A one-parameter family of analytic Markov maps with an intermittency transition

In this paper we introduce and study a one-parameter family of piecewise analytic interval maps having the tent map and the Farey map as extrema. Among other things, we construct a Hilbert space of analytic functions left invariant by the Perron-Frobenius operator of all these maps and study the transition between discrete and continuous spectrum when approaching the intermittent situation.


Introduction
Expanding maps of the unit interval have been widely studied in the last decades in that several problems concerning their statistical behaviour can be treated by the powerful technique of transfer operators and thermodynamic formalism [C], [Ba1], [Ma2]. On the other hand, in recent years an increasing interest has been carried on maps which are expanding everywhere but on a marginally unstable fixed point in a neighbourhood of which trajectories are considerably slowed down leading to an interplay of chaotic and regular dynamics characteristic of intermittent systems [PM], [Sch]. Several approaches have been proposed to extend the above mentioned techniques to this situation, in particular to characterize the nature possible phase transitions [P1], [PS] and that of the spectrum of the transfer operator [Rug], [Is]. In this paper we introduce a one-parameter family of piecewise analytic maps smoothly interpolating between the tent map and the Farey map and use it to investigate the passage between the uniformly expanding situation and the intermittent one in both perspectives: that of thermodynamics and that of spectral theory.
The paper is organized as follows: Section 2 is devoted to introduce the model and derive some of its properties along with those of an induced version of it. In Section 3 we discuss large deviation properties and show in particular how the free energy gets non-analytic in the intermittent limit. Section 4 deals with the spectral analysis of the transfer, or Perron-Frobenius, operator. We construct a Hilbert space of analytic functions where this operator gets a particularly expressive integral representation (which becomes symmetric in the intermittent limit) and study the mechanism with which a continuous component in the spectrum (intermittent situation) comes out of a purely discrete spectrum (expanding situation). Finally we extend the above construction to study a family of operator-valued power series by which we obtain a characterization of the analytic properties of the dynamical zeta functions [Ba2] for both the original map and its induced version discussed in Section 2.

Preliminaires
Let r ∈ [0, 1] be a real parameter and consider the family of piecewise real-analytic maps F r of the interval [0, 1] defined as where F r,0 (x) = (2 − r)x 1 − rx and F r,1 (x) = F r,0 (1 − x) = (2 − r)(1 − x) 1 − r + rx · (2.2) Some properties of this family are listed below: 1. For r = 0 we find the tent map whereas for r = 1 one has the Farey map (2.4) The latter provides an example of an intermittent map in that the fixed point at the origin is neutral (see below).
2. The left branch F r,0 satisfies F r,0 (0) = 0, i.e. the origin is always a fixed point, and is conjugated to the map T r defined as More precisely, we have, for all n ≥ 1, Notice that for r = 1 the map T r becomes the left translation x → x − 1. Moreover, for each r ∈ [0, 1] there is a unique point x 1 left fixed by F r,1 , i.e. F r,1 (x 1 ) = x 1 with 3. The derivative is given by This means that for r < 1 the map F r is uniformly expanding, i.e. |F ′ r | ≥ ρ > 1, thus providing an example of analytic Markov map [Ma2]. In particular we have On the contrary, for r = 1 one has |F ′ (x)| > 1 for x > 0 but F ′ (0) = 1.
Proof. Reasoning inductively in n one obtains and the claimed result follows upon applying (2.14). ♦ Note that (2.14) can be rewritten as which can be viewed as a generalized Abel equation (see [deB]), to which it actually reduces when r = 1. This suggests that Φ r,0 satisfies some non-linear fixed point equation. Indeed, let R r be the renormalisation operator acting as Proof. We have Remark 1 For r = 1 the scaling factors α and β in (2.20) are both equal to 2 and the function Φ 0,1 is but the fixed point of the Feigenbaum renormalisation equation with intermittency boundary conditions obtained in [HR].

Invariant measure, induced map, characteristic exponent
We let P denote the Perron-Frobenius, or transfer operator associated to the map F r (see [Ba1]). It acts on a function f : [0, 1] → I C as 21) or, more explicitly, A fundamental property of this operator is that if there is a measurable function f which satisfies the fixed point equation then f is the density of an absolutely continuous measure Theorem 2.1 The function e r (x) = K r /(1 − r + rx) where K r is a given positive constant, is a solution of equation (2.23) and thus represents the density of an absolutely continuous F r -invariant measure dν r (x) = e r (x)dx.
Proof. By virtue of (2.22) we have and therefore ν r ([0, 1]) → ∞ when r ր 1. In order to compare the F r -invariant measure ν r with a probability measure µ r invariant w.r.t. to an induced map G r to be defined below, we shall choose the value of K r so that ν r ([1/2, 1]) = 1. This renders In particular we have lim rց0 K r = 2 and lim rր1 K r = 1 log 2 · (2.26) Let A r = {A n } n∈IN be the countable partition of [0, 1] whose elements are the intervals A n = [c n , c n−1 ] with c 0 = 1 and c n = Φ n r,0 (1), n ≥ 1.
so that A n is the closure of the set {x ∈ X : τ (x) = n}. On the other hand, the return time function ℓ : X → IN in the interval A 1 is given by We now prove the following version of Kac's formula: the ν r -measure of the whole interval [0, 1], that is (2.24), equals the conditional expectation of the function ℓ on the interval A 1 (recall that we have set ν r (A 1 ) = 1).
) and the claim follows by induction. Therefore, and the last identity in (2.32) follows from (2.24). ♦ Remark 2 As already remarked, the one-parameter family F r is well suited to study the transition from a strongly chaotic behaviour, corresponding to the uniformly expanding situation with r < 1, to an intermittent behaviour, corresponding to the the tangent bifurcation point at r = 1. One interesting item in this study is the divergence type of the average duration < ℓ > of the laminar regime as r ր 1 (see, e.g., [Sch]). In our situation this is nothing but the expectation of the return time function ℓ, and by formula (2.32) it diverges logarithmically: Now, using (2.31) we can express the expected return time < ℓ > as the expected first passage time w.r.t. an absolutely continuous probability measure µ r obtained by pushing forward ν r with F r,1 , i.e.
Reasoning as in the proof of Lemma 2.2 one readily verifies that the converse relation is (2.36) In particular we have ν r (A n ) = l≥n µ r (A l ) and µ r (A n ) = µ r (F r,1 (B n )) = ν r (B n ), where B n is as in the proof of Lemma 2.2. The measure µ r will play the role of reference probability measure in our construction. If we set h r (x) = µ r (dx)/dx then by the foregoing we have (2.37) Using the explicit expressions for e r and Φ r,1 we get The induced map. The measure µ r is left invariant by a map G r obtained from F r by inducing w.r.t. the first passage time τ . Indeed, by the above we can write where G r : X → X denotes the map: The explicit expression for G r,n can be easily obtained from that of F r and reads with c n as in (2.28) and r ∈ [0, 1].
We now list some properties of G r which are relevant for our discussion.
Proposition 2.2 1. smoothness property: G r,n is a real analytic diffeomorphism of A n onto [0, 1]; Proof. Statement 1) is an immediate consequence of the definition. As for 2) notice that On the other hand, one can easily find a positive constant C 1 such that Moreover by an easy estimate using (2.44) one can find a constant C 2 ≥ 1 so that for any choice of x, y ∈ A n and any n ≥ 1. Hence, by the mean value theorem n−1 The assertion now follows putting together the above inequalities. ♦ These properties yield a uniform bound for the buildup of non-linearity in the induction process.
Corollary 2.1 Let x, y ∈ [0, 1] be such that G j r (x) and G j r (y) belong to the same atom A k j , for 0 ≤ j ≤ n and some n ≥ 1. There is a constant C > 0, independent of r, such that Then using the distortion property listed above we have Now, by the expanding property we can find a constant C > 0 such that, under the above hypotheses, |x − y| ≤ C L −1 |A k 0 | β n with β = 4 − 1 2 = 1/2. ♦ Putting together the above and ( [Wal], Theorem 22 (3)) we have the following Proposition 2.3 The probability measure µ r is the unique absolutely continuous invariant measure for the dynamical system ([0, 1], G r ). Moreover (G r , µ r ) is an exact endomorphism.
Characteristic exponents. Finally we show that the measures ν r and µ r have the same characteristic exponent. Set Then we have Proposition 2.4 For all r ∈ (0, 1) we have To prove the first identity we write, using (2.37), For the second identity we have, using (2.9): By (2.32) the first term in the r.h.s equals the first term in the r.h.s. of (2.48). Notice that this term has limits 2 log 2 and 0 when r ց 0 and r ր 1, respectively. Furthermore, we have The claimed formula now follows by putting together the above expressions and noting that

Free energy and large deviations
For r ∈ [0, 1) we shall consider the F r -invariant probability measure p r as well as its characteristic (or Lyapunov) exponent λ r given by and For β ∈ IR and n ≥ 1 we may then define the partition function Z n (β) as and consider the sequence of functions (3.52) The limit function is called free energy function of the characteristic exponent (see [BR], [C], [D]). Notice that f (0) = 0. If we set so that in particular then we get Now, for each r < 1 the transformation F r is a uniformly expanding Markov map and using standard arguments one sees that the sequence {f ′′ n (β) : n ≥ 1} is uniformly bounded on compact sets. This entails that the free energy function f (β) is convex and C 1 with f ′ (β) strictly increasing and given by (3.57) In particular we have f ′ (β)| β=0 = λ r · (3.58) One finds moreover that the function β → β f ′ (β) − f (β) is decreasing for β < 0 and increasing for β > 0. This relates f (β) to large deviation properties of the sequence of random variables S n (x). To see this, notice that 1 0 S n (x) p r (dx) = 0 for all n ≥ 1. The ergodic theorem then yields For each r < 1 the dynamical system ([0, 1], F r , p r ) satisfies the assumptions of the large deviation theorem which says that (see [C], [D]) In other words, the probability of finding a deviation of S n /n from its average value 0 decays exponentially with n. Notice that if we set α = f ′ (β) − λ r then the r.h.s. in (3.61) and (3.62) can be viewed as a Legendre transform We now derive upper and lower bounds for the free energy function. First, from |F ′ r | ≥ ρ > 1 it follows immediately that for β ≤ 0 we have f (β) ≤ β log ρ, the opposite inequality being valid for positive values of β. In addition, using either the convexity of f (β) or directly (3.52) we get the inequality f (β) ≥ λ r β, which is valid for all β ∈ IR. Whence, expressing λ r by means of Lemma 2.2 and Proposition 2.4 we get Lemma 3.3 For all β ≤ 0 and r ∈ [0, 1) we have Notice that lim rց0 log ρ = log 2 whereas lim rր1 log ρ = 0, moreover lim rց0 γ r = 0 and lim rր1 γ r = 0. Therefore from Lemma 3.3 we obtain the Remark 3 This result shows that for r = 1 the free energy has a discontinuity in its first derivative at β = 0. This can be interpreted in thermodynamic language as a second order phase-transition [P1], [FKO].

Transfer operators
The transfer operator P associated to the map F r has already been introduced in (2.22) and we write (4.65) Several interesting properties of the dynamics generated by F r are intimately related to sp (P), the spectrum of P (see [Ba1]). However, the latter depends crucially on the function space P is acting on, which is in general a Banach space. For smooth uniformly expanding maps and Banach spaces of sufficiently regular functions, e.g. the space C k of k-times differentiable functions on [0, 1] with k ≥ 0, the transfer operator is quasi-compact. This means that sp (P) is made out of a finite or at most countable set of isolated eigenvalues with finite multiplicity (the discrete spectrum) and its complementary, the essential spectrum. It has been proved in [CI] that for piecewise C ∞ expanding Markov maps of the unit interval the essential spectrum of P when acting on C k is a disk of radius r ess (P) = exp f (−k) , (4.66) where f (β) is the free energy function discussed in the previous Section. Putting together this result and Lemma 3.3 we obtain the following Theorem 4.2 For r ∈ [0, 1) the essential spectrum of P : C k → C k with k ≥ 0 is a disk of radius e −k (log ρ+γr) ≤ r ess (P) ≤ e −k log ρ .
The above bounds along with standard semicontinuity arguments [K] yield the following Corollary 4.3 For r = 1 and for each fixed k ≥ 0 the essential spectrum of P : C k → C k is the unit disk.

An invariant Hilbert space
From the above discussion it follows that if we want to understand the nature of the spectrum lying under the 'essential spectrum rug' we have to let P acting on increasingly smooth test functions as r approaches 1. In particular, Corollary 4.3 suggests that for r = 1 one should consider suitable spaces of analytic functions. In the following definition we shall introduce a Hilbert space of analytic functions which will be shown to be invariant under P for each r ∈ [0, 1].
with inner product and measure dm(t) = t e −t dt. (4.69) Remark 4 An alternative representation can be obtained by a simple change of variable when x is real and positive: ds e −s ψ(sx) with ψ(t) = t ϕ(t) . (4.70) Note that a function f ∈ H is analytic in the disk (here z = x + iy) A Hilbert space identical to H apart from a slightly different choice of the measure m was introduced in [Is] to study the operator P for r = 1 (the Farey map), whereas a generalized version of H has been used by Prellberg in [P2] to study the spectrum of the (thus again for the case r = 1).
Remark 5 The invariant densities e r (x) = K r /(δ + rx) and h r (x) = K r /(ρ + rx) can be represented as (4.73) respectively. For the limiting values r = 1 and r = 0 we get We point out that φ 1 is not in L 2 (m).

Lemma 4.4
The space H is invariant for P 0 and we have (4.77) Proof.
The following lemma will instead specify the action of P 1 on H. where N r : L 2 (m) → L 2 (m) is the operator acting as where J p denotes the Bessel function of order p. Proof.
where we have used the identity [GR] 1 u p+1 with p = 1, u = ρ x − δ and ψ(s) = s e −s ϕ(s). Finally, by the estimates (see [Er]) J 1 (x) ∼ x/2 as x ց 0 and J 1 (x) = O(x − 1 2 ) as x → ∞, along with the inequality 2δ/ρ < 1, which holds for all r ∈ [0, 1], one easily checks that N r : L 2 (m) → L 2 (m). ♦ We can summarize the above in the following Theorem 4.3 The space H is invariant for P and we have (4.81)

The spectrum of M r and P 0
We are now going to study the operator M r . First, note that it reduces to M 0 ϕ(t) = (1/2)ϕ(t/2) when r = 0, whereas for r = 1 yields the multiplication operator M 1 ϕ(t) = e −t ϕ(t). Moreover, for r ∈ [0, 1) its iterates are given by Assume that ϕ is analytic in some open neighbourhood of t = 0 and satisfies the eigenvalue equation (4.83) The above equation at t = 0 writes ϕ(0) = λ ρ ϕ(0), so that if ϕ(0) = 0 then λ = 1/ρ. In this case, by (4.82) we get (4.84) and therefore ϕ(t) = e − r δ t ϕ(0). If instead ϕ(0) = 0 we differentiate (4.83) to get which at t = 0 writes ϕ ′ (0)/ρ 2 = λϕ ′ (0). Therefore if ϕ ′ (0) = 0 then λ = 1/ρ 2 . In this case, differentiating (4.82)-(4.83) we get 1 ρ 2n ϕ ′ (t) = r δ ρ n 1 ρ n − 1 e − r δ t e r δ t ρ n ϕ t ρ n + e − r δ t e r δ t ρ n ρ 2n ϕ ′ t ρ n . (4.86) Taking the limit n → ∞ and noting that lim n→∞ ρ n ϕ(t/ρ n ) = t ϕ ′ (0) we obtain (4.87) which upon integration renders Iterating this argument we have that if ϕ satisfies (4.83) with ϕ (l) (0) = 0 for 0 ≤ l < k − 1 but ϕ (k−1) (0) = 0 for some k ≥ 1 then λ = ρ −k and ϕ(t) = t k−1 e − r δ t (up to a non-zero but otherwise arbitrary constant multiplicative factor). Denoting by 2 the norm in L 2 (m) we also have It is not hard to see that for all r ∈ [0, 1) the sequence {t k−1 e − r δ t } ∞ k=1 is a linearly independent family in L 2 (m), and by adapting ( [He], p.62, Thm.8) we have that the linear span of this family is dense in L 2 (m). Putting together the above along with standard arguments we have proved the following Proposition 4.5 For all r ∈ [0, 1) the operator M r : L 2 (m) → L 2 (m) is compact and its spectrum is given by sp (M r Each eigenvalue µ k is simple and the corresponding (normalized) eigenfunction ϕ k is given by Remark 6 For each fixed r ∈ [0, 1) the operator M r : L 2 (m) → L 2 (m) is actually trace-class. Its trace is easily computed: and satisfies lim We recall that for r = 1 we get the multiplication operator M 1 which is self-adjoint in L 2 (m), its spectrum is continuous and given by the closure of the range of the multiplying function, that is the interval [0, 1] (see, e.g., [DeV]). By the above Proposition we see how the continuous spectrum is approached as r ր 1: having fixed an interval [a, b] ⊆ (0, 1] an easy computation yields Moreover a simple calculation gives (4.96) Therefore by the above and Lemma 4.4 we have the following Corollary 4.4 The spectrum of P 0 when acting upon H is given by sp (P 0 ) = sp (M r ).

The spectrum of N r and P 1
We now turn to study the action of the operator N r on L 2 (m). We first note that N r can be viewed as the composition of the multiplication operator (which reduces to the identity for r = 1) and the symmetric integral operator Observing that the (associated) Laguerre polynomials L 1 k given by [GR] L 1 form a complete orthogonal basis in L 2 (m) and expanding the kernel of the integral operator defined above on this basis we get from which we see that N r has the representation (we keep using the symbol (ϕ 1 , ϕ 2 ) to denote the inner product in L 2 (m) as well) where e k , f k ∈ L 2 (m) are given by We find and therefore, for all r ∈ (0, 1], showing that the operator N r is nuclear in L 2 (m). Its trace can be computed as (4.108) Note that tr N 2 r ≤ tr N r , with strict inequality unless r = 0 where tr N 0 = tr N 2 0 = 1/3. This suggests that the eigenvalues of N r are alternately positive and negative. Indeed we have the following Proposition 4.6 For all r ∈ [0, 1] the spectrum of the operator N r : Each eigenvalue is simple and the corresponding (normalized) eigenfunction ψ k is given by Remark 7 Note that the eigenvalues of N r can be written in terms of the function β r as (4.111) In particular, lim rց0 β r = 0 so that for r = 0 we have ν k = (−1) k−1 (2) −k , moreover and thus for r = 1 we see that ν k = (−1) k−1 Proof of Proposition 4.6. The proof is based on the following Hankel transform (see [Er], vol 2) , (4.112) which for p = 1 can be rewritten in terms of the operator N r as . (4.113) To make the above identity an eigenvalue equation the following relations have to be satisfied 2b . (4.114) This renders (4.115) It is now easy to check that the only solution giving a function in L 2 (m) is that with b = (1 + √ 1 + 4ρ)/4 √ ρ and this choice yields the eigenvalues and the eigenfunctions given in the Proposition. Moreover, a standard evaluation of yields the claimed expression for the normalizing factor B k (see [Er], vol 1). ♦

Direct application of the Lemma 4.5 now yields
Corollary 4.5 The spectrum of P 1 when acting upon H is given for each r ∈ [0, 1] by sp (P 1 ) = sp (N r ). Each eigenvalue ν k , with k ≥ 1, is simple and the corresponding (normalized) eigenfunction ξ k is given by (4.117) 4.4 The spectrum of P First, putting together (4.93), (4.106) and Theorem 4.3 we deduce the following result, Theorem 4.4 For all r ∈ [0, 1) the transfer operator P when acting upon H is of the trace-class, with 4ρ (4.118) with δ = 1 − r and ρ = 2 − r.
Remark 8 Using (2.10) and setting set x 0 ≡ 0 one easily verifies that Moreover, a straightforward computation based on the integral and using (2.10) shows that where x 1 , defined in (2.8), is the unique fixed point of F r besides x 0 (obviously one would arrive at the same conclusion by directly summing the geometric series k≥1 µ k + k≥1 ν k with µ k and ν k given by (4.90) and (4.111), respectively). As a result we can rewrite (4.118) as . (4.122) This expression does'nt come unexpectedly: it is an instance of a trace formula valid for more general analytic Markov maps (see [Ma2], Sec. 7.3.1).
We are now ready to investigate the spectrum sp (P) on the space H. We first notice that from (4.113) with a = √ ρ and b = √ ρ/2 it follows that Therefore for all odd k the function L 1 k (2t) lies in the kernel of M r + N r . This is in agreement with (2.14) and (2.15) since we have which for k odd is an odd function w.r.t. x = 1/2. This implies that 0 is an eigenvalue of infinite multiplicity for P for all r ∈ [0, 1].
Let us first consider the two extremal cases r = 0 and r = 1. For r = 0 we the above theorem gives tr P = 4/3, suggesting the eigenvalues 2 −2n , n ≥ 0. To check, we first note that Therefore the set of polynomials with even degree is a subset of L 2 (m) which is invariant for M 0 + N 0 , and any polynomial with odd degree is mapped into this subset. Given k = 2n it is now a simple task to construct a polynomial eigenfunction φ 2n (of degree 2n) to the eigenvalue 2 −2n . The first three are Note moreover that for φ k (t) = k j=0 a j t j we have (B [φ k ])(x) = k j=0 a j (j + 1)! x j . These observations along with standard arguments yield the following Proposition 4.7 For r = 0 the spectrum of P when acting upon H is the set sp (P) = {2 −2n } n≥0 ∪ {0}. Each eigenvalue 2 −2n is simple and the corresponding eigenfunction is a polynomial of degree 2n.
At the opposite extremum we have the following result.
Proposition 4.8 For r = 1 the spectrum of P when acting upon H is the union of [0, 1] and a (possibily empty) countable set of real eigenvalues of finite multiplicity.
Proof. The assertion follows by noting that for r = 1 the operator P when acting on H is isomorphic to M 1 + N 1 , which is a self-adjoint compact perturbation of the (selfadjoint) multiplication operator M 1 . The assertion is now a consequence of Theorem 5.2 in [GK]. Note that although the function e 1 (x) = (log 2) −1 /x satisfies Pe 1 = e 1 it does not belong to H and therefore 1 / ∈ sp (P : H → H). ♦ We finally state the following Conjecture 1 For all 0 ≤ r < 1 the spectrum of P when acting upon H is a countable subset of [0, 1] densely filling [0, 1] as r ր 1. When r = 1 the spectrum is purely continuous (i.e. there are no eigenvalues).

Operator-valued functions and zeta functions
We may consider the operator-valued function Q z defined as Its relevance is twofold: first, expanding formally in powers of z we get and using (2.41) we see that for z = 1 the operator Q ≡ Q 1 is the transfer operator associated to the induced map G r . Second, it is related to P by the identity Now, as a consequence of (4.125) and Lemmas 4.4 -4.5, we have the following expression for the operator-valued function Q z when acting on H: Putting together the above we obtain Theorem 4.5 For all r ∈ [0, 1] the operator-valued function Q z when acting on the Hilbert space H is analytic for z ∈ I C \ Λ r , where Λ r = {ρ k } ∞ k=1 for r < 1 and Λ 1 = (1, ∞). For each z ∈ I C \ Λ r it defines a trace-class operator.
Remark 9 Introducing the operator we can rewrite (4.81) and (4.128) with z = 1 as 130) and (recall that Q ≡ Q 1 ) respectively. We thus see that the functions φ r and ψ r defined in (4.73) satisfy (4.132) so that φ r = (1 − M r ) −1 ψ r and ψ r = N r φ r . (4.133) We now consider the dynamical zeta functions ζ Fr and ζ Gr associated to the maps F r and G r , respectively, and defined by the following formal series [Ba2]: where the 'partition functions' Z n (F r ) and Z n (G r ) are given by Moreover, let us define the 'grand partition function' A straightforward extension of ( [Is], Proposition 4.3) to the present situation yields the identities ζ 2 (1, z) = (1 − z) ζ Fr (z) and ζ 2 (s, 1) = ζ Gr (s) (4.138) which are valid for all r ∈ [0, 1] and wherever the series expansions converge absolutely. Therefore the analytic properties of the dynamical zeta functions ζ Fr and ζ Gr can be deduced from those of ζ 2 (s, z). In turn, the latter can be studied as follows. For q = 0, 1, . . . define P 0,q f (x) := ρ (ρ + rx) 2 1+q · f x ρ + rx P 1,q f (x) := ρ (ρ + rx) 2 1+q · f 1 − x ρ + rx so that P 0,0 ≡ P 0 and P 1,0 ≡ P 1 . These operators are supposed to act upon the Hilbert space H q ⊆ H such that a function f ∈ H q can be represented as x e t ϕ(t) dm q (t), ϕ ∈ L 2 (m q ), (4.139) with dm q (t) = t 2q+1 e −t dt (4.140) A straightforward computation extending to non zero q values those performed in the previous Section yields P 0,q B q [ϕ ] = B q [M r,q ϕ ] and P 1,q B q [ϕ ] = B q [N r,q ϕ ] (4.141) where M r,q : L 2 (m q ) → L 2 (m q ) is defined as (M r,q ϕ)(t) := 1 ρ 1+q e − r ρ t ϕ t ρ .
(4.142) and N r,q : L 2 (m q ) → L 2 (m q ) is given by (N r,q ϕ)(t) := 1 ρ 1+q e We can now generalize (4.126) defining Q z,q := ∞ n=1 z n P 1,q P n−1 0,q . (4.144) In particular Q z,0 ≡ Q z . Now set Λ r,q = {ρ k+q } ∞ k=1 and note that Λ r,q ⊆ Λ r for all q ≥ 0. Reasoning as above and using (4.141) we have that for any given q = 0, 1 . . . the operator valued function z → Q z,q when acting on H q is analytic for z ∈ I C \ Λ r,q and for each z in this domain Q z,q defines a trace-class operator with (4.145) Furthermore, a straightforward adaptation of ( [Ma1], Corollaries 4 and 5) to our zdependent situation leads to the following expression for the grand partition function: Ξ n (z) = tr Q n z,0 − tr Q n z,1 . (4.146) This trace formula along with standard arguments (see [Ma1]) allow us to write the two-variables zeta function (4.137) as a ratio of Fredholm determinants, ζ 2 (s, z) = det (1 − s Q z,1 ) det (1 − s Q z,0 ) , is in the sense of Grothendieck [G]. Putting together the above we obtain the following result from which the analytic properties of the dynamical zeta functions associated to the maps F r and G r can be readily deduced via (4.138).
Theorem 4.6 1. for each s ∈ I C, the function ζ 2 (s, z), considered as a function of the variable z, extends to a meromorphic function in {z ∈ I C : z / ∈ Λ r }. Its poles are located among those z-values such that Q z : H → H has 1/s as an eigenvalue; 2. for each z ∈ I C \Λ r , the function ζ 2 (s, z), considered as a function of the variable s, extends to a meromorphic function in I C. Its poles are located among the inverses of the eigenvalues of Q z : H → H.
Remark 10 The first statement with s = 1 shows that for r = 1 the function ζ Fr (z) has a non-polar singularity at z = 1. This can be related to the non-analytic behaviour of the free energy at β = 0 discussed in Section 3.