Spectral Theory of Transfer Operators∗

Spectral theory of transfer operators∗

Aihua Fan†and Yunping Jiang‡

Abstract We give a survey on some recent developments in the spectral theory of transfer operators, also called Ruelle-Perron-Frobenius (RPF) operators, associated to expanding and mixing dynamical systems. Diﬀerent methods for spectral study are presented. Topics include maximal eigenvalue of RPF operators, smooth invariant measures, ergodic theory for chain of markovian projections, equilibrium states, spectral gaps for RPF operators, spectral decomposition and perturbation theory, central limit theorem, Hilbert metric and convergence speeds of RPF operators, and dynamical determinants and zeta functions.

Contents

1 Introduction. 64

2 Geometry of expanding and mixing dynamical systems 66

3 Maximal eigenvalues of RPF operators 70

4 Application: existence of smooth invariant measure 75

5 Chains of markovian projections 78

6 Gibbs distributions 84 ∗2000 Mathematics Subject Classiﬁcation. Primary 37C30, Secondary 37F15 †Partially supported by ChangJiang Scholarship program of China ‡Partially supported by NSF grants and PSC-CUNY awards and 100 men program of China

63 Complex Dynamics and Related Topics 64

7 Spectral gaps of RPF operators 91

8 Spectral decomposition and perturbation 94

9 Application: central limit theorem 100

10 Hilbert metric and convergence speed 103

11 Nuclear operators 114

12 Nuclear transfer operators and dynamical determinants 118

13 Dynamical Zeta functions 122

1 Introduction.

Let (X, d) be a compact metric space and f : X → X be a continuous n ∞ map. This gives us a topological dynamical system {f }n=0. We simply call f a dynamical system on X. Let ψ : X → C be a function. For the given f and ψ, we can define an operator L = Lf,ψ as X Lφ(y) = ψ(x)φ(x) x∈f −1(y) for φ in a suitable function space on X. The operator we just defined is called a transfer operator. If ψ is positive, i.e., ψ(x) > 0 for all x in X, then the operator is a positive operator, this means that it maps a positive function to a positive function. A positive transfer operator is also called a Ruelle-Perron-Frobenius (RPF) operator. When the given dynamical system f is the one-side shift on a symbolic space of finite type and when the given function ψ is positive and Höldercontinuous, Ruelle proved in [Ru1, Ru2] (see also [Bo]) that the RPF operator acting on the Höldercontinuous function space has a unique maximal positive eigenvalue ρ with a positive eigenfunction. He used this result when completing a mathematical understanding of the existence and uniqueness of the equilibrium measure (or the so-called Gibbs measure). Ruelle’s theorem represents an important result in modern thermodynamical formalism. Since then RPF operators have be- come a standard tool in many different areas in dynamical systems and other branches of mathematics and mathematical physics as well. Wal- ters [Wa] proved Ruelle’s theorem in more general settings. In Walters’ Complex Dynamics and Related Topics 65 paper, a dynamical system can be any expansive and mixing map and a function can be a positive summable variational function. There are many textbooks and articles about Ruelle’s theorem. We give a partial list [Ba1, Bo, Fa, FJ1, FJ2, FS, Ji, PP, Vi] in the literature. Ruelle’s theorem consists of two parts. The first part concerns the existence and simplicity of a unique maximal eigenvalue for an RPF operator acting on a Höldercontinuous function space. The second part concerns the existence and uniqueness of the Gibbs measure. The proof of the second part is based on the first part. A geometric proof of the first part for a given expanding and mixing dynamical system on a compact metric space and for a given positive Höldercontinuous function is given by Ferrero and Schmitt in [FS], where a remarkable point is that the Hilbert projective metrics introduced by Birkhoff in [Bi] are useful. In fact, Ferrero and Schmitt showed that the operator contracts the Hilbert projective metric on the convex cone of positive functions in the space of Höldercontinuous functions. Therefore, the existence and simplicity of a maximal eigenvalue follows directly from the contracting fixed point theorem. In [Ji], we found an elementary but elegant proof of the first part of Ruelle’s theorem. In [Fa], we gave a new probabilistic proof of the second part of Ruelle’s theorem for symbolic dynamical systems. By combining the ideas in [Ji] and in [Fa] (see also [DF]), we give a proof of Ruelle’s theorem in [FJ1, FJ2] for general dynamical systems with Dini continuous potentials. Furthermore, we studied in [FJ1, FJ2] the convergence speed of a RPF operators under general setting. Ruelle’s theorem becomes more and more important in the study of modern dynamical systems like searching Sinai-Bowen-Ruelle measures for deterministic and stochastic dynamical systems. In this article, we give a survey on some recent development on the spectral theory of transfer operators. Section 2 to Section 6 present a discussion of Ruelle’s theorem. In Section 2, we discuss some geometry of an expanding and mixing dynamical system on a compact metric space. In Section 3, we present an elementary but elegant proof of the first part in Ruelle’s theorem, that is, the existence and uniqueness of a maximal eigenvalue for a RPF operator. The study of the maximal eigenvalue for a RPF operator is not only beneficial for itself, it is also beneficial for the study of many other subjects. In Section 4, we give an application of the first part of Ruelle’s theorem to Krzyzewski-Szlenk’s Theorem [KS] which is about the existence and uniqueness of a smooth invariant measure for a C1+α expanding and mixing dynamical system on a compact C2 Riemannian manifold. In Sections 5 and 6, we present the theory of chains of markovian projections Complex Dynamics and Related Topics 66 and apply it to give a proof of the existence and uniqueness of the Gibbs distribution for a given expanding and mixing dynamical system acting on a general metric space. Section 7 is devoted to the spectral decomposition of a bounded operator. In Section 8, we discuss the relation of the maximal eigenvalue and the rest of spectra for a RPF operator. This relation will be further discussed in Section 10 by a new method called Hilbert metric. The existence of a gap between the maximal eigenvalue and the rest of spectra for a RPF operator provides many important properties for the underline dynamical systems, two of which are discussed in Section 9 and Section 10, one is the central limit theorem and the other one is the convergence speed. Another topic closely related to transfer operators is dynamical determinants and dynamical zeta functions. In Sections 11-13, we give an introduction to this topic. The reader who is interested in this topic may refer to [Ba2, BL, Fr, PP, Rg, Ru3] for further study.

2 Geometry of expanding and mixing dynamical systems

Let X be a compact metric space with metric d. Let f : X → X be n ∞ a continuous map. The iterates {f }n=0 of f give rise to a dynamical system on X. We simply call f a dynamical system. For each n ≥ 0, we deﬁne dn as i i dn(x, y) = max {d(f (x), f (y))} 0≤i≤n and call it the n-Bowen metric. We also call

Bn(x, r) = {y ∈ X ; dn(x, y) < r} the n-Bowen ball centered at x of radius r > 0. The 0-Bowen metric and a 0-Bowen ball are just the original metric d and a ball for d. The Bowen metrics, which describe how orbits are close, will be very useful for studying the dynamics. The dynamical system f is said to be locally expanding if there are constants λ > 1 and b > 0 such that d(f(x), f(x0)) ≥ λd(x, x0), x, x0 ∈ X with d(x, x0) ≤ b. The couple (λ, b) is called a primary expanding parameter. We say f is mixing if for any open set U of X, there is an integer n > 0 such that f n(U) = X. Complex Dynamics and Related Topics 67

Remark 2.1. In general one can deﬁne locally expanding as that there are three constants C, b > 0 and λ > 1 such that d(f n(x), f n(x0)) ≥ Cλnd(x, x0)

0 0 for any x, x ∈ X with dn(x, x ) ≤ b. Suppose f is locally expanding and mixing with a primary expanding constant (λ, b). We are going to present some geometric properties for such a dynamics f, as stated in the following propositions. Proposition 2.1. The map f is locally homeomorphic. More precisely, f : B(x, b) → f(B(x, b)) is homeomorphic for any x ∈ X. Proof. It is clear that f|B(x, b) is injective. Since f is continuous on the closed ball B(x, b) and since the closed ball is compact, the inverse of f|B(x, b) is also continuous. But f : B(x, b) → f(B(x, b)) is bijective, so it is homeomorphic. Proposition 2.2. There is a constant 0 < a < b such that for any y ∈ X −1 with f (y) = {x1, ··· , xn}, there are local inverses g1, ··· , gn of f deﬁned n on B(y, a) such that gi(y) = xi and {gi(B(y, a))}i=1 are pairwise disjoint. −1 Moreover, there is a constant integer n0 ≥ 1 such that #(f (y)) ≤ n0 for all y ∈ X. Proof. First #(f −1(y)) is ﬁnite for each y ∈ X because otherwise f would not be homeomorphic around a limit point of f −1(y). Let

d(y) = inf d(xk, xj) 1≤k6=j≤n be the shortest distance between the preimages of y. It is clear that d(y) > b. For 0 < r ≤ b/2, f : B(xi, r) → f(B(xi, r)) is homeomorphic n for each 1 ≤ i ≤ n. Since the open set ∩i=1f(B(xi, r)) contains y, it must contain a suﬃciently small closed ball B(y, ry) with ry > 0 such that the inverse giy mapping y to xi satisﬁes

giy : B(y, ry) → giy(B(y, ry)) ⊂ B(xi, r).

Since B(xi, r) are disjoint, giy(B(y, ry)) are disjoint. Now take a ﬁnite number of balls {B(yj, ryj )} such that {B(yj, ryj /2)} form a cover of X. Let 1 a = min{ry }. 2 j j Complex Dynamics and Related Topics 68

Then it satisﬁes the proposition. In fact, for any y ∈ X, we have y ∈

B(yj, ryj /2) for some j. Then B(y, a) ⊆ B(yj, ryj ). Let

gi = giyj |B(yj, a), 1 ≤ i ≤ n.

Then g1, ···, gn are local inverses of f on B(y, a) such that gi(y) = xi n and {gi(B(y, a))}i=1 are pairwise disjoint Since #(f −1(y)) is a locally constant function of y. It is then bounded because of the compactness of X, that means that we have a constant −1 integer n0 ≥ 1 such that #(f (y)) ≤ n0 for all y ∈ X. We call (λ, a) where a is a number in Proposition 2.2 an expanding parameter for f. Note that any (λ0, a0) where 0 < a0 < a and 1 < λ0 < λ is also an expanding parameter. In this chapter (λ, a) will always mean an expanding parameter for f. For any 0 < r ≤ a and y ∈ X, let g be an inverse branch of f on B(y, r). Let x = g(y). For any z, w ∈ B(y, r), 1 d(g(z), g(w)) ≤ d(z, w). λ This implies that r g(B(y, r)) ⊆ B(x, ). λ

Moreover, B1(x, r) = g(B(y, r)) and f : B1(x, r) → B(y, r) is homeomorphic. Furthermore, we have Proposition 2.3. For any 0 < r ≤ a and x ∈ X, the map

n n f : Bn(x, r) → B(f (x), r) is homeomorphic. Proof. Let x, f(x), ··· , f n(x) be a ﬁnite orbit of f. Then we have the n local inverses, h1, ···, hn, of f,such that

h x ←−h1 f(x) ←−h2 f 2(x) ←−h3 · · · ←−n−1 f n−1(x) ←−hn f n(x).

From the argument before this proposition, we know that

n n−1 hn : B(f (x), r) → B1(f (x), r) is homeomorphic. This implies that

n n−1 hn(B(f (x), r)) = B1(f (x), r). Complex Dynamics and Related Topics 69

Suppose we already know that

n n−k hn−k+1 ◦ · · · ◦ hn(B(f (x), r)) = Bk(f (x), r) for 1 ≤ k < n − 1. Then

n n−k hn−k ◦ · · · ◦ hn(B(f (x), r)) = hn−k(Bk(f (x), r)).

n−k n−k This implies that z ∈ hn−k(Bk(f (x), r)) if and only if f(z) ∈ Bk(f (x), r) and if and only if

d(f i(f(z)), f i(n − k(x))) < r, 0 ≤ i ≤ k.

So we have

d(f(z), f n−k(x)) < r, ··· , d(f k+1(z), f n(x)) < r.

This combining with the fact that 1 r d(z, f n−k−1(x)) ≤ d(f(z), f n−k(x)) < λ λ gives us that

n n−k−1 hn−k ◦ · · · ◦ hn(B(f (x), r)) = Bk+1(f (x), r). Now the induction implies

n h1 ◦ h2 ◦ · · · ◦ hn(B(f (x), r)) = Bn(x, r),

n n n that is f (Bn(x, r)) = B(f (x), r). Because f on Bn(x, r) is also injective, n n f : Bn(x, r) → B(f (x), r) is homeomorphic. Proposition 2.4. For any 0 < r ≤ a, there is an integer p = p(r) ≥ 1 such that f p(B(x, r)) = X for any x ∈ X.

Proof. Let {B(yi, r/2)} be a ﬁnite ball cover of X. For each i, there is an integer pi = p(yi) > 0 such that r f pi (B(y , )) = X i 2 Let p = max{pi}. i Complex Dynamics and Related Topics 70

r For any y ∈ X, we have y ∈ B(yi, 2 ) for some i. Then r B(y, r) ⊃ B(y , ) i 2 and r f p(B(y, r)) ⊇ f p−pi (f pi (B(y , ))) ⊇ f p−pi (X) = X. i 2

Proposition 2.5. For any 0 < r ≤ a, let p = p(r) be the integer in Proposition 2.4 and let n0 be the integer in Proposition 2.2, then

−(n+p) p 1 ≤ #(f (y) ∩ Bn(x, r)) ≤ n0 for any x, y ∈ X and any n ≥ 1.

n n Proof. Since f : Bn(x, r) → B(f (x), r) is a homeomorphism,

n+p p n f (Bn(x, r)) = f (B(f (x), r)) = X.

−(n+p) −p This implies that f (y)∩Bn(x, r) 6= ∅. On the other hand, #(f (y)) ≤ p −p n n0 and every z ∈ f (y)∩B(f (x), r) has exact one preimage in Bn(x, r) under f n. So −(n+p) p 1 ≤ #(f (y) ∩ Bn(x, r)) ≤ n0.

3 Maximal eigenvalues of RPF operators

Usually, one proves the existence of positive eigenfunction of a RPF operator by using the Schauder ﬁxed point theorem. In this section, we present a direct proof just basing on the Ascoli-Arzela theorem. Suppose f is a locally expanding and mixing map with an expanding parameter (λ, a). Let R denote the real line. Let C0 = C0(X, R) be the space of all continuous functions φ : X → R with the supremum norm

||φ|| = max |φ(x)|. x∈X Let 0 < α ≤ 1 and let Cα = Cα(X, R) be the space of all α-H¨older continuous functions φ in C0, that is, φ ∈ C0 satisfying |φ(x) − φ(y)| [φ]α = sup α < ∞, 0

where [φ]α is called the local H¨olderconstant for φ. For two functions φ1 and φ2, we write φ1 ≥ φ2 if φ1(x) ≥ φ2(x) for all x in X. A function φ is called positive if φ > 0. A positive function in Cα is called a potential. For two constants K, s > 0, deﬁne

α α α CK,s = CK,s(X, R) = {φ ∈ C ; φ ≥ s, [log φ]α ≤ K}.

Recall that [log φ]α ≤ K means that

φ(x) ≤ φ(y)eKd(x,y)α .

The following lemma is a consequence of Ascoli-Arzela Theorem.

α Lemma 3.1. Any bounded sequence in CK,s has a convergent subsequence 0 α in C whose limit is in CK,s. Let ψ be a potential. The Ruelle-Perron-Frobenius (RPF) operator with weight ψ is deﬁned as X Lφ(y) = ψ(x)φ(x). x∈f −1(y)

It is clear that L(C0) ⊆ C0 and

L : C0 → C0 is a bounded linear operator. Moreover for any φ ∈ Cα, consider y, y0 ∈ X 0 −1 0 0 −1 0 with d(y, y ) ≤ a. Let {x1, ··· , xk} = f (y) and {x1, ··· , xk} = f (y ) 0 0 be the corresponding inverse images of y and y such that d(xi, xi) ≤ λ−1d(y, y0) for all 1 ≤ i ≤ k. Then

Xk Xk 0 0 |Lφ(y) − Lφ(y)| = | ψ(xi)φ(xi) − ψ(xi)φ(xi)| i=1 i=1 ¯ Xk ³ ´¯ ¯ 0 0 0 ¯ = ¯ ψ(xi)(φ(xi) − φ(xi)) + φ(xi)(ψ(xi) − ψ(xi)) ¯. i=1 So n [Lφ] ≤ 0 (||ψ||[φ] + ||φ||[ψ] ) < ∞. α λα α α Thus L(Cα) ⊆ Cα and L : Cα → Cα Complex Dynamics and Related Topics 72 is a bounded linear operator. It is well known that being a positive operator on C0 (i.e. Lφ ≥ 0 when φ ≥ 0), L has its spectral radius as a spectral point. We will prove that it is actually an eigenvalue of L : Cα → Cα and there is a corresponding strictly positive eigenfunction. For the purpose of the study of eigenvalues of L, we can normalize the weight ψ such that minx∈X ψ(x) = 1. Henceforth, we will always assume α that ψ is a normalized element in CK0,1 for some constant K0 > 0 in the α rest of this section. Let 0 < s < 1 and K > K0/(λ − 1) > 0 be two ﬁxed constants. Lemma 3.2. For any φ ≥ 0 in Cα with ||φ|| = 1, there is an integer N = N(φ) > 0 such that

n α L φ ∈ CK,s, ∀n ≥ N. Proof. Since ||φ|| = 1, there is a point y in X such that φ(y) = 1. We thus have a neighborhood U of y such that φ(y0) > s for all y0 in U. n Since f is mixing, there is an integer n1 > 0 such that f (U) = X for −n all n ≥ n1. Therefore for any z in X, f (z) ∩ U is non-empty for all n n ≥ n1. Thus L φ(z) ≥ s. 0 0 −1 For any y and y in X with d(y, y ) ≤ a, let {x1, ··· , xk} = f (y) 0 0 −1 0 and {x1, ··· , xk} = f (y ) be the corresponding inverse images of y and 0 0 −1 0 y such that d(xi, xi) ≤ λ d(y, y ) for all 1 ≤ i ≤ k. Let

0 n1 K = [log L φ]α. Then Xk n1 0 0 n1 0 L(L φ)(y ) = ψ(xi)L φ(xi) i=1 Xk 0 α n1 0 0 α ≤ ψ(xi) exp (K0d(xi, xi) )L φ(xi) exp (K d(xi, xi) ) i=1 0 −α 0 α n1 ≤ exp ((K0 + K )λ d(y, y ) )L(L φ)(y) for all y and y0 in X with d(y, y0) ≤ a. Inductively, for Xn −αi 0 −αn Kn = K0( λ ) + K λ , i=1 we have n n1 0 0 α n n1 L (L φ)(y ) ≤ exp (Knd(y, y ) )L (L φ)(y) Complex Dynamics and Related Topics 73

0 0 for all y and y in X with d(y, y ) ≤ a. It is clearly that Kn tends to α K0/(λ − 1) as n goes to inﬁnity. So there is an integer n2 > 0 such that for any n ≥ n2

Ln(Ln1 φ)(y0) ≤ exp (Kd(y, y0)α)Ln(Ln1 φ)(y)

0 0 for all y and y in X with d(y, y ) ≤ a. Then N = n1 + n2 satisﬁes the lemma. Lemma 3.2 implies that if t > 0 is an eigenvalue of L : Cα → Cα with α a non-zero eigenfunction φ ≥ 0. Then this eigenfunction must be in CK,s. Therefore, we are led to ﬁnd positive eigenvalues of L : Cα → Cα with α non-negative eigenfunctions in CK,s. From the proof of Lemma 3.2, we have seen that

α α L(CK,s) ⊆ CK,s

−α α because (K0 + K)λ < K for K > K0/(λ − 1). Deﬁne

α S = {t ∈ R ; t > 0, there is a φ in CK,s such that Lφ ≥ tφ} Lemma 3.3. The set S is a nonempty bounded subset on the real line R

α Proof. Take a function φ in CK,s. Then for any x and y in X, X ³ X φ(x) ´ s Lφ(y) = ψ(x)φ(x) = ψ(x) φ(y) ≥ φ(y). φ(y) ||φ|| x∈f −1(y) x∈f −1(y)

Thus s/||φ|| ∈ S. So, S is nonempty. Let X m = sup ψ(x). y∈X x∈f −1(y)

α For any φ in CK,s, let φ(y) = ||φ||. Then X X Lφ(y) = ψ(x)φ(x) ≤ φ(y) ψ(x) ≤ mφ(y) x∈f −1(y) x∈f −1(y)

Therefore, any t > m will not be in S. Thus S is bounded. Theorem 3.1 (Ruelle). The linear operator L : Cα → Cα has a unique maximal positive eigenvalue whose corresponding eigenspace has dimension one. Complex Dynamics and Related Topics 74

∞ Proof. Take δ = sup S > 0. Then there is a sequence {tn}n=1 in S α converging to δ. Let φn be a corresponding functions in CK,s such that Lφn ≥ tnφn. Let us normalize φn with minx∈X {φn(x)} = s. Then ∞ α ∞ {φn}n=1 is a bounded sequence in CK,s. From Lemma 3.1, {φn}n=1 has 0 α a convergent subsequence in C whose limit is in CK,s. Let us assume, ∞ without loss of generality, that {φn}n=1 itself converges to φ0. Then

Lφ0 ≥ δφ0.

We now show that Lφ0 = δφ0. Suppose there is a point y in X such that

Lφ0(y) > δφ0(y). Then there is a neighborhood U of y such that

0 0 Lφ0(y ) − δφ0(y ) > 0 for all y0 in U. Since f is mixing, there is an integer n > 0 such that f n(U) = X. Then n L (Lφ0 − δφ0) > 0, that is, n n L(L φ0) > δL φ0. n Therefore for φ = L φ0, we have a t > δ such that Lφ ≥ tφ. This contradicts to the maximal property of δ. This proved that

Lφ0 = δφ0. Now let us show that δ is simple, i.e., the eigenspace

α Eδ = {φ ∈ C ; Lφ = δφ} has dimension one. Suppose φ is any function in Eδ. Let φ(x) a = min x∈X φ0(x) and φ1 = φ − aφ0.

Then φ1 is in Eδ and φ1 ≥ 0. Moreover, there is a point y in X such that −1 φ1(y) = 0. Then φ1(x) = 0 for all x in f (y) because X Lφ1(y) = φ(x) = 0. x∈f −1(y) Complex Dynamics and Related Topics 75

∞ −n Inductively, we have φ1 = 0 on Xy = ∪n=0f (y). Since f is mixing, Xy is a dense subset in X. So φ1 ≡ 0 on X, that is, φ ≡ aφ0. The reminder is to prove that δ is the biggest eigenvalue but it is easy as follows. Suppose t 6= δ is an eigenvalue of L : Cα → Cα. Then there is a non-zero function φ in Cα with ||φ|| = 1 such that Lφ = tφ. So N α L|φ| ≥ |t||φ|. There is an integer N > 0 such that L |φ| is in CK,s and also L(LN |φ|) ≥ |t|LN |φ|. Thus |t| is a number in S, so |t| ≤ δ. If |t| < δ, then we have nothing to prove. If |t| = δ, by using the mixing property as we did in the previous two paragraphs, we have |φ| = aφ0 for some a > 0. This implies φ = ±aφ0 and t = δ.

4 Application: existence of smooth invariant measure

As an application of Theorem 3.1, we are going to prove the existence of smooth invariant measure for differentiable expanding dynamical systems. This result is due to Krzyzewski-Szlenk. Suppose M is an m-dimensional compact C2 Riemannian manifold where m ≥ 1 is an integer. Let f : M → M be a C1 map. We say f is C1+α for 0 < α ≤ 1 if the determinant J(f) of the Jacobi matrix Jac(f) of f is an α-Höldercontinuous function defined on M. A probability measure ν on M is called f-invariant if ν(f −1(A)) = ν(A) for all Lebesgue measurable subsets A of M. Let dy denote the Lebesgue measure on M. A probability measure ν is called a smooth measure if there is a continuous function ρ defined on M such that Z ν(A) = ρ(y)dy A for all Lebesgue measurable sets A in M. The function ρ is called the density function of the smooth measure ν.

1 Lemma 4.1. Suppose f : M → M is a C map such that J(f)(y) 6=R 0 for all y in M and suppose ν is a smooth probability measure with ν = ρdy. Then ν is a f-invariant measure if and only if X ρ(x) = ρ(y) J(f)(x) x∈f −1(y) for all y in M. Complex Dynamics and Related Topics 76

Proof. Since J(f)(y) 6= 0 for all y in M there is a constant a1 > 0 such that for any connected domain U with diameter less than or equal 2a1, f on each component V of f −1(U) is injective and has the local inverse g : V → U such that fg = identity. If ν is a f-invariant measure, then ν(f −1(U)) = ν(U) for all Lebesgue measurable subsets U of M. In particular, take U as the ball of centered y and radius 0 < ² < a1 and −1 denote V1, ···, Vk as the components of f (U). Then Xk Z Z ρ(x)dx = ρ(y)dy. i=1 Vi U By the mean value theorem and let ² tend to zero, we get X ρ(x) = ρ(y). J(f)(x) x∈f −1(y) Now assume X ρ(x) = ρ(y) J(f)(x) x∈f −1(y) for all y in M. For any ball U with radius a1 , let V1, ···, Vk be the −1 components of f (U). Then f on each Vi is injective and has the local inverse. So Z Z X ρ(x) ν(U) = ρ(y)dy = dy J(f)(x) U U x∈f −1(y) Z Z X ρ(x) Xk = dy = ρ(x)dx J(f)(x) x∈f −1(y) U i=1 Vi Xk −1 = ν(Vi) = ν(f (U)). i=1

Theorem 4.1 (Krzyzewski-Szlenk). Suppose f : M → M is a C1+α locally expanding map for some 0 < α ≤ 1. Then f has a unique smooth f-invariant probability measure with an α-H¨oldercontinuous density function. Proof. Since f is C1 and locally expanding, J(f)(y) 6= 0 for all y in M. Let ||J(f)|| ||J(f)|| = max J(f)(y) and ψ = . y∈M J(f) Complex Dynamics and Related Topics 77

α Then ψ is an function in CK0,1 for some K0 > 0. Consider the RPF operator with weight ψ X Lφ(y) = ψ(x)φ(x). x∈f −1(y) Theorem 3.1 implies that there is a unique maximal positive eigenvalue δ with a positive eigenfunction in Cα and the corresponding eigenspace Ris one-dimensional. Let ρ be the one in the eigenspace normalized by M ρdy = 1. Then Lρ(y) = δρ(y), that is, X ρ(x) = δ ρ(y), J(f)(x) 0 x∈f −1(y) for δ0 = δ/||J(f)||. Integrate on both sides of the last equation, we have that the right hand side is δ0. Now let us calculate the left hand side. Cut M into path connected pieces M1, ···, Mn such that

1. M = M1 ∪ M2 ∪ · · · ∪ Mn,

2. the Lebesgue measure of each Mi ∩ Mj is zero for i 6= j,

−1 3. f on each component of f (Mi) is injective, 1 ≤ i ≤ n. j −1 Let Mi , 1 ≤ j ≤ ki be the components of f (Mi) for 1 ≤ i ≤ n. Then 0 n ki j j j M = ∪i=1 ∪j=1 Mi and the Lebesgue measure of each Mi ∩Mi0 is zero for 0 −1 j i 6= i . Let us use xij to denote the point in f (y) ∩ Mi for any y ∈ Mi, where 1 ≤ i ≤ n and 1 ≤ j ≤ ki. Therefore, Z Z X ρ(x) Xn X ρ(x) dy = dy J(f)(x) J(f)(x) M x∈f −1(y) i=1 Mi x∈f −1(y) Z Xn Xki ρ(x ) = ij dy J(f)(x ) i=1 j=1 Mi ij Z Xn Xki = ρ(xij)dxij M j Zi=1 j=1 i = ρ(y)dy = 1. M R So we have that δ0 = 1 (that is, δ = ||J(f)||) and that ν = ρdy is a smooth f-invariant measure following LemmaR 4.1. Uniqueness follows the fact that if ν = ρdy is a smooth f-invariant measure, than ρ is in the eigenspace of L with respect to the eigenvalue δ. Complex Dynamics and Related Topics 78

5 Chains of markovian projections

In this section we present the theory of chains of markovian projections and G-measure theory [DF]. This theory includes RPF operators as a special case. Let X be a compact Hausdorff space. Let FX be the standard σ- algebra generated by all open sets in X. Let C0 = C(X, R) still be the space of all real valued continuous functions on X, equipped with the supremum norm || · ||. Let M = M(X) = (C0)∗ be the dual space of C0. By Riesz representation theorem, M is identical to the space of all measures on X with respect to FX . Let M0 ⊂ M be the family of all probability measures. We use Z < µ, φ >= φ dµ X to mean the integral of a function φ with respect to a measure µ in M. Definition 5.1. A linear map P : C0 → C0 is said to be a projection if P 2 = P . It is said to be markovian if P 1 = 1 and if P φ ≥ 0 whenever φ ≥ 0. Let us first give some basic properties about projections and markovian projections. Let Ker(P ) = {φ ∈ C0; P φ = 0} and let Im(P ) = P (C0). Proposition 5.1. Let P and Q are two projections. Then L (1) C0 = Ker(P ) Im(P ). (2) φ ∈ Im(P ) if and only if P φ = φ. (3) PQ = Q if and only if Im(Q) ⊆ Im(P ). (4) QP = Q if and only if Ker(P ) ⊆ Ker(Q). Proof. (1) means that for each φ ∈ C0 we can write φ = φ0 + φ00 in a unique way with φ0 ∈ Ker(P ) and φ00 ∈ Im(P ). This is true because φ = (φ − P φ) + P φ. (2) is a consequence of (1). Suppose φ ∈ Im(Q) and PQ = Q. By (2), we have Qφ = φ. So P φ = P Qφ = Qφ = φ, i.e., φ ∈ Im(P ). Conversely, for any φ ∈ C0, there is another map φ0 ∈ C0 such that Qφ = P φ0. Then P Qφ = P 2φ0 = P φ0 = Qφ. For (4), suppose QP = Q and φ ∈ Ker(P ). Then it is clear that Qφ = 0. Conversely, for φ ∈ C0 we decompose it as φ = (φ − P φ) + P φ. By assumption, φ − P φ ∈ Ker(Q), so Qφ = QP φ. Complex Dynamics and Related Topics 79

For an operator P : C0 → C0, let P ∗ : M → M be its dual operator. The proof of the next proposition is not hard. The reader may do it as an exercise. Proposition 5.2. Let P and Q be markovian projection on C0. We have 1) ||P || = ||P ∗|| = 1. 2) P ∗2 = P ∗.

∗ 3) P (M0) ⊆ M0. 4) PQ = Q if and only if Q∗P ∗ = Q∗.

∞ 0 A sequence of markovian projections P = {Pn}n=1 defined on C is called a chain of markovian projections (abbreviated as CMP) if it satisfies PmPn = Pm, 1 ≤ n ≤ m. For such a chain, define

∗ Gn = {µ ∈ M0 ; Pn µ = µ}, 1 ≤ n < ∞ ∗ Because M0 is a weakly compact convex subset of M and Pn : M0 → M0, from the Schauder-Tychonoﬀ theorem (see [DS]), Gn 6= ∅. Let ∞ G∞ = ∩n=1Gn.

If G∞ is non-empty, then every element in it is called a G-measure with respect to P. If G∞ contains only one element, then we call the given CMP uniquely ergodic.

Theorem 5.1. For any CMP P, G∞ 6= ∅. Actually, for any sequence ∞ ∗ ∞ {µn}n=1 in M0, any weak limit of {Pn µn}n=1 is an element in G∞. ∗ ∗ ∞ Proof. Since ||Pn µn|| = 1, there is a weak limit of {Pn µn}n=1. Let ν be ∗ such a weak limit. Then there is a subsequence Pni µni tends to ν weakly as i goes to inﬁnity. This implies that for any φ ∈ C0,

∗ ∗ ∗ ∗ lim < Pn Pn µni , φ >= lim < Pn µni ,Pnφ >=< ν, Pnφ >=< Pn ν, φ > . i→∞ i i→∞ i On the other hand,

∗ ∗ < Pn ν, φ >= lim < Pn µni ,Pnφ >= lim < µni ,Pni Pnφ > i→∞ i i→∞ ∗ = lim < µni ,Pni φ >= lim < Pn µni , φ >=< ν, φ > . i→∞ i→∞ i ∗ So Pn ν = ν for all n ≥ 1, that is, ν ∈ G∞. Complex Dynamics and Related Topics 80

Theorem 5.1 says that G∞ is weakly compact. It is clear that G∞ is convex, i.e., tµ1 + (1 − t)µ2 ∈ G∞ if µ1, µ2 ∈ G∞ and 0 ≤ t ≤ 1. Theorem 5.2. Suppose P is a CMP deﬁned on C0. Then the following are equivalent.

(1) The CMP is unique ergodic.

0 (2) for every φ ∈ C , Pnφ converges uniformly on X to a constant.

0 (3) for every φ ∈ C , Pnφ converges pointwise on X to a constant.

Proof. It is clear (2) implies (3). Assume (3), for any µ ∈ G∞, the constant is < µ, φ > because by the Lebesgue theorem

∗ < µ, φ >=< Pn µ, φ >= lim < µ, Pnφ >=< µ, lim Pnφ >= lim Pnφ. n→∞ n→∞ n→∞

0 Therefore, for any µ, ν ∈ G∞ and φ ∈ C ,

< µ, φ >=< ν, φ >= lim Pnφ. n→∞ It implies that µ = ν, so (1) holds. Now assume (1) and suppose µ is 0 the unique element in G∞. Suppose (2) is false. There exists φ ∈ C such that Pnφ does not converge to < µ, φ > uniformly. So we have a ∞ constant ² > 0 and a subsequence {ni}i=1 of integers and a sequence of ∞ points {xi}i=1 such that

|Pni φ(xi)− < µ, φ > | ≥ ²

for all i ≥ 1. Let δxi be the Dirac measure concentrating at xi. Then

∗ Pni φ(xi) =< Pni δxi , φ > .

∗ ∞ By Theorem 5.1, any weak limit ν of {Pni δxi }i=1 is in G∞. But | < ν, φ > − < µ, φ > | ≥ ².

It contradicts with the assumption of (1).

∞ A CMP P = {Pn}n=1 is said to be compatible if it satisﬁes

(a) Pn(φχ) = χPnφ if χ ∈ ImPn.

(b) PnPm = Pm(= PmPn) if m ≥ n. Complex Dynamics and Related Topics 81

∞ Suppose P = {Pn}n=1 is a compatible CMP. Let F0 be the standard σ-algebra on R generated by all open sets. For a function φ : X → R, −1 let Fφ = φ (F0) be the pull-back σ-algebra on X. Given a family of functions Γ, we use FΓ to mean the minimal σ-algebra containing all σ- ∞ algebras Fφ for φ ∈ Γ. Let Fn = FImPn for all n ≥ 1. Then {Fn}n=1 is a decreasing sequence of sub-σ-algebras in FX , i.e.,

· · · ⊆ Fn+1 ⊆ Fn ⊆ · · · ⊆ F1 ⊆ FX .

This is because from Proposition 5.1

0 · · · ⊆ ImPn+1 ⊆ ImPn ⊆ · · · ⊆ ImP1 ⊆ C .

∞ Let F∞ be the σ-algebra generated by the limit of {Fn}n=1 which is deﬁned as ∞ F∞ = ∪n=1 ∩m≥n Fm(= ∩n≥1 ∪m≥n Fm).

A G-measure µ is called P-ergodic if µ|F∞ is trivial, i.e., µ(A) = 0 or 1 for any A ∈ F∞. 0 Let µ ∈ M0 be a probability measure. For any φ ∈ C and n ≥ 1, we have a measure µn deﬁned on the sub-σ-algebra Fn by Z

µn(A) = φdµ, A ∈ Fn. A

It is clear that µn is absolutely continuous with respect to µ|Fn. By the Radon-Nikodym theorem there exists a unique (modulo sets of measure 1 zero) L (X, Fn, µ)-function denoted by E(φ|Fn) and called the condi- tional expectation of φ given Fn, such that Z

µn(A) = E(φ|Fn)dµ, A ∈ Fn. A The function is deﬁned uniquely a.e. by R R i A E(φ|Fn)dµ = A φdµ, A ∈ Fn, 1 ii E(φ|Fn) ∈ L (X, Fn, µ).

The operator E(·|Fn) enjoys the following properties:

1 0 ∞ iii For all φ ∈ L (X, FX , µ) and φ ∈ L (X, Fn, µ),

0 0 E(φφ |Fn) = φ E(φ|Fn). Complex Dynamics and Related Topics 82

The proof of the following theorem can be found in [Pa, pp. 30]

Theorem 5.3 (Decreasing martingale theorem). If

···Fn+1 ⊂ Fn ⊂ · · · ⊂ F1 ⊆ FX

∞ is a decreasing sequence of sub-σ-algebras such that ∩n=1Fn = F∞, then 1 1 E(φ|Fn) → E(φ|F∞) a.e. and in L (X, FX , µ) when φ ∈ L (X, FX , µ).

Now assume that µ ∈ G∞ is a G-measure for P. Then we have that 0 for any φ ∈ ImPn,

0 0 ∗ 0 0 < µ, φ Pnφ >=< µ, Pn(φφ ) >=< Pn µ, φφ >=< µ, φφ >, so Pnφ = E(φ|Fn), µ − a.e..

Following Theorem 5.3, the limit of Pnφ exists µ-a.e. and, furthermore, a P-invariant measure µ is P-ergodic iﬀ limn→∞ Pnφ =< µ, φ > µ-a.e. for any φ ∈ C0 (see [Pa, pp. 21]). Then we have the following classical ergodicity theorem.

∞ Theorem 5.4. Suppose P = {Pn}n=1 is a compatible CMP.

1. If µ1 and µ2 in G∞ are P-ergodic, then either µ1 = µ2 or µ1 ⊥ µ2.

2. µ ∈ G∞ is P-ergodic iﬀ µ is an extremal point in G∞.

0 Proof. (1) Suppose µ1 6= µ2. There exists φ ∈ C such that

< µ1, φ >6=< µ2, φ > .

Deﬁne

A1 = {x ∈ X; lim Pnφ =< µ1, φ >} and A2 = {x ∈ X; lim Pnφ =< µ2, φ >}. n→∞ n→∞

We have µ1(A1) = 1 and µ2(A1) = 0, and µ1(A2) = 0 and µ2(A2) = 1. This implies that µ1 ⊥ µ2. (2) Suppose µ ∈ G∞ is P-ergodic and µ = tµ1 + (1 − t)µ2 with µ1, µ2 in G∞ and 0 < t < 1. For any A ∈ F∞, µ(A) = 0 or 1 because of the ergodicity of µ. Then µ1(A) = µ2(A) = 0 or 1 because 0 < t < 1. That means that µ1 and µ2 are also P-ergodic. According to (1), if µ1 6= µ2 we can ﬁnd A ∈ F∞ with µ1(A) = 1 and µ2(A) = 0. Consequently Complex Dynamics and Related Topics 83

µ(A) = t. This contradicts the ergodicity of µ. Therefore µ1 = µ2 and µ is extremal. Conversely, suppose µ ∈ G∞ is not P-ergodic. Let A ∈ F∞ such that 0 < t = µ(A) < 1. Deﬁne 1 1 µ = µ1 , µ = µ1 . 1 t A 2 1 − t X\A

We are now going to show µ1, µ2 ∈ G∞. For any n ≥ 1 1 < P ∗µ , φ >=< µ, 1 P φ >, φ ∈ C0. n 1 t A n

Since A ∈ F∞ ⊆ Fn, 1 1 1 < µ, 1 P φ > = sup < µ, φ0P φ >= sup < P ∗µ, φφ0 > t A n t n t n 1 1 = sup < µ, φ0φ >=< 1 µ, φ > t t B

0 0 where the supremum is taken over {φ ≤ 1A, φ ∈ ImPn}. This implies ∗ that Pn µ1 = µ1 for all n ≥ 1, that is, µ ∈ G∞. Similarly, we can prove that µ2 ∈ G∞. But

µ = tµ1 + (1 − t)µ2, 0 < α < 1 which implies that µ is not an extremal point. The space M is a locally convex topological space and metrizable. So G∞ is a compact metrizable convex subset in M. Let EG∞ be the set of P-ergodic µ in G∞. Then the above theorem says that EG∞ consists of all extremal points in G∞. The relation between G∞ and EG∞ can be obtained from the Choquet representation theorem (see [OR, pp. 1-32] for the proof).

Theorem 5.5 (Choquet representation theorem). For each µ ∈ G∞, there exists a Borel probability measure m on G∞, supported on the set EG∞ of extremal points, so that Z

µ = νdm(ν), µ ∈ G∞. G∞ Complex Dynamics and Related Topics 84

6 Gibbs distributions

Suppose that a physical system of n states {1, 2, ··· , n} with their energies E1, ···, En and that the system is put in contact with a much larger “heat source” which is at temperature T . Energy is therefore allowed to pass between the original system and the heat source. Suppose the temperature T of the heat source remains constant. It is a physical fact derived in statistical mechanics that the probability pj that state j occurs is given by the Gibbs distribution

e−βEj pj = Pn −βEi i=1 e 1 where β = kT and k is a physical constant. This is the starting point for the thermodynamical formalism. In this section, we combine Theorem 3.1 and the theory of chains of markovian projections (Section 5) to develop a thermodynamical formalism for more general systems. We will keep use the same notations as those in Sections 2, 3 and 5. Suppose f is a locally expanding and mixing dynamical system with an expanding parameter (λ, a) and suppose ψ > 0 ∈ Cα, 0 < α ≤ 1, is a potential. Deﬁne nY−1 i Gn(x) = ψ(f (x)), x ∈ X, n ≥ 1. i=0 Let X Lφ(y) = ψ(x)φ(x) x∈f −1(y) be the RPF operator with weight ψ. Let δ > 0 be the maximal eigenvalue and h > 0 ∈ Cα be a corresponding eigenvector of L. Let L∗ : M → M be the dual operator of L.

Theorem 6.1 (Ruelle). There is a unique probability measure ν = νψ ∈ ∗ M0 such that L ν = δν and for any 0 < r ≤ a/2, there is a constant C = C(r) > 0 such that

−1 ν(Bn(x, r)) C ≤ −n ≤ C (Gibbs Property) δ Gn(x) R for all x ∈ X and n ≥ 1. And moreover, take h such that X h dν = 1. Then for any φ ∈ C0, lim δ−nLnφ =< ν, φ > h n→∞ Complex Dynamics and Related Topics 85 uniformly. The inequality in the theorem is called the Gibbs Property and the probability measure µ = hν is called the Gibbs measure for (f, ψ). To prove Theorem 6.1, we ﬁrst normalize the operator L. Take h(x) ψ˜(x) = ψ(x). δh(f(x)) Then it is still a positive function in Cα. So we can consider the RPF operator X ˜ ˜ Lφ(x) = Lψ˜φ(x) = ψ(y)φ(y). y∈f −1(x) The important feature of L˜ is that L˜1 = 1. We call it a normalized RPF operator. Let L˜∗ be the dual operator of L˜ acting on the space M. Let nY−1 ˜ ˜ i Gn(x) = ψ(f (x)), x ∈ X, n ≥ 1. i=0 Then we have h G˜ = G n δnh ◦ f n n and relations between L and L˜ and between L∗ and L˜∗, Lnφ = δnhL˜n(φh−1) and L∗nν = δnh−1L˜∗n(hν) From these relations, Theorem 6.1 follows from the following statement for normalized RPF operators. Theorem 6.2. Suppose L is a normalized RPF operator, i.e., L1 = 1. ∗ Then there is a unique probability measure µ ∈ M0 such that L µ = µ and for any 0 < r ≤ a/2, there is a constant C = C(r) > 0 such that the Gibbs property holds, i.e., µ(B (x, r)) C−1 ≤ n ≤ C (Gibbs Property) Gn(x) for all x ∈ X and n ≥ 1. Moreover, for any φ ∈ C0 lim Lnφ =< µ, φ > n→∞ Complex Dynamics and Related Topics 86

Suppose L is normalized in the rest of this section. Deﬁne X n n Pnφ(x) = L φ(f (x)) = Gn(y)φ(y). y∈f −n(f n(x))

0 It is a linear operator from C into itself with Pn1 = 1 and Pnφ > 0 whenever φ > 0. Moreover, we have the following facts.

Lemma 6.1. For any m ≥ n ≥ 1, PnPm = PmPn = Pn.

Proof. Let us show that PmPn = Pm. X PmPnφ(x) = Gm(y)Pnφ(y) y∈f −m(f m(x)) X X = Gm−n(w)Gn(y)Pnφ(y) w∈f −(m−n)(f m(x)) y∈f −n(w) X X X = Gm−n(w)Gn(y) Gn(z)φ(z) w∈f −(m−n)(f m(x)) y∈f −n(w) z∈f −n(f n(y)) X X X = Gn(y) Gm−n(w)Gn(z)φ(z) w∈f −(m−n)(f m(x)) y∈f −n(w) z∈f −n(w) X ³ X ´³ X ´ = Gn(y) Gm(z)φ(z) w∈f −(m−n)(f m(x)) y∈f −n(w) z∈f −n(w) X X = Gm(z)φ(z) w∈f −(m−n)(f m(x)) z∈f −n(w) X = Gm(z)φ(z) z∈f −m(f m(x))

= Pmφ(x). P We use the fact that y∈f −n(w) Gn(y) = 1. This also implies that Pn is a 2 projection, i.e., Pn = Pn. Similar arguments imply that PnPm = Pm.

0 Lemma 6.2. For any φ ∈ C and χ ∈ ImPn, Pn(φχ) = χPnφ. P Proof. Suppose χ(x) = y∈f −n(f n(x)) Gn(y)β(y). Then X X Pn(φχ)(x) = Gn(z)φ(z) Gn(y)β(y) z∈f −n(f n(x)) y∈f −n(f n(z)) X X = Gn(y)Gn(z)φβ y∈f −n(f n(x)) z∈f −n(f n(x)) Complex Dynamics and Related Topics 87

X X = Gn(y)β(y) Gn(z)φ(z) y∈f −n(f n(x)) z∈f −n(f n(x))

= χ(x)Pnφ(x).

∞ Lemmas 6.1 and 6.2 say that P = {Pn}n=1 is a compatible CMP. So we can apply the theory of CMP in Section 5 to give a proof of Theorem 6.2. ∗ Let Pn be the dual operator of Pn. Remember that G∞ is the set ∗ of common fixed points of Pn ’s and an element of G∞ is called a G- measure. An element µ such that L∗µ = µ is called a ψ-measure. Because L(φ ◦ f) = φ, any ψ-measure µ is f-invariant, i.e., µ(f −1(A)) = µ(A) for any µ-measurable sets A. So a ψ-measure is a G-measure. Since M0 ∗ is a weakly compact convex subset of M and L : M0 → M0, by the Schauder-Tychonoff fixed point theorem there is at least one ψ-measure (and G-measure). Now we use the results in Section 5 to prove the uniqueness of G-measure.

0 Lemma 6.3. For any φ in C , Pnφ converges to a constant if and only if Lnφ converges to the same constant. And moreover, the constant has to be < µ, φ > for any ψ-measure (or G-measure) µ. Proof. Since X n n Pnφ(x) = ψ(y)φ(y) = (L φ)(f (x)) y∈f −n(f n(x)) and f : X → X is surjective, it is clear that

n kPnφ(x) − ck = kL φ(x) − ck.

n Therefore, Pnφ converges to c if and only if L φ converges to c. Suppose Pnφ converges to c. Then

∗ c = lim Pnφ = lim < µ, Pnφ >= lim < Pn µ, φ >=< µ, φ > . n→∞ n→∞ n→∞

Lemma 6.4 (Naive distortion lemma). There is a constant C > 0 such that for any n ≥ 0 and any x, y ∈ X with dn(x, y) ≤ a, G (x) C−1 ≤ n ≤ C. Gn(y) Complex Dynamics and Related Topics 88

i i Proof. Let xi = f (x) and yi = f (y) for 0 ≤ i ≤ n. Then

n−i d(xi, yi) ≤ λ d(xn, yn).

So Xn−1 | log Gn(x) − log Gn(y)| ≤ | log ψ(xi) − log ψ(yi)| i=0 [ψ] Xn−1 ≤ α d(x , y )α A i i i=0 [ψ] Xn−1 ≤ α λ−α(n−i)d(x , y )α A n n i=0 ≤ C0 where A = minx∈X ψ(x) and [ψ]α is the H¨olderconstant for ψ, and

[ψ] aαλα C = α . 0 A(λα − 1)

Therefore, let C = eC0 , we have

G (x) C−1 ≤ n ≤ C. Gn(y)

Proof of Theorem 6.2. We prove the Gibbs Property ﬁrst. Let µ be a G- measure. Let r be a real number such that 0 < 2r ≤ a. For any x ∈ X, let φ be a function such that

1Bn(x,r) ≤ φ ≤ 1Bn(x,2r) where 1B denotes the characteristic function of a set B. Then we have Z Z Z ∗ µ(Bn(x, r)) ≤ φ dµ = φ dPn µ = Pnφ dµ where X X

Pnφ(y) = Gn(z)φ(z) ≤ Gn(z)1Bn(x,2r)(z). z∈f −n(f n(y)) z∈f −n(f n(y)) Complex Dynamics and Related Topics 89

From Lemma 6.4 and Proposition 2.3, there is a constant C > 0 such that −n n #(f (f (y)) ∩ Bn(x, 2r)) ≤ C and Gn(z) ≤ CGn(x), z ∈ Bn(x, 2r). Thus we get 2 µ(Bn(x, r)) ≤ C Gn(x). On the other hand, we have Z Z Z ∗ µ(Bn(x, 2r)) ≥ φ dµ = φ dPn+pµ ≥ Pn+pφ dµ where p is an integer in Proposition 2.4 in Section 2 and X Pn+pφ(y) = Gn+p(z)φ(z) z∈f −n−p(f n+p(y)) X

≥ Gn+p(z)1Bn(x,r)(z). z∈f −n−p(f n+p(y))

Proposition 2.5 says that there is at least one term in the sum is non- zero. This and Lemma 6.4 imply that there is a positive constant, we still denote it as C, such that

p µ(Bn(x, 2r)) ≥ CGn+p(x) ≥ CA Gn(x),

s where A = minx∈X ψ(x). Let s be the least integer such that λ ≥ s 2. Then we have Bn(x, r) ⊃ Bn+s(x, λ r) ⊃ Bn+s(x, 2r). By the last inequality, we get p+s µ(Bn(x, r)) ≥ CA Gn(x). Therefore, we have a positive constant depending on r only, which we still denote it as C, such that µ(B (x, r)) C−1 ≤ n ≤ C. Gn(x) Following Section 5 and the Gibbs Property, only remaining thing to be proven is that a G-measure is unique. From Theorem 5.5, we only need to prove that a P-ergodic G-measure is unique. Theorem 5.4 says that any two P-ergodic G-measures are either equal or totally singular. Now we use the Gibbs Property to prove that any two P-ergodic G-measures Complex Dynamics and Related Topics 90

µ and ν are mutually absolutely continuous, that is, there is a constant C > 0 such that C−1ν(U) ≤ µ(U) ≤ Cν(U) for any open set U of X. Let us prove it as follows. Fix a real number r, 0 < 2r ≤ a. Let {x1, ··· , xm} be a 2r-net in (X, d), this means that the balls {B(xi, r)}1≤i≤m are disjoint and the balls {B(xi, 2r)}1≤i≤m form a cover of X. Deﬁne

A1 = B(x1, 2r) \ (B(x2, r) ∪ · · · ∪ B(xm, r))

Ai = B(xi, 2r) \ (A1 ∪ · · · ∪ Ai−1), 2 ≤ i ≤ m.

m Then we get a partition Q0 = {Ai}i=1 of X satisfying

B(xi, r) ⊆ Ai ⊆ B(xi, 2r), 1 ≤ i ≤ m.

−n kni For every n ≥ 1 and every 1 ≤ i ≤ m, denote f (xi) = {zj}j=1. n Let gjn be the inverse of f : Bn(zj, 2r) → B(xi, 2r). Define Anij = −n gjn(Ai). We call Anij a n-component of f |Q0. Let Qn be the set of all −n n components of f |Q0. It is again a partition of X and satisfies that for any A ∈ Qn, Bn(cA, r) ⊆ A ⊆ Bn(cA, 2r) n where cA ∈ A such that f (cA) = xj. The point cA is called the center of A. It is worth to note that for n > k ≥ 1 and for any A ∈ Qn, (n−k) f (A) ∈ Qk. However Qk may not be a refinement of Qn. (So they are not Markov partitions.) Let U be an arbitrary open set in X. For n ≥ 1, let Qn(U) be the family of all elements A of the partition Qn such that the n-Bowen ball Bn(cA, r) is entirely contained in U. Let [ Vn = A.

A∈Qn(U) This is a Borel subset of U which is a countable union of disjoints sets. From the Gibbs Property, we get X X µ(Vn) = µ(A) ≤ µ(Bn(cA, 2r)) A∈Q (U) A∈Q (U) nX n X 2 ≤ C Gn(cA) ≤ C ν(Bn(cA, r)) A∈Q (U) A∈Q (U) Xn [n 2 2 2 ≤ C ν(A) = C ν( A) = C ν(Vn)

A∈Qn(U) A∈Qn(U) Complex Dynamics and Related Topics 91

Then we have µ(U) ≤ C2ν(U) by using Fatou lemma and the fact that

U = lim inf Vn. n→∞ Similarly ν(U) ≤ C2µ(U). Therefore, a G-measure is unique. Let µ be the unique G-measure. Then following Theorem 5.2, Pnφ →< µ, φ > as n → ∞ for any φ ∈ C0. Therefore, Lnφ →< µ, φ > as n → ∞ (Lemma 6.3). This completes the proof.

7 Spectral gaps of RPF operators

It will be proved in this section that all the spectral points other than the maximal eigenvalue of L acting on α-H¨oldercontinuous functions is contained in a disk centered at 0 of radius strictly smaller than the maximal eigenvalue. As we shall see, there are many consequences of this fact. 0 Let L be the RPF operator in the previous section. Recall that CC = C0(X, C) is the Banach space of all continuous complex valued functions φ : X → C, equipped with the supremum norm ||φ|| = max |φ(x)|. x∈X α α Let 0 < α ≤ 1 and let CC = C (X, C) be the space of all α-H¨older 0 0 complex valued continuous functions φ in CC, that is, φ ∈ CC satisfying |φ(x) − φ(y)| [φ]α = sup α < ∞, 0

Lφ = Lφ1 + iLφ2. 0 0 Thus we have that L0 : CC → CC is a bounded linear operator. The α space CC equipped with the norm

||φ||α = ||φ|| + [φ]α α α is a Banach space. Then Lα = L : CC → CC is a bounded linear operator (refer to the argument between the statements of Lemma 3.1 and Lemma 3.2). The following is a directly consequence of Theorem 6.1. Complex Dynamics and Related Topics 92

0 0 Corollary 7.1. The maximal eigenvalue δ of L : CR → CR is the spectrum radius of 0 0 L0 = L : CC → CC. Proof. The spectrum radius can be calculated as

1 n n ρ(L0) = lim sup ||L0 || n→∞ where n n ||L0 || = sup ||L0 φ||. 0 φ∈CC,||φ||≤1 0 For any φ ∈ CC with ||φ|| ≤ 1, from Theorem 6.1,

−n n ||δ L0 φ|| ≤ | < ν, φ > | · ||h|| + 1 ≤ ||h|| + 1 for n large. So −n n δ ||L0 || ≤ ||h|| + 1 and 1 1 n n n ||L0 || ≤ (||h|| + 1) δ for n large. So ρ(L0) ≤ δ. But δ is a spectrum point, we have ρ(L0) = δ. Furthermore, we have

Theorem 7.1. The maximal eigenvalue δ is the spectrum radius of

α α Lα = L : CC → CC.

The rest of spectrum is in a disk of center 0 with radius strictly less than δ.

This is a direct consequence of the relation between L. More precisely, it follows from the next corollary which is a consequence of Theorem 6.2.

Corollary 7.2. Suppose L is a normalized RPF operator, i.e., L1 = 1. Then the maximal eigenvalue 1 is the spectrum radius of

α α Lα = L : CC → CC.

The rest of spectrum is in a disk of center 0 with radius strictly less than 1. Complex Dynamics and Related Topics 93

Proof. The spectrum radius can be calculated as

1 n n ρ(Lα) = lim sup ||Lα||α , n→∞ where n n ||Lα||α = sup ||L0 φ||α. α φ∈CC ,||φ||α≤1 α For any φ ∈ CC with ||φ||α ≤ 1 and x, y ∈ X with d(x, y) ≤ a, let −1 −1 f (x) = {x1, ··· , xn} and f (y) = {y1, ··· , yn} such that d(xi, yi) ≤ 1 λd(x,y) . Then Xn |Lαφ(x) − Lαφ(y)| ≤ |ψ(xi)φ(xi) − ψ(yi)φ(yi)| i=1 Xn Xn ≤ |ψ(xi) − ψ(yi)||φ(yi)| + ψ(xi)|φ(xi) − φ(yi)|. i=1 i=1

1 Pn Since d(xi, yi) ≤ λd(x,y) and since i=1 ψ(xi) = 1, we have

[ψ] n 1 [L φ] ≤ α 0 ||φ|| + [φ] . α α λα λα α It follows that 1 ||L φ|| ≤ C ||φ|| + ||φ|| , α α 1 λα α α α where C1 = 1 − 1/λ + [ψ]αn0/λ . Inductively, suppose 1 ||Ln−1φ|| ≤ C ||φ|| + ||φ|| . α α n−1 λα(n−1) α Then 1 ||Lnφ|| ≤ C ||L φ|| + ||L φ|| α α n−1 α λα(n−1) α α 1 1 ≤ C ||φ|| + C ||φ|| + ||φ|| n−1 λα(n−1) 1 λαn α 1 = C ||φ|| + ||φ|| , n λαn α

α(n−1) α α where Cn = Cn−1 + (1/λ C1 ≤ C = C1λ /(λ − 1). Therefore we have that 1 ||Lnφ|| ≤ C||φ|| + ||φ|| α α λαn α Complex Dynamics and Related Topics 94 for all n ≥ 1. Let α⊥ α CC = {φ ∈ CC ; < µ, φ >= 0}. α α⊥ L Then CC = CC C because φ = (φ− < µ, φ >)+ < µ, φ >. To prove the rest of spectrum of Lα is in a disk of center 0 with radius less than 1, we only need to prove that the spectrum of

α⊥ α⊥ α⊥ Lα|CC : CC → CC is strictly less than 1. We prove this as follows. Suppose n, k > 0. Then 1 ||Ln+kφ|| ≤ C||Lk φ|| + ||Lk φ|| α α α λαn α α 1 1 ≤ C||Lk φ|| + C ||φ|| + ||φ|| . α λαn λα(n+k] α

α⊥ 0 Since {φ ∈ CC ; ||φ||α ≤ 1} is a compact set in CC (because it is a uniformly bounded and equicontinuous family), following Theorem 6.2, for any 0 < τ < 1 there are m, k > 0 such that

m+k ||Lα φ||α ≤ τ

α⊥ m+k for all φ ∈ CC with ||φ||α ≤ 1. So ||Lα ||α ≤ τ. Therefore,

1 1 n α⊥ n m+k lim sup ||L |CC ||α ≤ τ < 1. n→∞ This completes the proof. In Section 10, we will prove an estimate on the gap between the maximal eigenvalue and the rest of spectrum of Lα by using a new method (see Theorem 10.2 and Corollary 10.3).

8 Spectral decomposition and perturbation

In this section, we present two standard results on bounded linear operators. One is the spectral decomposition and the other is a perturbation theorem. A projection on a Banach space B is a bounded linear operator P such that P 2 = P. Complex Dynamics and Related Topics 95

It is clear that P is a projection if and only if I − P is a projection. The space B is a direct-sum of two closed subspaces B1 and B2, written as B = B1 ⊕ B2, if B1 and B2 are complementary in the following sense \ B = B1 + B2, B1 B2 = ∅.

To specify a projection P is essentially the same thing to specify a direct- sum decomposition B = B1 ⊕ B2. In fact, if P is a projection, then

B = R(P ) ⊕ N(P ) where R(P ) denotes the range of P and N(P) denotes the null space of P . Notice that R(P ) is closed because of the fact P 2 = P . So does N(P ) = R(1 − P ). If B = B1 ⊕ B2 is written into a direct-sum of two closed subspaces, then the operator P : B → B deﬁned by

x = x1 ⊕ x2 → P x = x1 is a projection having B1 as its range. Notice that P is continuous because of the closed graph theorem. We may speak of the projection P onto B1 along B2. It is easy to see that a projection P commutes with a bounded operator A if and only if its range and null set are invariant under A, i.e.

AR(P ) ⊂ R(P ), AN(P ) ⊂ N(P ).

The main core of spectral decomposition is the following: to any splitting of the spectrum σ(A) = S1 ∪ S2 into two disjoint closed subsets, there corresponds a direct-sum splitting of the space B = B1 ⊕ B2 into two closed invariant spaces, i.e.

AB1 ⊂ B1,AB2 ⊂ B2 such that the spectra of the restrictions

A|B1 : B1 → B1,A|B2 : B2 → B2 Complex Dynamics and Related Topics 96

are respectively S1 and S2. The splitting is not only invariant under A but also invariant under any operator which commutes with A. All these determine uniquely the splitting B = B1 ⊕ B2. The construction of the space splitting is trivial in the case of a diagonalizable operator in a ﬁnite-dimensional space: Bi is spanned by all eigenvectors with eigenvalues in Si (i = 1, 2). Unfortunately, this construction can not be generalized. However, the projection onto B1 along B2 can be written as follows I 1 T = (λI − A)−1dλ 2πi γ where γ is a contour enclosing all points of the set S1 but none of the points of S2. In order to see that T is really the projection, we have only to verify

(i) T x = x for any eigenvectors with eigenvalue in S1;

(ii) T x = 0 for any eigenvectors with eigenvalue in S2. In fact, since A is diagonalizable, i.e. A = QDQ−1 with D diagonal, we have (λI − A)−1 = Q(λI − D)−1Q−1. Now we may check (i) and (ii) by the Cauchy integral formula. This representation of a projection in a space of finite dimension can be generalized to the general case as we will state in the following theorem. Let γ be a cycle which is defined as a formal sum of closed paths in C and z ∈ C be a point not on γ. Define the winding number of γ around z as the integer I 1 dλ . 2πi γ λ − z Theorem 8.1. Let A be a bounded linear operator on a Banach space B. Let σ(A) = S1 ∪ S2 be a decomposition of the spectrum of A into two disjoint closed subsets. For any cycle γ in the complement of the spectrum which winds once around each point of S1 and zero times around each point of S2, let I 1 P = R(λ, A)dλ 2πi γ where R(λ, A) = (λI − A)−1, called the resolvent of A. Then Complex Dynamics and Related Topics 97

(1) The integral depends only on S1 and S2, not on γ; (2) The operator P is a projection which commutes with every bounded operator commuting with A;

(3) The spectrum of A|R(P ) is S1 and the spectrum of A|N(P ) is S2; (4) The properties of P listed above uniquely determines P .

Proof. (1) The independence of γ is a consequence of the Cauchy integral formula. By the way we point out that if S2 is empty and γ is a large circle centered at the origin, for λ on γ we have the Neumann series

X∞ An R(λ, A) = . λn+1 n=0 Integrating term by term gives P = I. (2) Take another cycle γ0 with the same properties as γ, such that γ0 is inside γ in the sense that γ winds once around each point of γ0 but γ0 winds zero times around each point of γ. We may take γ0 to lie in a small neighborhood of S1. Then we have I I 1 1 P 2 = R(λ, A)dλ · R(λ0,A)dλ0 2πi 2πi 0 γ I I γ 1 0 0 = 2 R(λ, A)R(λ ,A)dλdλ . (2πi) γ γ0 Applying the identity

U −1 − V −1 = U −1(V − U)V −1 to U = λI − A and V = λ0I − A allows us to write

R(λ, A) − R(λ0,A) = −(λ − λ0)R(λ, A)R(λ0,A).

So, we have

I I 0 2 1 R(λ, A) − R(λ ,A) 0 P = − 2 0 dλdλ (2πi) 0 λ − λ I γ γ I I I 1 0 1 0 1 1 0 = 2 R(λ ,A) 0 dλdλ − 2 R(λ, A) 0 dλ dλ. (2πi) γ0 γ λ − λ (2πi) γ γ0 λ − λ Complex Dynamics and Related Topics 98

Since γ0 est inside γ, we have I I 1 1 0 0 dλ = 2πi, 0 dλ = 0. γ λ − λ γ0 λ − λ Finally we have P 2 = P , as desired. It is clear from the deﬁnition of P that P commutes with any bounded operator which commutes with A. (3) In particular, P commutes with A, so the range and the null space of the projection P are invariant under A. Let A1 : R(P ) → R(P ) be the restriction of A on R(P ) and A2 the restriction of A on N(P ). The following inclusion is clear by the deﬁnition of spectrum [ [ σ(A1) σ(A2) ⊂ σ(A) = S = S1 S2.

We are going to prove σ(A1) ⊂ S1 by showing that λI − A1 is invertible for λ ∈ S2. Actually we can construct its inverse explicitly as a contour integral: I 1 R(z, A) B = dz 2πi γ λ − z where γ as in the statement of the theorem. In fact, since B commutes with A, it also commutes with P . Hence I 1 (λ )I + (zI − A) (λI − A)B = z R(z, A)dz 2πi λ − z Iγ µ ¶ 1 1 = R(z, A) + dz 2πi λ − z γ I 1 1 = P + dz 2πi γ λ − z = P.

Thus if P x = x,(λI − A)Bx = x. We also have B(λI − A)x = x because A and B commute. Thus the restriction of B on R(P ) is the inverse of the restriction on R(P ) of λI − A. Interchanging the roles of S1 and S2, we get σ(A2) ⊂ S2. (4) It remains to prove the uniqueness of P . Let Pe be another projection such that (i) Pe commutes with every operator commuting with A; e e (ii) the spectrum of A restricted to R(P ) (resp. N(P )) is S1 (resp. S2). It follows that Complex Dynamics and Related Topics 99

(a) P (I − Pe) is also a projection; (b) it commutes with A.

Since the range of P (I − Pe) is contained in both R(P ) and N(I − Pe), the spectrum of A restricted on the range of P (I − Pe) is contained in e both S1 and S2. So, the spectrum of P (I − P ) is empty since S1 and e e S2 are disjoint. So, P = P P . Interchanging the roles of P and P gives Pe = PPe . Finally we get P = Pe.

Assume now that S1 = {λ} consists of a single eigenvalue isolated (by γ) from the rest of the spectrum of A. We consider the projection P associated to {λ}. If A0 is a bounded operator suﬃciently close to A, then the spectrum of A0 also consists of an eigenvalue {λ0} isolated by γ from the rest of the spectrum of A0. The projection P 0 associated to {λ0} is called the eigenprojection of λ0. The map A0 → P 0 is analytic in a neighborhood of A. Let us state more precisely the following theorem for the case where λ is a simple isolated eigenvalue.

Theorem 8.2 (Perturbation theorem). Let A be an element of the Banach algebra L(B) of bounded linear operators on a Banach space B. Suppose that A has a simple isolated eigenvalue λ with eigenvector v. Then for any ² > 0 there exists δ > 0 such that if B ∈ L(B) with kB − Ak < δ, then B has a simple isolated eigenvalue λ(B) and a corresponding eigenvector v(B) with the following properties

(1) λ(A) = λ and v(A) = v;

(2) The maps B → λ(B) and B → v(B) are analytic in kB − Ak < δ;

(3) for any B with kB − Ak < δ, λ(B) is inside the open disk D(λ, ²) centered at λ of radius ² and σ(B) \{λ(B)} is outside the closed disk D(λ, ²).

If λ is an isolated eigenvalue of A of finite multiplicity m, then for suf- ficiently close operator B of A the spectrum of B inside an neighborhood 0 0 of λ (delimited by γ) will consist of eigenvalues λ1, ··· , λm, associated to which is the projection π0 of B, and the map B → π will still be analytic. 0 However the individual eigenvalues λj are not well defined maps of B. Complex Dynamics and Related Topics 100

9 Application: central limit theorem

As application of the fact of spectral gaps (Theorem 7.1), we prove that for any H¨oldercontinuous function φ and any Gibbs measure µ associated to a H¨olderpotential, the central limit theorem holds on the probability space (X, µ) for the process

n−1 Snφ(x) = φ(x) + φ(fx) + ··· + φ(f x) (see [FSc]). Recall that f : X → X is a continuous dynamical system on a compact metric space (X, d). Let C(X) = C(X, C) be the space of real or complex continuous functions deﬁned on X. Two functions g1, g2 ∈ C(X) are said to be cohomologous if

g1 = g2 + u ◦ f − u + c for some u ∈ C(X) and some constant c. Then we write g1 ³ g2. Func- tions of the form u◦f −u+c are called coboundaries. The relation ”³” is an equivalence relation in C(X). The quotient space C˜(X) = C(X)/ ³, like C(X), is also a linear space. If functions φ1, . . . , φn ∈ C(X), regarded as functions in the linear space C˜(X), are linearly independent, we say that they are cohomologously independent. This is equivalent to saying that any linear combination ξ1φ1 + ··· + ξdφd (ξj’s being not all zero) is not cohomologous to a constant. Let µ be the Gibbs measure (also called equilibrium state) associated to a real valued α-H¨olderfunction ψ. The measure µ depends only on the cohomology class of ψ. Thus we may assume that the potential ψ is normalized in the sense that X eψ(y) = 1 (∀x ∈ X). f(y)=x

d Let φ = (φ1, . . . , φd) be a R –valued α-H¨oldercontinuous function. d Let z = (z1, . . . , zd) ∈ C . Consider the transfer operator

X Pd ψ(y)+ ziφi(y) Lzu(x) = e i=1 u(y). f(y)=x

For fixed z, the operator Lz is defined by a complex valued potential. It acts on the space of α-Höldercontinuous functions. We denote its spectral radius by ρ(z) and write P (z) = log ρ(z). Complex Dynamics and Related Topics 101

By Theorem 7.1 and Theorem 8.2, the function P (z) is analytic in a neighborhood U of 0 ∈ Cd, the eigenspace associated to ρ(z) is simple and it is possible to take an eigenvector v(z) such that v(z) is analytic in U and v(0) = 1. Thus we have

P (z) Lzv(z) = e v(z)(z ∈ U).

For a Rd-valued function deﬁned on X, we write

Xn−1 j SnΦ(x) = Φ(f x). j=0

Theorem 9.1. Let µ be the equilibrium state associated to a normal- d ized Höldercontinuous potential ψ. Let Φ = (φ1, . . . , φd) be a R –valued Höldercontinuous function such that EµΦ = 0. Suppose that the components of Φ are cohomologously independent. Then S Φ √n → N (0,P 00(0)) in distribution n where P 00(0) is the Hessian matrix of the second order derivative of P (z) at zero, which is positive definite.

Proof. Recall that

P (z) Lzv(z) = e v(z)(z ∈ U).

The partial derivatives of the both sides of this equation are respectively µ ¶ ∂ ∂v L v(z) = L φ v(z) + (z) ∂z z z j ∂z j µ j ¶ ∂ ∂P ∂v eP (z)v(z) = eP (z) v(z) + (z) . ∂zj ∂zj ∂zj

Write L = L0. Taking z = 0 leads to µ ¶ ∂v ∂P ∂v L φj + (0) = (0) + (0). ∂zj ∂zj ∂zj Integrating with respect to µ, we get ∂P (0) = Eµφj = 0. ∂zj Complex Dynamics and Related Topics 102

Differentiate a second time: µ ¶ ∂2 ∂v ∂v ∂2 Lzv(z) = Lz φkφjv(z) + φk (z) + φj (z) + (z) ∂zk∂zj ∂zj ∂zk ∂zk∂zj µ ¶ ∂2 ∂P ∂P ∂P ∂v ∂2P ∂2v eP (z)v(z) = eP (z) v(z) + + + ∂zk∂zj ∂zk ∂zj ∂zk ∂zj ∂zk∂zj ∂zk∂zj Taking z = 0 and then integrating, we get ∂2P ∂v ∂v (0) = Eµ(φkφj) + Eµφj + Eµφk. ∂zk∂zj ∂zk ∂zj Similarly, from the equality n nP (z) Lz v(z) = e v(z)(z ∈ U) we can obtain ∂2P ∂v ∂v n (0) = Eµ(SnφkSnφj) + EµSnφj + EµSnφk. ∂zk∂zj ∂zk ∂zj 00 ∂2P d Let P (0) be the matrix ( (0) ). Let ξ = (ξ1, . . . , ξd) ∈ R . Multiply ∂zk∂zj both sides of the above equality by ξkξj and then sum over k and j. We obtain t 00 2 nξ P (0)ξ = Eµ(Snhφ, ξi) + 2h∇v, ξi · EµSnhφ, ξi. Divide by n then take the limit. According to the Birkhoff Theorem, we have t 00 1 2 ξ P (0)ξ = lim Eµ (Snhφ, ξi) . n→∞ n It can be proved that ξtP 00(0)ξ > 0 if and only if hφ, ξi is not cohomologous to a constant. Consequently P 00(0) is positive definite if and only if hφ, ξi is not cohomologous to a constant for any ξ 6= 0. √1 Consider now the characteristic function of n Snφ Z Z √ −1 n √ fn(ξ) := exp ihξ, ( n) Snφidµ = Liξ/ n1dµ. X X Write µ ¶ µ ¶ iξ iξ 1 = v √ + 1 − v √ . n n we get µ ¶ √ iξ f (ξ) = enP (iξ/ n)v √ + o(1). n n Taking the second order Taylor development of P(z) at zero yields that d ξtP 00(o)ξ for any ξ ∈ R , fn(ξ) tends to exp(− 2 ). Thus we can conclude by using the Lévycontinuity theorem. Complex Dynamics and Related Topics 103

10 Hilbert metric and convergence speed

In this section, we will present a new method, called Hilbert projective metric method, to prove Theorem 3.1. This method further provides us an estimate on the spectral gap as well. See [Bi, FS]. Let ν be the unique probability measure in Theorem 6.1 and deﬁne

0 ρ(φ) =< ν, φ >, φ ∈ CC.

0 Then ρ is a functional from CC to C. From Theorem 6.1, we know that 0 −n n for φ ∈ CC, δ L φ converges to ρ(φ)h uniformly as n goes to inﬁnity. An important question in thermodynamical formalism is how fast does δ−nLnφ converge to ρ(φ)h? We discuss this question in this section (refer to [FJ1, FJ2] for some further study). Deﬁne

α α CK = {φ ∈ C ; φ > 0 and [log φ]α ≤ K} ∪ {0}.

α α One can check that CK is a convex cone in C , this means that (1) for α α any φ ∈ CK and any real number t ≥ 0, tφ ∈ CK and (2) for any φ1 and α α φ2 in CK and any 0 ≤ t ≤ 1, tφ1 + (1 − t)φ2 is in CK . α α Suppose ψ is in CK0 for some K0 > 0. Let K > K0/(λ − 1) be a −α ﬁxed constant and deﬁne τ = (K + K0)λ /K < 1. Then we have Lemma 10.1. α α L(CK ) ⊆ CτK . α Proof. Suppose φ 6= 0 ∈ CK and x, y ∈ X with d(x, y) ≤ a. Then X X Lφ(x) = ψ(z)φ(z) and Lφ(y) = ψ(w)φ(w). z∈f −1(x) w∈f −1(y)

d(x,y) We can arrange z’s and w’s such that d(z, w) ≤ λ . So X Lφ(x) = ψ(z)φ(z) z∈f −1(x) X α α ≤ ψ(w)eK0d(z,w) φ(w)eKd(z,w) w∈f −1(y) ³ ´ −α α ≤ Lφ(y) e(K0+K)λ d(x,y) ³ ´ = Lφ(y) eτKd(x,y)α .

α This implies that Lφ ∈ CτK . Complex Dynamics and Related Topics 104

The real valued function space Cα equipped with the norm

||φ||α = ||φ|| + [φ]α

α is a Banach space. The convex cone CK is closed in it. So we can deﬁne α a so-called Hilbert projective metric with respect to the convex cone CK as follows (refer to [Bi]) α α First let ¹ be the partial order in C deﬁned as φ1 ¹ φ2 if φ2−φ1 ∈ CK . The partial order ¹ is integral, meaning that if φn ¹ φ for all n and φn converges uniformly to φ0, then φ ¹ φ0. Let

A = A(φ1, φ2) = sup{t > 0; tφ1 ¹ φ2} and B = B(φ1, φ2) = inf{t > 0; φ2 ¹ tφ1}.

Note that A may be 0 and B may be ∞, however, if both φ1 and φ2 are not zero, then both A and B are ﬁnite numbers. The Hilbert projective α metric with respect to CK is deﬁned as B Θ(φ , φ ) = log . 1 2 A The following lemma gives an explicit formula for Θ.

α Lemma 10.2. Let φ1, φ2 ∈ CK . If both φ1 and φ2 are not zero, then

Kd(x,y)α Kd(z,w)α e φ1(x) − φ1(y) e φ2(z) − φ2(w) Θ(φ1, φ2) = log sup Kd(z,w)α Kd(x,y)α . d(x,y),d(z,w)≤a e φ1(z) − φ1(w) e φ2(x) − φ2(y)

Proof. By the deﬁnition, B Θ(φ , φ ) = log 1 2 A where Aφ1(x) ≤ φ2(x) ≤ Bφ1(x), x ∈ X, and

Kd(x,y)α φ2(y) − Aφ1(y) ≤ e (φ2(x) − Aφ1(x)), d(x, y) ≤ a, and

Kd(z,w)α Bφ1(w) − φ2(w) ≤ e (Bφ1(z) − φ2(z)), d(z, w) ≤ a. Complex Dynamics and Related Topics 105

Then φ (x) A ≤ 2 ≤ B, x ∈ X, φ1(x) and Kd(x,y)α e φ2(x) − φ2(y) A ≤ Kd(x,y)α , d(x, y) ≤ a, e φ1(x) − φ1(y) and Kd(z,w)α e φ2(z) − φ2(w) B ≥ Kd(z,w)α , d(z, w) ≤ a. e φ1(z) − φ1(w) We have

Kd(x,y)α φ2(x) e φ2(x) − φ2(y) A = min { inf , inf Kd(x,y)α } x∈X φ1(x) d(x,y)≤a e φ1(x) − φ1(y) and Kd(z,w)α φ2(z) e φ2(z) − φ2(w) B = max { sup , sup Kd(z,w)α }. z∈X φ1(z) d(z,w)≤a e φ1(z) − φ1(w)

Let x0 ∈ X such that φ (x ) φ (x) 2 0 = min 2 . φ1(x0) x∈X φ1(x)

Then for x ∈ X with d(x, x0) ≤ a,

α α Kd(x,x0) φ2(x0) φ2(x) Kd(x,x0) e φ1(x0) − φ1(x) e φ2(x0) − φ2(x) φ1(x0) φ1(x) φ2(x) Kd(x,x )α = Kd(x,x )α ≤ . e 0 φ1(x0) − φ1(x) e 0 φ1(x0) − φ1(x) φ1(x)

Take a sequence {xn} tending to x0 in X as n goes to inﬁnity. Then

α α Kd(x,y) Kd(xn,x0) e φ2(x) − φ2(y) e φ2(x0) − φ2(xn) φ2(xn) inf Kd(x,y)α ≤ Kd(x ,x )α ≤ . d(x,y)≤a e φ1(x) − φ1(y) e n 0 φ1(x0) − φ1(xn) φ1(xn) As n goes to inﬁnity,

Kd(x,y)α e φ2(x) − φ2(y) φ2(x0) inf Kd(x,y)α ≤ . d(x,y)≤a e φ1(x) − φ1(y) φ1(x0) We get Kd(x,y)α e φ2(x) − φ2(y) A = inf Kd(x,y)α d(x,y)≤a e φ1(x) − φ1(y) Complex Dynamics and Related Topics 106

Similarly we can get

Kd(z,w)α e φ2(z) − φ2(w) B = sup Kd(z,w)α . d(z,w)≤a e φ1(z) − φ1(w)

Thus

Kd(x,y)α Kd(z,w)α e φ1(x) − φ1(y) e φ2(z) − φ2(w) Θ(φ1, φ2) = log sup Kd(z,w)α Kd(x,y)α . d(x,y),d(z,w)≤a e φ1(z) − φ1(w) e φ2(x) − φ2(y)

Let ∆ = sup Θ(φ1, φ2) α φ1,φ2∈CτK and ³∆´ Λ = tanh . 4

Let k0 be the minimal positive integer such that there are k0 balls of radius a covering X.

Lemma 10.3. ³1 + τ ´ ∆ ≤ 2 log + 2(1 − τ + 2k )Kaα. 1 − τ 0 Thus 0 < Λ < 1.

α Proof. For any φ1, φ2 ∈ CτK , ³ ´ ³ ´ Kd(z,w)α Kd(z,w)α φ2(z) Kd(z,w)α −τKd(z,w)α e φ2(z)−φ2(w) = e − φ2(z) ≤ e −e φ2(z). φ2(w) In the same way, we get that ³ ´ Kd(x,y)α Kd(x,y)α τKd(x,y)α e φ2(x) − φ2(y) ≥ e − e φ2(x) and ³ ´ Kd(x,y)α Kd(x,y)α −τKd(x,y)α e φ1(x) − φ1(y) ≤ e − e φ1(x), ³ ´ Kd(z,w)α Kd(z,w)α τKd(z,w)α e φ1(z) − φ1(w) ≥ e − e φ1(z). Complex Dynamics and Related Topics 107

This implies that

Θ(φ1, φ2) Kd(x,y)α −τKd(x,y)α Kd(z,w)α −τKd(z,w)α (e − e ) (e − e ) φ2(z) φ1(x) ≤ log sup Kd(x,y)α τKd(x,y)α Kd(z,w)α τKd(z,w)α d(x,y),d(z,w)≤a (e − e ) (e − e ) φ2(x) φ1(z) −(1+τ)Kd(x,y)α −(1+τ)Kd(z,w)α (1 − e ) (1 − e ) φ2(z) φ1(x) = log sup −(1−τ)Kd(x,y)α −(1−τ)Kd(z,w)α . d(x,y),d(z,w)≤a (1 − e ) (1 − e ) φ2(x) φ1(z)

Since Z 0 e−C t ≤ 1 − e−t = eξdξ ≤ t, 0 ≤ t ≤ C −t and since φ2(z) φ1(x) α , ≤ ek0Ka , φ2(x) φ1(z) we have

α α ³ ´ (1 + τ)Kd(x, y) (1 + τ)Kd(z, w) α 2 Θ(φ , φ ) ≤ ek0Ka 1 2 (1 − τ)Kd(x, y)αe−(1−τ)Kaα (1 − τ)Kd(z, w)αe−(1−τ)Kaα ³ ´ 1 + τ 2 α α = e2(1−τ)Ka e2k0Ka . 1 − τ Therefore, ³1 + τ ´ ∆ ≤ 2 log + 2(1 − τ + 2k )Kaα. 1 − τ 0

α Lemma 10.4. For any φ1, φ2 ∈ CK ,

Θ(Lφ1, Lφ2) ≤ ΛΘ(φ1, φ2).

Proof. By the deﬁnition, B Θ(φ , φ ) = log 1 2 A α and φ2 − Aφ1 and Bφ1 − φ2 are both in CK . From Lemma 10.1,

Θ(L(φ2 − Aφ1), L(Bφ1(x) − φ2)) ≤ ∆.

Let A0 and B0 be two corresponding numbers in the deﬁnition of

Θ(L(φ2 − Aφ1), L(Bφ1(x) − φ2)). Complex Dynamics and Related Topics 108

Then we have B 0 ≤ e∆ A0 and A0L(φ2 − Aφ1)) ¹ L(Bφ1 − φ2) ¹ B0L(φ2 − Aφ1). This gives us that

B + B0A B + A0A Lφ1 ¹ Lφ2 ¹ Lφ1. 1 + B0 1 + A0 So

(B + A0A)(1 + B0) Θ(Lφ1, Lφ2) ≤ log (B + B0A)(1 + A0) eΘ(φ1,φ2) + A 1 + A = log 0 − log 0 eΘ(φ1,φ2) + B 1 + B Z 0 0 Θ(φ1,φ2) ξ (B0 − A0)e = ξ ξ dξ 0 (e + A0)(e + B0) 1 − A0 B0 ≤ q Θ(φ1, φ2) (1 + A0 )2 B0

≤ ΛΘ(φ1, φ2).

It is easy to check that the functional ρ satisﬁes 1. ρ(sφ) = sρ(φ) for any s ≥ 0 and φ ∈ Cα and

2. if φ1 ¹ φ2, then ρ(φ1) ≤ ρ(φ2). Use these properties, we have

α Lemma 10.5. For any φ1 and φ2 in CK satisfying ρ(φ1) = ρ(φ2) 6= 0,

Θ(φ1,φ2) ||φ2 − φ1|| ≤ (e − 1)||φ1||. Proof. From the deﬁnition, B Θ(φ , φ ) = log 1 2 A where Aφ1 ¹ φ2 ¹ Bφ1. Complex Dynamics and Related Topics 109

We get Aρ(φ1) ≤ ρ(φ2) ≤ Bρ(φ1). So A ≤ 1 ≤ B. Thus φ B 2 Θ(φ1,φ2) ||φ2−φ1|| ≤ || −1||||φ1|| ≤ (B−1)||φ1|| ≤ ( −1)||φ1|| ≤ (e −1)||φ1||. φ1 A

Lemma 10.6. Suppose φ ∈ Cα, φ > 0. Then there is an integer N = N(φ) > 0 such that n α L φ ∈ CK for all n ≥ N.

Proof. Let [log φ]α be the H¨olderconstant of log φ. From the proof of Lemma 3.2 we see that

−α [log Lφ]α ≤ ([log φ]α + K0)λ .

In general, K [log Lnφ] ≤ K (λ−α + ··· + λ−nα) + K λ−nα → 0 as n → ∞. α 0 φ λα − 1

α Since K > K0/(λ − 1), we can ﬁnd an integer N > 0 satisfying the lemma.

α Theorem 10.1. Suppose φ ∈ C . Then there is a constant C0 > 0 independent of φ and an integer N = N(φ) > 0 such that

−n n n−N ||δ L φ − ρ(φ)h|| ≤ C0Λ ||φ||, n ≥ N.

Proof. Suppose ||φ|| > 0. Otherwise it is trivial. Take

2||φ|| b = . minx h(x)

Then φ+bh > 0. From Lemma 10.6, there is an integer N = N(φ+bh) > 0 such that n α L (φ + bh) ∈ CK , n ≥ N. Complex Dynamics and Related Topics 110

For n ≥ 0, Z ρ(δ−nLn(φ + bh)) = δ−n Ln(φ + bh) dν Z = δ−n (φ + bh) dL∗nν Z = (φ + bh)dν = ρ(φ + bh) = ρ(φ) + b. Because φ ||φ|| |ρ(φ)| = |ρ( h)| ≤ , h minx h(x) ρ(φ) + b 6= 0. From Lemma 10.4, ||δ−nLnφ − δ−mLmφ|| = ||δ−nLn(φ + bh) − δ−mLm(φ + bh)|| ³ ´ ≤ eΘ(Ln(φ+bh),Lm(φ+bh)) − 1 ||δ−mLm(φ + bh)||

k α for any m > n ≥ N. Since L (φ + bh) ∈ CK for k ≥ N, Θ(Ln(φ+bh), Lm(φ+bh)) ≤ Λn−N Θ(LN (φ+bh), Lm−n+N (φ+bh)) ≤ Λn−N ∆. Thus ||δ−nLnφ − δ−mLmφ|| ≤ (eΛn−N ∆ − 1)||δ−mLm(φ + bh)||. Let m → ∞. We get −n n Λn−N ∆ n−N ||δ L φ − ρ(φ)h|| ≤ (e − 1)(ρ(φ) + b)||h|| ≤ C0Λ ||φ|| for n ≥ N, where C0 > 0 is a constant independent of φ. Remember that µ = hν is the Gibbs measure for (f, ψ) and is f- invariant. For any φ ∈ C0, deﬁne ¯ Z Z ¯ ¯ 2¯ Φ(n) = ¯ φ · φ ◦ f n dµ − ( φ dµ) ¯.

It is called the correlation for φ. Since L∗nν = δnν, Z Z φ · φ ◦ f n dµ = δ−n φ · φ ◦ f nh dL∗nν Z = δ−n Ln(φ · h · φ ◦ f n) dν Z = (δ−nLn(φh))φ dν. Complex Dynamics and Related Topics 111

Thus for n ≥ N, Z −n n n−N 2 n−N 2 Φ(n) ≤ ||δ L (hφ)−ρ(hφ)h||·||φ|| dν ≤ C0||h||Λ ||φ|| = CΛ ||φ|| where C = C0||h|| is a constant independent of φ. This gives us that Corollary 10.1. The decay of correlation is exponential. More precisely, there is a constant C > 0 such that for any φ ∈ Cα, there is an integer N = N(φ) > 0 such that

Φ(n) ≤ CΛn−N ||φ||2, n ≥ N.

Assume L is normalized, i.e., L1 = 1. Let

Cα⊥ = {φ ∈ Cα ; ρ(φ) = 0}.

Then Cα = Cα⊥ ⊕ R since φ = (φ − ρ(φ)) + ρ(φ). For φ ∈ Cα⊥, Theorem 10.1 says that there is an integer N = N(φ) > 0 such that

n n−N ||L φ|| ≤ C0Λ ||φ||, n ≥ N.

This is useful in the proof of the next lemma, from which will deduce an estimate on the essential spectral radius of L : Cα → Cα (see Theo- rem 10.2. Let κ = max{λ−α, Λ} < 1. Lemma 10.7. There is a constant C > 0 independent of φ such that

n n−N [L φ]α ≤ C · (n + 1 − N) · κ ||φ|| for all n ≥ N. Proof. Suppose x, y ∈ X with d(x, y) ≤ a. Let f −1(x) = {z} and f −1(y) = {w}. We can arrange z’s and w’s such that d(z, w) ≤ λ−1d(x, y). Then X |Lφ(x) − Lφ(y)| = | (ψ(z)φ(z) − ψ(w)φ(w))| z∈f −1(x),w∈f −1(y) X ≤ |ψ(z) − ψ(w)||φ(z)| + ψ(w)|φ(w) − φ(z)|. z∈f −1(x),w∈f −1(y) Complex Dynamics and Related Topics 112

So [ψ] n 1 [Lφ] ≤ α 0 ||φ|| + [φ] . α λα λα α −1 −α Let C2 = [ψ]αn0C0Λ λ . Then for n > N,

n n−1 [L φ]α = [L(L φ)]α [ψ] n 1 ≤ α 0 ||Ln−1φ|| + [Ln−1φ] λα λα α [ψ] n 1 ≤ α 0 C Λn−1−N ||φ|| + [Ln−1φ] λα 0 λα α 1 ≤ C Λn−N ||φ|| + [Ln−1φ] . 2 λα α Further, 1 1 [Lnφ] ≤ C Λn−N ||φ|| + (C Λn−1−N ||φ|| + [Ln−2φ] ) α 2 λα 2 λα α 1 ≤ 2C κn−N ||φ|| + [Ln−2φ] 2 λ2α α 1 ≤ · · · ≤ (n − N)C κn−N ||φ|| + [LN φ] 2 λ(n−N)α α ≤ C · (n − N + 1) · κn−N ||φ|| where C = max{C2,C1} > 0. Lemma 10.8. For any φ ∈ Cα⊥ and for any ξ ∈ R such that |ξ| > κ, the series Kφ = ξ−1φ + ξ−2Lφ + ξ−3L2φ + ··· + ξ−(n+1)Lnφ + ··· converges in the supremum norm and its sum Kφ belongs to Cα⊥. Thus K : Cα⊥ → Cα⊥. Proof. The convergence of the series in C0 follows directly from The- orem 10.1. Because ρ(Lnφ) = 0 for all n > 0, ρ(Kφ) = 0. From Lemma 10.7, X∞ −n−1 n [Kφ]α ≤ |ξ| [L φ]α n=0 NX−1 X∞ −n−1 n −n−1 n−N ≤ |ξ| [L φ]α + C |ξ| κ (n − N + 1) < ∞. n=0 n=N Therefore, Kφ is in Cα⊥. Complex Dynamics and Related Topics 113

α α⊥ Now consider complex valued function spaces CC and CC . Both of them are Banach spaces under the norm ||φ||α. Then α Lφ = Lφ1 + iLφ2 for φ = φ1 + iφ2, φ1, φ2 ∈ C . α α α⊥ α⊥ Thus we have L : CC → CC and L : CC → CC . Similarly, K is also α⊥ α⊥ deﬁned on CC and for any φ ∈ CC and any ξ ∈ C such that |ξ| > κ, the series Kφ = ξ−1φ + ξ−2Lφ + ξ−3L2φ + ··· + ξ−(n+1)Lnφ + ··· α⊥ α⊥ α⊥ converges in the maximal norm and belongs to CC . So K : CC → CC . Now we prove a stronger result than Theorem 7.1. Theorem 10.2. Suppose L is a normalized RPF operator, i.e., L1 = 1. α⊥ α⊥ α⊥ Then the spectrum radius of L|CC : CC → CC is less than or equal to κ. Proof. From Lemma 10.8, for ξ ∈ C such that |ξ| > κ, α⊥ K(ξI − L) = (ξI − L)K = I on CC . α⊥ α⊥ So ξI − L : CC → CC is bijective. By the Banach theorem, it is α⊥ invertible. This implies that ξ is not a spectrum point of L|CC . This proves the theorem. α⊥ Corollary 10.2. For any φ ∈ CC and any ² > 0, there is an integer N = N(φ, ²) > 0 such that n n ||L φ||α ≤ (κ + ²) ||φ||α, n ≥ N. Proof. From the spectrum formula, 1 n α⊥ n lim sup ||L |CC ||α ≤ κ. n→∞ We have an integer N = N(φ, ²) > 0 such that n n ||L φ||α ≤ (κ + ²) ||φ||α, n ≥ N.

α α Corollary 10.3. For a normalized RPF operator L : CC → CC, the maximal eigenvalue is 1 and the rest of its spectrum is inside the closed disk centered 0 of radius κ. From the relation between an RPF operator and its normalization, Corollary 10.3 implies that α α Corollary 10.4. For any RPF operator L : CC → CC, let δ be its maximal positive simple eigenvalue. Then the rest of its spectrum is inside the closed disk centered 0 of radius κδ. Complex Dynamics and Related Topics 114

11 Nuclear operators

Suppose B and D are two Banach spaces. We denote by B0 the dual space of B, i.e., the space of all bounded linear functionals from B to C. Let L(B, D) be the space of bounded linear operators from B to D. If B = D, we write L(B) = L(B, D). For x0 ∈ B0 and y ∈ D and λ ∈ C, denote by x0 ⊗y the linear operator x0 ⊗ y(z) = x0(z)y defined for z ∈ B and taking values in D. It is a bounded linear operator. A bounded linear operator N is said to have 0 n 0 n n finite rank if there are {xi}i=1 ⊂ B and {yi}i=1 ⊂ D and {λi}i=1 ⊂ C such that Xn 0 N = λixi ⊗ yi. i=1 When D = B, the operator N defined above belongs to L(B). Then we can define its trace as Xn 0 tr(N) = λixi(yi). i=1 The finite rank operator N has different representations. However as the following lemma shows, the trace is independent of representations. 0 0 First remark that if {yi} are independent and xi are such that xi(yj) = δij, then Xn Xn 0 tr(N) = λi = xj(N(yi)). i=1 i=1 Lemma 11.1. The trace tr(N) of a linear operator of finite rank in L(B) is independent of its representations.

Proof. Let {e1, ··· , em} be a linearly independent set in B such that ||ei|| = 1 and such that {y1, ··· , yn} ⊂ span{e1, ··· , em}. Then

yi = ai1e1 + ··· + aimem, i = 1, ··· , n.

0 0 0 Let ei ∈ B be the linear functional satisfying that ei(ej) = δij. Then Xn Xm Xn Xm Xm ³ Xn ´ 0 0 0 N(ek) = λixi(ek) aijej = λiaijxi(ek)ej = λiaijxi(ek) ej. i=1 j=1 i=1 j=1 j=1 i=1 It follows that ³ ´ Xn 0 0 ek N(ek) = λiaikxi(ek). i=1 Complex Dynamics and Related Topics 115

So Xn Xn Xm 0 0 tr(N) = λixi(yi) = λixi( aikek) i=1 i=1 k=1 Xn Xm Xm Xn Xm ³ ´ 0 0 0 = λiaikxi(ek) = λiaik)xi(ek) = ek N(ek) . i=1 k=1 k=1 i=1 k=1 This implies that the trace Tr(N) is independent of representations.

∞ A sequence a = {λn}n=1 of complex numbers is in lp, 0 < p ≤ 1 if

³ X∞ ´ 1 p p ||a||p = |λn| < ∞; n=1 and it is in l∞ if ||a||∞ = sup max |λi| < ∞. n≥0 1≤i≤n A linear operator N in L(B, D) is called a nuclear operator of order 0 ≤ q < 1 if X∞ 0 N = λnxn ⊗ yn n=1 ∞ p ∞ where {λn}n=1 is in l for any p > q, {yn}n=1 is a sequence of elements in 0 ∞ D with norms ||yn||D = 1, and x = {xn}n=1 is a sequence of elements in 0 0 B with also norms ||xn||B0 = 1. In other words, N is a nuclear operator if and only if it can be decomposed as the product N = ABC where C : B → l∞, B : l∞ → ∩qcompact operator may not be nuclear because all eigenvalues of a compact operator may accumulate to 0 very slowly.

Theorem 11.1. Suppose B, D, and E are three Banach spaces. Suppose N ∈ L(B, D) and M ∈ L(D, E) are bounded linear operators. If one of N and M is nuclear of order 0 ≤ q < 1, then MN is a nuclear operator of order q.

Proof. Suppose N is nuclear and has a representation X∞ 0 N = λnxn ⊗ yn. n=1 Complex Dynamics and Related Topics 116

It is easy to see that X∞ 0 MN = λnxn ⊗ M(yn) n=1 X∞ M(y ) = λ ||M(y )||x0 ⊗ n n n n ||M(y )|| n=1 n for all n > 0 such that M(yn) 6= 0. Therefore, we have X∞ ˜ 0 MN = λnxn ⊗ y˜n n=1 ˜ wherey ˜n = M(yn)/||M(yn)|| or 0 and λn = λnM(yn). Since X∞ X∞ ˜ p p p |λn| ≤ |λn| ||M|| < ∞ n=1 n=1 for any q < p ≤ 1, we get that MN is a nuclear operator of order q. Similarly, we can prove that MN is a nuclear operator of order q when M is nuclear of order q.

P∞ For a nuclear operator in L(B) having its representation N = n=1 λnyn⊗ 0 xn, we could define its “formal” trace X 0 “tr”(N) = λnxn(yn). n∈N A Banach space B is said to have approximation property if there is a ∞ sequence {en}n=1 of independent unit elements in B such that any element x in B can be written as a linear combination of these independent elements: X∞ x = anen, n=1 ∞ where {an}n=1 is a sequence of complex numbers (such a system {en}) is called a Schauder basis). If B has approximation property and N is a nuclear operator of order 0 ≤ q < 1, then it may be proved that the trace defined above is independent of representations (the same proof for finite rank operators). Not every Banach space has approximation property. However, we have Complex Dynamics and Related Topics 117

Theorem 11.2 (Grothendick [Gr]). If N ∈ L(B) is a nuclear operator of order 0, then “tr”(N) is unique, this means that it is independent of representations, and we have a well-deﬁned trace formula X∞ X 0 tr(N) = λnxn(yn) = ρ n=1 ρ∈sp(N) and a well-deﬁned determinant

³ X∞ zn ´ Y det(I − zN) = exp − tr(N n) = (1 − τz), n n=1 τ∈sp(N) where sp(N) means the spectrum of N. Moreover, det(I − zN) is an entire function of C.

Remark 11.1. To prove that det(I − zN) is entire, Grothendick proved that ∞ X ^n det(I − zN) = (−z)n N n=0 V where nN means the n-fold exterior product of the linear operator N. Then following the Hadamard Inequality, he showed that

^n − n − n log n |tr( N)| ≤ Cn 2 = Ce 2 .

So det(I − zN) is an entire function.

Now let us look at an example of nuclear operators. Let U and V be two bounded Jordan domains in the complex plane such that U is compact and U ⊂ U ⊂ V . Suppose Cω(V ) (resp. Cω(U)) is the space of all complex analytic functions on U (resp. V ) and suppose C0(V ) (resp. Cω(U)) is the space of all complex valued continuous functions on U (resp. V ). Let A(U) = Cω(U) ∩ C0(U) and A(V ) = ω 0 C (V ) ∩ C (V ). Deﬁne the restriction operator RU : A(V ) → A(U) by

RU (φ) = φ|U , ∀φ ∈ A(V ).

Example 11.1. The restriction operator RU ∈ L(A(V ),A(U)) is a nuclear operator of order 0. Complex Dynamics and Related Topics 118

Proof. Let Dr = {z ∈ C; ||z|| < r} for any 0 < r ≤ 1. First let us consider a special case that U = Dr for some 0 < r < 1 and V = D1, we want to prove that Rr = RDr : A(D1) → A(D1) is nuclear of order 0. For any φ in A(D1), we have I I φ(u) du X∞ φ(u) du X∞ φ(z) = · = zn−1 = λ x0 (φ) ⊗ y (z) u − z 2πi un 2πi n n n ∂D1 n=1 ∂D1 n=1

n−1 n−1 where yn(z) = (z/r) , λn = r and I 0 φ(u) du xn(φ) = n · . ∂D1 u 2πi

0 n−1 Notice that ||xn||A(D1) = 1 and yn = (z/r) is a function deﬁned on Dr P∞ p(n−1) with ||yn(z)||A(Dr) = 1 and that n=1 r < ∞ for all 0 < p ≤ 1. So, the operator X∞ 0 Rr = λnxn ⊗ yn(z) n=1 is nuclear of order 0. Return to the general case. Since U and V are Jordan domains, let f : D1 → V be a conformal mapping (Riemann mapping). Then f can ∗ be continuously extended to D1. Let f φ = φ ◦ f for φ ∈ A(V ). Let 0 < −1 r < 1 such that f (U) ⊂ Dr and let g : Dr → U be another conformal mapping (Riemann mapping). Also g can be continuously extended to −1 −1 ∗ −1 Dr. Let g∗φ = φ ◦ g for φ ∈ A(Dr). Let (g ◦ f ) φ(z) = φ(g ◦ f (z)) for φ ∈ A(U). Then

−1 ∗ ∗ RU = (g ◦ f ) ◦ g∗ ◦ Rr ◦ f .

−1 ∗ ∗ Since RDr is nuclear of order 0 and (g ◦f ) , g∗, and f are all bounded, RU is nuclear of order 0.

12 Nuclear transfer operators and dynamical determinants

Consider two complex numbers c, θ, |θ| < 1. Deﬁne an operator Nθ,c : A(D1) → A(D1) by

Nθ,cφ(w) = cφ(θw), φ ∈ A(D1). Complex Dynamics and Related Topics 119

Theorem 12.1. The operator Nθ,c is nuclear of order 0.

Proof. Consider the restriction operator RD|θ| : A(D1) → A(D|θ|) which is nuclear of order 0 ( Example 11.1), and the bounded operator N˜ : A(D|θ|) → A(D1) deﬁned by ˜ Nφ(w) = cφ(θw), φ ∈ A(D|θ|).

Then, by Theorem 11.1, the composition ˜ Nθ,c = N ◦ RD|θ| . is nuclear of order 0.

2 How to calculate the trace Tr(Nθ,c)? Consider the bases 1, w, w , ···, n w , ··· of A(D1) and the Taylor expansion X∞ n φ(w) = anw n=0

n n n for φ in A(D1). Notice that Nθ,c(w ) = cθ w for n = 0, 1, ···. Thus n ∞ n ∞ {w }n=0 are all eigenvectors of Nθ,c with eigenvalues {cθ }n=0. So the trace is equal to X∞ c tr(N ) = cθn = . θ,c 1 − θ n=0 It is clear that the determinant Y∞ n det(1 − zNθ,c) = (1 − cθ z) n=0 is an entire function. Let V and U are Jordan domains in the complex plane C with U ⊂ V . Suppose g : V → U is conformal. Then g has a unique ﬁxed point zg which is attractive (following hyperbolic geometry). Let ψ ∈ A(V ). Deﬁne Ng,ψ : A(V ) 7→ A(V ) by

Ng,ψφ = ψ ◦ g · φ ◦ g, φ ∈ A(V ).

Lemma 12.1. The operator Ng,ψ is nuclear of order 0 and

ψ(zg) tr(Ng,ψ) = 0 1 − g (zg) Complex Dynamics and Related Topics 120

Proof. The restriction operator RU : A(V ) → A(U) is nuclear of order 0. Since ˜ Ng,ψ = N ◦ RU , where Nφ˜ = ψ ◦ g · φ ◦ g, φ ∈ A(V ). is bounded, N is nuclear of order 0. For any φ ∈ A(V ), I g(z) dz Ng,ψφ(w) = φ(z) . ∂V z − g(w) 2πi

Since for w ∈ U and z ∈ ∂V , z − g(w) 6= 0, I g(z) dz ψ(zg) tr(Ng,ψ) = = 0 . ∂V z − f(z) 2πi 1 − g (zg)

Remark 12.1. In general if K(w, z) is a kernel function such that I Nφ(w) = K(w, z)φ(z)dz : A(V ) → A(V ) ∂V is an integral operator (called the standardH Fredholm operator). The trace of N can be calculated by tr(N) = ∂V K(z, z)dz (see [DS]). Finally let us consider some conformal dynamics and its dynamical determinants. Suppose that V is a Jordan domain and U1, ···, Ud are sub-Jordan domains of V such that [d \ U k ⊂ V and Ui Uj = ∅, 1 ≤ i 6= j ≤ d. k=1

Sd Consider the dynamics f : k=1 Uk → V . Suppose that each f|Uk : Uk → V is conformal. Let ψ ∈ A(V ). The transfer operator L for f with weight ψ is deﬁned as X Lφ(z) = ψ(w)φ(w), φ ∈ A(V ). w∈f −1(z) Complex Dynamics and Related Topics 121

−1 Let gk = (f|Uk) and zk be the unique ﬁxed point of gk in Uk, 1 ≤ k ≤ d. Deﬁne Nk : A(V ) → A(V ) by

Nkφ = ψ ◦ gk · φ ◦ gk.

Then we have Xd L = Nk. k=1

Since each Nk is nuclear of order 0, so is L and

Xd Xd ψ(zk) tr(L) = tr(Nk) = 0 . 1 − g (zk) k=1 k=1 k

Let wn = i0i1 ··· i0 be a sequence of 1’s, ···, d’s of length n. Let

Nwn = Nin−1 ◦ · · · ◦ Ni0 .

Deﬁne

ψwn = ψ ◦ gi0 · ψ ◦ gi0 ◦ gi1 ··· ψ ◦ gi0 ◦ gi1 ◦ · · · ◦ gin−1 and

gwn = gi0 ◦ g1 ◦ · · · ◦ gin−1 . We have

Nwn φ = ψwn · φ ◦ fwn . Furthermore, X n L = Nwn wn and X n tr(L ) = tr(Nwn ), wn where the summations are over all sequence of 1’s, ···, d’s of length n.

Let zwn be the unique ﬁxed point of gwn in Uwn = gwn (V ). Then we have

ψwn (zwn ) tr(Nwn ) = 0 . 1 − gwn (zwn ) Complex Dynamics and Related Topics 122

Theorem 12.2 (Ruelle). The operator L is nuclear of order 0 and the determinant

³ X∞ n X ´ Y z ψwn (zwn ) det(I − zL) = exp − 0 = (1 − zτ) n 1 − g (zwn ) n=1 wn τ∈sp(L) is an entire function and the roots of det(I − λL) are exactly one over the eigenvalues of L (with the same multiply), where sp(L) means the spectrum of L.

Proof. Because

X∞ zn det(I − zL) = exp(− tr(Ln)) n n=1 and X n ψwn (zwn ) tr(L ) = 0 . 1 − g (zwn ) wn Now from Theorem 11.2, we have that

³ X∞ n X ´ Y z ψwn (zwn ) det(I − zL) = exp − 0 = (1 − τz) n 1 − g (zwn ) n=1 wn τ∈sp(L) is an entire function. It is clear from the last formula that the roots of det(I − zL) are exactly one over the eigenvalues of L (with the same multiply).

13 Dynamical Zeta functions

We consider the same dynamics in the last section. Recall that V , U1, ···, Ud are Jordan domains such that

d ∪k=1U k ⊂ V and Ui ∩ Uj = ∅, 1 ≤ i 6= j ≤ d.

d The dynamics f : ∪k=1Uk → V is deﬁned so that and each f|Uk : Uk → V is conformal. Let \∞ Λ = f −n(V ). n=0 Complex Dynamics and Related Topics 123

Let C0(Λ) be the space of all continuous functions φ :Λ → C. Suppose ψ is in C0(Λ). For any n > 0, let

nY−1 k Gn(x) = ψ(f (x)), x ∈ Λ. k=0

−1 Remember that gk = (f|Uk) and that for wn = i0i1 ··· in−1, a sequence of 1’s, 2’s, ···, d’s of length n,

gwn = gi0 gi1 ··· gin−1 : V → Uwn = gwn (V ).

The map gwn has a unique ﬁxed point zwn in Λwn = Λ ∩ Uwn because it is a contracting map. For each n > 0, deﬁne X

Zn(ψ) = Gn(zwn ) wn where wn runs over all sequences of 1’s, 2’s, ···, d’s of length n. The 0 functional Zn(ψ): C (Λ) → C is called a partition function. The zeta function of (f, ψ) is deﬁned as

³ X∞ zn ´ ξ(z, ψ) = exp Z (ψ) . n n n=1 The number 1 P (ψ) = lim sup log |Zn(ψ)| n→∞ n is called the topological pressure of the observable log ψ. The convergence radius of ξ(z, ψ) is eP .

Remark 13.1. If ψ > 0, we can deﬁne a measure µn on Λ as

Gn(zwn ) dµn = on Λwn . Zn(ψ)

All weak limits of {µn} will be f-invariant probability measures on Λ and are called Gibbs measures for (f, ψ) (refer to §6).

Now let us see some examples to calculate zeta functions. Complex Dynamics and Related Topics 124

n n Example 13.1. Suppose ψ = 1. Then Zn(1) = #(F ix(f )) = d . Let A be the d × d matrix with all entries 1. Then it is easy to see n Zn(1) = tr(A ). So

³ X∞ zn ´ 1 ξ(z, ψ) = exp tr(An) = n det(I − zA) n=1 where I means the d × d identity matrix. The pressure is 1 P = lim log Zn(1) = log d. n→∞ n

Example 13.2. Suppose ψ is a function only depending on i0i1, that is,

ψ(x) = ai0i1 , x ∈ Λi0i1

Let A = (ai0i1 ) be the d × d-matrix whose i0i1-entry is ai0i1 . One can see Pd Pd Pd easily that Z1(ψ) = i=1 aii = tr(L) and Z2(ψ) = i=1 j=1 aijaji = 2 n tr(A ). In general, Zn = tr(A ). Therefore,

³ X∞ zn ´ 1 ξ(z, ψ) = exp tr(An) = . n det(I − zA) n=1 Now let us calculate the zeta function for a function ψ ∈ A(V ). Theorem 13.1 (Ruelle). Suppose ψ ∈ A(V ). Then the zeta function ξ(z, ψ) can be extended to a meromorphic function of z and its pole are exactly the inverses of all eigenvalues of the transfer operator Lψ (counted by multiplicity).

Proof. The transfer operator L = Lψ can be written as

Xd Lφ(z) = ψ(gk(z))φ(gk(z)). i=1 Furthermore, for any n ≥ 1, we have X n L φ(z) = ψwn (z)φ(gwn (z)), wn where wni0i1 ··· in−1 runs over all strings of 1’s, ···, d’s of length n and

gwn = gi0 ◦ gi1 ◦ · · · ◦ gin−1 Complex Dynamics and Related Topics 125 and

ψwn = ψ ◦ gi0 · ψ ◦ gi0 ◦ gi1 ····· ψ ◦ gi0 ◦ gi1 ◦ · · · ◦ gin−1 . As we already know, Ln is nuclear of order 0 and its trace is X n ψwn (zwn ) tr(L ) = 0 . 1 − gw (zwn ) wn n

n 0 The diﬀerence between tr(L ) and Zn(ψ) is the factor 1 − gwn (zwn ). However, it is quite easy to remove the factor as follows. Deﬁne L(0) = L and Xd 0 L(1)φ(z) = ψ(gk(z))gk(z)φ(gk(z)). i=1 Then X n 0 L(1)φ(z) = ψwn (z)gwn (z)φ(gwn (z)) wn and ³ ´ X 0 n ψwn (zwn )gwn (zwn ) tr L(1) = 0 . 1 − gw (zwn ) wn n So ³ ´ ³ ´ n n Zn(ψ) = tr L(0) − tr L(1) . This gives us that

³ X∞ zn ³ ³ ´ ³ ´´´ ξ(z, ψ) = exp tr Ln − tr Ln n (0) (1) ³ n=1 ³ ´´ P∞ zn n exp − n=1 n tr L(1) = ³ P ³ ´´ ∞ zn n exp − n=1 n tr L(0) det(I − zL ) = (1) . det(I − zL(0))

Since det(I −zL(1)) and Det(I − zL(0)) are both holomorphic functions of z, ξ(z, ψ) is a meromorphic function of z. It is clearly the poles of ξ(z, ψ) are exactly one over the eigenvalues of L (counted by multiplicity). Complex Dynamics and Related Topics 126

References

[Ba1] V. Baladi, Positive Transfer Operators and Decay of Correlations, Vol. 16, World Scientiﬁc, Singapore (2000).

[Ba2] V. Baladi, Periodic orbits and dynamical spectra, Ergod. Th. & Dynam. Sys., 18(1998), 255-292.

[Bi] G. Birkhoﬀ, Extensions of Jentzch’s theorem, Trans. Amer.Math.Soc., 85 (1957), 219-227.

[Bo] R. Bowen, Equilibrium states and the ergodic theory of Anosov diﬀeomorphisms, LNM 470, Springer, Berlin, 1975.

[BL] R. Bowen and O. E. Lanford III, Zeta functions of restrictions of the shift transformation, Proc. Symp. Pure Math., 14 (1970), 43-50.

[DF] A.H. Dooley and A. H. Fan, Chains of markovian projections and (G, Γ)-measures, Trends in Probability and Related Analysis, Eds. N. Kono and N.R. Shieh, World scientiﬁc, 1997, 101-116.

[DS] M. Dunford and J.T. Schwartz, Linear operators, Interscience, 1957.

[Fa] A.H. Fan, A proof of the Ruelle theorem, Reviews Math. Phys., Vol. 7, 8 (1995), 1241-1247.

[FJ1] A. H. Fan and Y. P. Jiang, On Ruelle-Perron-Frobenius operators- I. Ruelle’s theorem, Commun. Math. Phys. , Vol. 223 (2001), Issue 1, 125-141.

[FJ2] A. H. Fan and Y. P. Jiang, On Ruelle-Perron-Frobenius operators- II. Convergence speed, Commun. Math. Phys. , Vol. 223 (2001), Issue 1, 143-159.

[FSc] A. H. Fan and J. Schmeling, On fast Birkhoﬀ averaging, Math. Proc. Camb. Phil. Soc., vol. 134, part. 3 (2003).

[FS] P. Ferrero and B. Schmitt, Ruelle’s Perron-Frobenius theorem and projective metrics, Coll. Math. Soc., J´aneBolyai, vol 27 (1979), 333-336. Complex Dynamics and Related Topics 127

[Fr] D. Fried, Meromorphic zeta functions for analytic ﬂows, Comm. Math. Phys., 174 (1995), 161-190.

[Gr] A. Grothendieck, Produits tensoriels topologiques et espaces nucl´eaires, Mem. Am. Math. Soc., 16 (1955).

[Ji] Y. P. Jiang, A Proof of existence and simplicity of a maximal eigenvalue for Ruelle-Perron-Frobenius operators, Letters in Math- ematical Physics, 48 (1999), 211-219.

[KS] K. Krzyzewski and W. Szlenk, On invariant measures for expanding diﬀerentiable mappings, Studia Math., T. XXXIII (1969), 83- 92.

[Pa] W. Parry, Topics in Ergodic Theory, Cambridge University Press, Cambridge-London-New York, 1981.

[PP] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Ast´erisque, 187-188, 1990.

[OR] E. Odell and H. Rosenthal, Functional Analysis, Lecture Notes in Mathematics, 1332, Springer-Verlag, 1988.

[Rg] H. H. Rugh, Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems, Ergod. Th. & Dynam. Sys. 16 (1996), 805-819.

[Ru1] D. Ruelle, Statistical mechanics of a one-dimensional lattice gas, Commun. in Math. Phys., Vol 9 (1968), 267-278.

[Ru2] D. Ruelle, A measure associated with Axiom A attractors, Amer. J. Math., Vol 98 (1976), 619-654.

[Ru3] D. Ruelle, Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval, CRM Monograph Series, Vol 4 (1994), Amer. Math. Soc. Providence, NJ.

[Vi] M. Viana, Stochastic dynamics of deterministic systems, preprint, IMPA, Brazil, 1997.

[Wa] P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc., 236 (1978), 121-153. Complex Dynamics and Related Topics 128

[You] L.S. Young, Developments in chaotic dynamics, Notices of Amer. Math. Soc., Vol. 45, no. 10 (1998), 1318-1332.

Aihua FAN: LAMFA CNRS UMR 6140, Facultéde Mathématiques et Informatique, Universitéde Picardie, 39 Rue Saint Leu, 80039 Amiens, France & Department of Mathematics, Wuhan University, 420073 Wuhan, China Email: [email protected] Yunping JIANG: Department of Mathematics, Queens College of CUNY, Flushing, NY 11367 and Department of Mathematics, CUNY Graduate School, 365 Fifth Avenue, New York, NY 10016 and the 100 Man Program in the Academy of Mathematics and System Sciences, Chi- nese Academy of Sciences, Beijing 100080, P. R. China Email: [email protected]