D ynamics of C ohomological E xpanding M appings I : First and Second Main Results

Let f : V −→ V be a Cohomological Expanding Mapping 1 of a smooth complex com- 1 pact homogeneous manifold with dim C ( V ) = k ≥ 1 and Kodaira Dimension ≤ 0 . We 2 study the dynamics of such mapping from a probabilistic point of view, that is, we de- 3 scribe the asymptotic behavior of the orbit O f ( x ) = { f n ( x ) , n ∈ N or Z } of a 4 generic point. Using pluripotential methods, we construct a natural invariant canonical 5 probability measure of maximum Cohomological Entropy µ f such that χ − m 2 l ( f m ) ∗ Ω → 6 µ f as m → ∞ for each smooth probability measure Ω on V . Then we study the 7 main stochastic properties of µ f and show that µ f is a measure of equilibrium, smooth, er- 8 godic, mixing, K-mixing, exponential-mixing and the unique measure with maximum Co- 9 homological Entropy. We also conjectured that µ f := T + l ∧ T − k − l , dim H ( µ f ) = Ψ h χ ( f ) 10 and dim H ( Supp T + l ) ≥ 2( k − l ) + log χ 2 l ψ l . 11

14 When X is a compact metric space, we can apply Birkhoff's theorem to continuous functions ϕ and deduce that for ν almost all Clearly, mixing implies ergodicity. It is not difficult to see that ν is mixing if, and only if, for any test functions ϕ, ψ on L ∞ (ν) or on L 2 (ν), we have lim n→∞ ν, (ϕ • g n )ψ = ν, ϕ ν, ψ .
The Quantity I n (ϕ, ψ) := | ν, (ϕ • g n )ψ − ν, ϕ ν, ψ | is called the correlation on time n of ϕ and ψ. Thus, mixing is equivalent to the convergence of I n (ϕ, ψ) to 0. We say that ν is K-mixing if for each ψ ∈ L 2 (ν) sup ϕ L 2 (ν) ≤1 χ i (f ) = ρ f * Definition 1. 6. We say that f is a Cohomological Expanding Mapping when f is dynamically compatible 24 (that is (f n ) * = (f * ) n ) and there is l ∈ {1, ..., k} such that : We will write χ i for χ i (f ) and ξ n l for ξ n l (f ) if there is no confusion. 26 27 Let (M, F, m) be a probability space and g : M → M be a measurable map that preserves m, that is, m is g * -invariant: g * m = m. The measure m is ergodic if for any measurable set A such that g −1 (A) = A, we have m(A) = 0 or m(A) = 1. This is equivalent to the property that m is extremal on the convex set of invariant probability measures (if m is mixing, so it is ergodic). When m is ergodic, Birkhoff's theorem implies that if ψ is an observable on L 1 (m) then lim n→∞ 1 n ψ(x) + ψ(g(x)) + · · · + ψ(g n−1 (x)) = m, ψ for m -almost all x. 28 Suppose now that m, ψ = 0. Then, the previous limit is equal to 0. The theorem of limit central (TLC), when it occurs, provides the speed of this convergence. We say that ψ satisfies the TLC if there is a constant σ > 0 such that 1 √ n ψ(x) + ψ(g(x)) + · · · + ψ(g n−1 (x)) converges in distribution for the Gaussian random variable N(0, σ) of mean 0 and variance σ. Remember that ψ is a coboundary whether there is a function ψ on L 2 (µ) such that ψ = ψ − ψ • g. In that case, it is easy to see that lim n→∞ 1 √ n ψ(x) + ψ(g(x)) + · · · + ψ(g n−1 (x)) = lim n→∞ 1 √ n ψ (x) − ψ (g n (x)) = 0 in distribution. Therefore, ψ does not satisfy the TLC (sometimes it is said that ψ satisfies the TLC by The TLC can be deduced from strong mixing, see [11,46,48]. In the following result, Et(ψ|F n ) indicates 3 the expectation of ψ in relation to F n , that is, ψ → Et(ψ|F n ) is the orthogonal projection of L 2 (m) in the 4 subspace generated by the measurable functions F n . 5 Theorem 1.7 (Gordin). Consider the decreasing sequence F n := g −n (F), n ≥ 0, of algebras. Let ψ be a function with real value on L 2 (m) such that m, ψ = 0. Suppose that n≥0 Et(ψ|F n ) L 2 (m) < ∞.
So, the positive number σ defined by is finite. It vanishes if and only if ψ is a coboundary. Furthermore, when σ = 0, then ψ satisfies the TLC 6 with variance σ.

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Note that σ is equal to the limit of n −1/2 ψ + · · · + ψ • g n−1 L 2 (m) . The last expression is equal to 8 ψ L 2 (m) if the family (ψ • g n ) n≥0 is orthogonal on L 2 (m). 9 We refer to [47,49] for the notion of Lyapunov exponent. 10 Definition 1.8. An invariant positive measure is hyperbolic if its Lyapunov exponents are non-zero.

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A function quasi-p.s.h. on V is a function of V on [−∞, ∞), which is locally the sum of a plurisubharmonic Next, we will consider the dynamics of f with ξ −1 l (f ) > 1.

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Here is the first Main Result. Entropy ≥ log χ 2l independent of ν as m → ∞ so that then for each Hermitian metric ω on V, µ f is Hölder continuous on PSH 0 (ω).

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The Hölder continuity of µ f on PSH 0 (ω) for f holomorphic implies that µ f is moderate in the sense that there are constants ε, M > 0 such that for each ϕ ∈ PSH 0 (ω), we have Here is the second Main Result. 6 Theorem 1. 10. Let V, f, χ 2l , µ f be as in Theorem 1.9. So µ f is exponential mixing in the sense that for each constant 0 < α ≤ 1, there is a constant A α such that for each m ≥ 0, each ψ ∈ L ∞ (V) and every function Hölder continuous ϕ of order α. In particular, µ f is 7 K-mixing. 8 If a real function Hölder continuous ϕ is not a coboundary, i.e, there is not ψ ∈ L 2 (V) with ϕ = ψ • f − ψ, and satisfies µ, ϕ = 0, then µ f satisfies the central limit theorem, which means that there is a constant σ > 0 such that for each interval I ⊂ R, we have The expression µ f , (ψ • f m )ϕ − µ f , ψ µ f , ϕ is called the Correlation of order m between the observ-9 ables ϕ and ψ. The measure µ f is said mixing if this correlation converges to 0, when m tends to infinity, 10 for smooth observables (or equivalently, observables continuous, limited or L 2 (µ f ) ).

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Remember that f * ϕ is defined by where the points on f −1 (x) are counted with multiplicities (there are exactly χ 2k points). Also define the Perron-Frobenius Operator by Λϕ := χ −1 2k f * ϕ. As µ f is totally invariant, this is the adjoint operator of f * on L 2 (µ f ). In this section, we will prove the Theorem 1.9. For a current T of order 0 defined in a manifold V, we denote 14 by T V the mass of T on V. Let's write (resp. ) for ≤ (resp. ≥) module a multiplicative constant 15 independent of involving terms in inequality. 16 Theorem 2.1 (Theorem 1.9 " First Main Result "). Let V be a smooth compact complex homogeneous 17 manifold with dim C (V) = k ≥ 1 and Kodaira dimension ≤ 0 and f : V −→ V a Cohomological 18 Expanding Mapping. Let ν be a complex measure with density L 2k+1 on V such that ν(V) = 1. Let 19 ω be a (1, 1)-strictly positive Hermitian form on V. So the sequence 1 there is a function U R on L 1+1/(2k) (B r ) so that the following three properties are verified: is a sequence of (1, 1)-closed real currents of order 0 of uniformly limited mass, converging 1 weakly to R on B so U Rn → U R on L 1+1/(2k) (B r ).

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Proof. The new point is the estimate for the norm L 1+1/(2k) of the potential U R and its continuity on R.
3 These properties will be obtained by carefully examining the steps in the usual construction of U R , cf [15, p.  Let R be a (1, 1)-real current closed on B. Let x ∈ C k be the canonical coordinate system. Let ρ be a smooth function supported compactly on B and B ρdx = 1. For y ∈ B, let A y : B → B be the diffeomorphism defined by is a smooth closed form that is cohomologous to R. Precisely, by the formula of homotopy, we have Since R is a smooth closed form on B, we can use an explicit formula (cf [15, p. 13]) to define a smooth form L 2 = L 2 (R ) on B such that This combined with (2.1) shows that for L 3 := L 1 + L 2 , we have and L 3 continuously depends on R. So if (R n ) n∈N is a sequence of (1, 1)-currents of order 0 with uniformly 8 limited mass, converging towards R so L 3 (R n ) is also of uniformly limited mass and converges to L 3 (R).

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Since R is a (1, 1)-real form, L 3 is a 1-real form. We decompose L 3 in the sum of one (1, 0)-form and a are currents of order 0. We deduce from (2.2) that For a bidirectional reason and the fact that R = dL 3 , we have∂L (y). (2.5) We do not give explicit formulas here for K 1 , K 2 but we emphasize only that K 1 , K 2 are the products of x − y −2k+1 with smooth forms on C k .

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Denote by I 1 , I 2 the first and second integrals, respectively, on the right side of (2.5). We havē By the type of singularity of K 1 and the fact that L (0,1) 3 is of order 0, we see that I 1 is a form with coefficients by (2.4) and I 3 : We deduce from this and (2.2) that .

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It remains to prove the property of continuity of U R . We saw that I 3 , L 3 are continuous on R. We just need to check this property to I 1 . Let (R n ) be the sequence as defined above. Let's show that I 1 (R n ) → I 1 (R) on L 1+1/(2k) (B). For the continuity property above of L 3 , we have that S n := ρL (0,1) 3 (R n ) is of uniformly limited mass and converges to S := ρL where K 1 (x, y) is a smooth form. For every small constant ε > 0, let which is a continuous form. Since ε → 0, we have K 1,ε (·, y) → K 1 (·, y) on L 1+1/(2k) (B) uniformly on y ∈ B. So when n → ∞, converges uniformly to 0 as ε is fixed because K 1,ε is continuous. We deduce that . This completes the proof.  . We use a specific case of this result: each analytic proper subset of a compact manifold V is 3 pluripolar, cf Lemma 2.11 above. 4 Now consider that V is compact. Let µ 0 be a smooth probability measure on V. We use this measure to The function · W is a norm on W because if dd c ϕ = 0 then ϕ must be a constant. However, we do not know whether these two norms are equivalent in this case. 10 We introduce the topology on W in the following way: we say that ϕ n ∈ W converges to ϕ ∈ W when 11 n → ∞ if ϕ n → ϕ as current and ϕ n W is uniformly limited. 12 We have the following compactness result.

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Lemma 2.6. Let V be a compact complex manifold . There is a constant c so that for each function weakly (2.10) Therefore, ϕ − τ j can be represented by a pluriharmonic function on W j . For simplicity, we identified this 25 function with (ϕ − τ j ). We deduce that ϕ ∈ L 1+1/(2k) (V). 26 We now assume, on the contrary, that (2.9) is not valid, it means that there is a sequence of non-null functions weakly quasi-p.s.h. ϕ n with V ϕ n dµ 0 = 0 and Multiplying ϕ n by a positive constant, we can assume that So we have 28 dd c ϕ n ≤ 1/n.
(2.12) Note that we still have V ϕ n dµ 0 = 0. Let τ n j be the function τ j for ϕ n in place of ϕ. Put T n := dd c ϕ n .

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These currents of order 0 are of uniformly limited mass and converge to 0 by (2.12). The Lemma 2.2 tells 30 us that τ n j converges to 0 on L 1+1/(2k) (W j ), for each W j W j . We can also provide that (W j ) continue 31 to be a cover of V. For simplicity, we can assume that W j = W j for each j.

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Now remember that ϕ n − τ n j is pluriharmonic on W j . The last function is of L 1+1/(2k) -norm limited on W j because of (2.10) and (2.11). The average equality for pluriharmonic functions implies that (ϕ n − τ n j ) is of C l -norm uniformly limited on compact subsets of W j on n ∈ N for each l ∈ N. We deduce that, extracting a subsequence, we can assume that ϕ n − τ n j converging uniformly to a pluriharmonic function τ ∞ j on compact subsets of W j when n → ∞. Since τ n j L 1+1/(2k) (Wj ) → 0, we get that This produces this function τ ∞ := τ ∞ j on W j for each j is a well-defined pluriharmonic function on V. 1 Since V is compact, τ ∞ is a constant. This combined with V ϕ n dµ 0 = 0 gives τ ∞ = 0. We proved that 2 ϕ n → 0 on L 1+1/(2k) (V), consequently ϕ n L 1+1/(2k) → 0, a contradiction. Therefore, (2.9) is verified.

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To prove the second desired statement, we again use the function τ j above. We have that ϕ − τ j is pluri-4 harmonic on W j and by (2.9), the L 1+1/(2k) -norm of ϕ is also A. Then the L 1+1/(2k) -norm of the 5 pluri-harmonic function (ϕ − τ j ) is A. It follows that its C l -norm is also A . Therefore, we can extract 6 a convergent subsequence of (ϕ − τ j ) for ϕ ∈ W on C l . This combined with the L 1+1/(2k) continuity of τ j 7 on T implies the desired statement. This completes the proof. be a continuous linear endomorphism of the last vector space. Define W P to be the set of ϕ ∈ W for which 12 P ϕ ∈ W.

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Lemma 2.7. There is a constant c such that for any ϕ ∈ W P . In particular, there is a constant c such that Proof. The Inequality (2.14) is a direct consequence of (2.13) and of Lemma 2.6. Now suppose there is a Applying compactness property in Lema 2.6 for the sequence (P ϕ n ) n∈N , we see that by extracting a sub- 20 sequence from ϕ n if necessary, the sequence P ϕ n converges on L 1+1/(2k) for a function weakly d.s.h ϕ ∞ .

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Consequently, (2.16) Therefore ϕ ∞ is a constant. As the convergence on L 1 implies the convergence almost always of a subse-23 quence, we can also assume that P ϕ n converges almost always to ϕ ∞ .

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On the other hand, the inequality of (2.15) allows us to use the compactness property in the Lemma 2.6 25 again for (ϕ n ). Therefore, we can extract a subsequence of (ϕ n ) converging to ϕ ∞ := 0 on L 1+1/(2k) and 26 almost always. Thus P ϕ n converges almost always to P ϕ ∞ because of the continuity of P. It follows that 27 ϕ ∞ = P ϕ ∞ = 0, note here P (0) = 0 by the linearity of P. This is a contradiction because of (2.16). Thus

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(2.13) follows. The last desired statement follows directly from the arguments above. This completes the 29 proof.

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Let a ∈ C * , r be a constant on (0, |a|) and δ > 0 a constant. Assume that P (1) = a, where 1 is the constant function equal to 1 on V. Define W ∞ P,r,δ to be the set of all ϕ ∈ B such that P n ϕ ∈ W for each n ≥ 0 and dd c (P n ϕ) ≤ δr n for each n ≥ 0, here P 0 denotes the identity map. By the linearity of P, every constant function belongs to W ∞ P,r,δ . We equip W ∞ P,r,δ with the topology induced from there on W. Note that W ∞ P,r,δ is closed on W and r −m P m (W ∞ P,r,δ ) ⊂ W ∞ P,r,δ for every positive integer m. So W ∞ P,r,δ ∩ W 0 is compact and P m (W ∞ P,r,δ ) is contained in the complex 32 vector subspace ‹ W ∞ P,r,δ of W generated by W ∞ P,r,δ .

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Consider a sequence ϕ m → ϕ on W ∞ P,r,δ . Let b nm , ϕ nm respectively the b n and ϕ n for ϕ m in place of ϕ.
14 By the last statement of the Lemma 2.7, b nm → b n when m → ∞ for each n and (2.19) still applies to 15 b nm , ϕ nm in place of b n , ϕ n . We infer that b nm → b n and a −n ϕ nm → 0 on L 1+1/(2k) when m → ∞.

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Let V be a complex compact manifold and f be a meromorphic self-map on V. Denote by Γ the graph of f 19 on V × V and π 1 , π 2 the restrictions to Γ of natural projections of V × V for the first and second components 20 respectively.

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Let Φ be a form with measurable coefficients on V. We say that Φ ∈ L 1 if its coefficients are L 1 functions 22 (in relation to the Lebesgue measure on V). If Ω is a dense open subset of Zariski of V such that π 2 is a 23 unrestricted cover on Ω, the form f * Φ := (π 2 | π −1 2 (Ω) ) * (π * 1 Φ) is a measurable form on Ω. Consequently f * Φ 24 is a measurable form on V independent of Ω. We can verify that f * : B → B is continuous. Consequently, 25 f * is an example of the map P considered above.

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If f * Φ ∈ L 1 , then we can define f * Φ to be a current of order 0 induced by f * Φ on V. This definition is 27 independent of the choice of Ω. Note that the pull-back by f of smooth functions or smooth forms is always 28 on L 1 . The following is similar to the results on [9, 23].
5 Note that f * η has finite mass on V. We infer that f * ϕ ∈ W. In other words, ϕ ∈ W f * ∩ W. Applying this to 6 f n instead of f and using the formula that (f n ) * ϕ = f * (f n−1 ) * ϕ as functions in some suitable open dense 7 subset of V, we get (2.23). This completes the proof. for each n ≥ 1.

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Proof. Replacing η by a strictly positive smooth form that dominates it, we can assume that η > 0. Let ω be a metric of Gauduchon on V, this means that ω is a Hermitian metric and dd c ω k−1 = 0, cf [?]. Let Γ n be the graph of f n and π 1,n , π 2,n the natural maps of Γ n for the first and second components of V × V. By Lemma 2.9, the current dd c (f n ) * ϕ + (f n ) * η is positive. So, using dd c ω k−1 = 0 gives This combined with the definition of χ 2l−1 (f ) gives The desired inequality follows immediately. This completes the proof.

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We come now to the end of the proof of the first main result. because we only need to consider integrals on a dense open subset of Zariski of V. Applying Proposition 2.8 for P , we get a continuous functional µ P on ‹ W ∞ P,r,δ such that χ −m 2l (f m ) * ν, ϕ → µ P , ϕ , for each ϕ ∈ ‹ W ∞ P,r,δ . Choosing ν ≥ 0, we see that µ P , ϕ ≥ 0 if ϕ ≥ 0. Let µ f be the probability measure is obtained by applying the pull-back f * for convergence using f * µ f = χ 2l µ f , we deduce that the cohomological entropy of µ f is at least log χ 2l . 20 Suppose now that f is holomorphic. To prove that µ f is Hölder continuous on PSH(ω), we use a known 21 idea of [24]. Without loss of generality, we can assume that ω L ∞ ≤ 1. Let ϕ, ψ be two functions quasi-22 p.s.h. on PSH(ω). Remember that they are on W ∞ P,r,δ .
Let b n (ϕ), b n (ψ) be as in the proof of the proposition 2. 8. Let J f be the Jacobian of f. We have Using (2.20) gives which implies that µ f is Hölder continuous in that case. It remains to treat the case 2 J f L ∞ ≥ χ 2l . We have . We see that Consequently, µ f is also Hölder continuous in this case. This completes the proof. decay for µ f . 6 Lemma 2.11. Any proper analytic subset V of a complex compact manifold V is a pluripolar set on V. 7 Proof. We use here the idea in [22] where the authors prove the same result when V is Kähler. Suppose 8 now that V is smooth and codimV ≥ 2 (otherwise the problem is trivial). Let σ : V → V be the explosion 9 of V along V. Denote by " V the exceptional hypersurface. 10 Let ω be a positive-defined Hermitian form on V. Let ω h be a form of Chern of O(− " V ) whose restriction to 11 each fiber of " V ≈ P(E) is strictly positive. Choosing ω if necessary, we can assume that ω := σ * ω + ω h > 12 0. Since σ * ω h = σ * ω − ω, the closed current σ * ω h is quasi positive. Thus, there is a function quasi-p.s.h.

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ϕ on V such that 14 σ * ω h = dd c ϕ + η (2.28) for some smooth closed form η. Multiplying ω h by a strictly positive constant, we have σ cross RegV (or their inverse images) so that the strict transformation " V of V is smooth.
14 Lemma 3.4. For each ϕ ∈ W 1,2 * ,f , if m = (m 0 , m 1 ) is a representative of size of ϕ, then m is also a size 15 representative of |ϕ|, ϕ + and ϕ − . 16 We already know that the pushforward of a function quasi-p.s.h. by f is a function weakly d.s.h. The 3) for some constant c independent of ϕ. unbranched cover on f −1 (Ω). We have f * ϕ ∈ L 1 loc (Ω) and for any compact K on Ω and some constant c independent of ϕ. Note that V\Ω is a proper analytical subset of V, Thus , is of Hausdorff (2k − 1)-dimensional and zero measure. On Ω, by Cauchy-Schwarz inequality, we have It follows that d(f * ϕ) ∈ L 2 (Ω). For this and by Lemma 3.1, we get f * ϕ ∈ W 1,2 . Thus, i∂(f * ϕ) ∧∂(f * ϕ) has no mass on V\Ω. It follows that Lemma 3.6. m n := (χ n 2l m 0 , m 1 + n) is also a size representative of |ϕ n |, ϕ + n and ϕ − n . for n ≥ 1 and some possible different constant A. Now we are in a situation very similar to the one in the last section. Using arguments similar to those in the last section, we can show that lim n→∞ χ −n 2l (f n ) * ω k , ϕ exists and denote by b ∞ (ϕ) its limit. In fact, we have It follows that for some constant A independent of ϕ. Clearly, if ϕ is a function quasi-p.s.h limited , b ∞ is equal to the same number defined in the last section. So we have for function quasi-p.s.h limited ϕ. Let W 1,2 * * ,f the subset of W 1,2 * ,f consisting of functions that are continuous 9 outside a closed pluripolar set. Note that f * preserves W 1,2 * * ,f because f is a covering outside an analytical 10 subset of V. We now affirm that 11 Lemma 3.7. For ϕ ∈ W 1,2 * * ,f , we have µ f , ϕ = b ∞ (ϕ).

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Proof. The proof is similar to that on [21, Lemma 5.5]. We proved first that ϕ is µ f -integrable. We assume for a moment that ϕ ≥ 0. Let V be a closed pluripolar set such that ϕ is continuous outside of V.
(3.9) Note that, as before, we have |c n | ≤ AA n (ϕ) for some constant A independent of ϕ. On the other hand, f * preserves W 1,2 * * ,f , thus ϕ n ∈ W 1,2 * * ,f and so is |ϕ n |. By Lemma 3.6, (χ n 2l m 0 , m 1 + n) is a size representative of |ϕ n | if (m 0 , m 1 ) is a size representative of ϕ. Arguing as in the proof of Lemma 3.7 gives that χ −n 2l | µ f , |ϕ n | | ≤ AA n (ϕ) for some constant A independent of ϕ. Hence the desired inequality follows. This completes the proof.

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End of Proof of Theorem 1.10. The central limit theorem for µ f is a direct consequence of its correlation decay as shown in [21]. Therefore, it remains to prove the property of the correlation decay. By Theorem 3.8, for each C 1 function ϕ on V, we have This combined with the interpolation inequality for functional in Banach spaces C 1 , C 0 provides the desired 16 correlation decay for µ f , cf [21].