), (2k)n1 S α n X b |α|=n1 where the ”remainder” R (x,g) is n R (x,g) = n 1 1 G(αβx,τ(n)(αβx)g) G (τ ...τ g) . (2k)n1 αX=n1 (2k)n2 βX=n2(cid:16) − α β1 βn2 (cid:17) | | | | Notethatforallx Σ+,wehavebytranslation invarianceof Haarmeasure, ∈ (cid:13)G(αβx,τ(n)(αβx)g)−Gα(τβ1...τβn2g)(cid:13)L2(G) ≤ kGkθ,Gdθ(αβx,ξα), (cid:13) (cid:13) and(cid:13)thus (cid:13) Rn(x,g) L2(G) θn1 G θ,G. k k ≤ k k On the other hand, using Lemma 2.1, we have 1 (cid:13)(cid:13)(2k)n1 TSn2(Gα)(g)−Z Z Gdµdm(cid:13)(cid:13) (cid:13)(cid:13) |αX|=n1 (cid:13)(cid:13)L2(G) (cid:13) (cid:13) 1 (cid:13) (cid:13) ≤ (2k)n1 |αX|=n1(cid:13)(cid:13)(cid:13)TSn2(Gα)−ZGG(ξα,g)dm(g)(cid:13)(cid:13)(cid:13)L2(G)+O(kGkθ,Gθn1). By the spectral ga(cid:13)p property (1), (cid:13) (cid:13)(cid:13)TSn2(Gα)−ZGG(ξα,g)dm(g)(cid:13)(cid:13)L2(G) ≤ 2ρn2kGαkL2(G) ≤ 2ρn2kGkθ,G, (cid:13) (cid:13) the(cid:13)refore we have obtained, unifo(cid:13)rmly in x Σ+, ∈ Ln(G)(x,g) Gdµdm = O( G (θn1 +ρn2)), θ,G (cid:13) −Z Z (cid:13) k k (cid:13) (cid:13)L2(G) (cid:13)b (cid:13) and the(cid:13)proof ends by choosing n =(cid:13)[n/2], n = n n . (cid:3) 1 2 1 − In the proof, we have used no specific information about the group G, except that T has a spectral gap. Therefore the main theorem extends S without modification to the case of G = SU (C) and more generally any d compact connected simple Lie group by [1]. WealsopointoutthattherateofmixingobtainedisO (max √θ,√ρ )n , { } (cid:16) (cid:17) whichcanbemadeexplicitifT hasanexplicitspectralgap,see[8]forsome S examples arising from quaternionic lattices. On the other hand, if Γ = τ ,...,τ ,τ 1,...,τ 1 h 1 k 1− k− i LOCALLY CONSTANT SKEW EXTENSIONS 7 is free, it follows from a result of Kesten [10] that the L2-spectrum of T S contains the full continuous segment √2k 1 √2k 1 − ,+ − , (cid:20)− k k (cid:21) n which suggests that the rate of mixing cannot exceed O √2k 1 . k− (cid:16)(cid:16) (cid:17) (cid:17) 2.3. Central Limit Theorem. The CLT, has stated in the introduction follows from the paper of Liverani [11], section 2 on non invertible, onto § maps. Following his notations we have for all F L2(dmdµ), ∈ T(F) = F σ, ◦ while the L2-adjoint T is T = L. Given F W2 with Fdmdµ = 0, ∗ ∗ b ∈ θ,G we know that : R R b (1) we have by Theorem 1.1 ∞ (F σℓ)Fdµdm < , (cid:12)Z Z ◦ (cid:12) ∞ Xℓ=0(cid:12) (cid:12) (cid:12) b (cid:12) (2) by Proposition 2.2(cid:12), the series (cid:12) Lℓ(F), Xℓ N ∈ b converges absolutely almost surely (actually in L1(dmdµ)). We can therefore apply Therorem 1.1 from [11], which says that the CLT holdsandthat thevarianceσ isvanishingifand only ofF is acoboundary. F Alternatively, since we have a spectral gap for L 1 one could also use a spectral method and perturbation theory to prove the CLT, see [9] for a b survey on this approach for proving central limit theorems. Acknowledgments. Frédéric Naud is supported by ANR GeRaSic and Institut universitaire de France. References [1] Yves Benoist and Nicolas de Saxcé. A spectral gap theorem in simple Lie groups. Invent. Math., 205(2):337–361, 2016. [2] J. Bourgain and A. 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Eur. Math. Soc. (JEMS), 1(1):51–85, 1999. [9] Sébastien Gouëzel. Limit theorems in dynamical systems usingthespectral method. InHyperbolicdynamics,fluctuationsandlargedeviations,volume89ofProc.Sympos. Pure Math., pages 161–193. Amer. Math. Soc., Providence, RI,2015. [10] HarryKesten.Symmetricrandomwalksongroups.Trans.Amer.Math.Soc.,92:336– 354, 1959. [11] CarlangeloLiverani.Centrallimittheoremfordeterministicsystems.InInternational Conference on Dynamical Systems (Montevideo, 1995), volume 362 of Pitman Res. Notes Math. Ser., pages 56–75. Longman, Harlow, 1996. [12] David Ruelle. Flots qui ne mélangent pas exponentiellement. C. R. Acad. Sci. Paris Sér. I Math., 296(4):191–193, 1983. [13] Peter Sarnak. Some applications of modular forms, volume 99 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1990. Laboratoire de Mathématiques d’Avignon, Campus Jean-Henri Fabre, 301 rue Baruch de Spinoza, 84916 Avignon Cedex 9. E-mail address: [email protected]