RUELLE TRANSFER OPERATORS WITH TWO COMPLEX PARAMETERS AND APPLICATIONS

For a C Axiom A flow φt : M −→ M on a Riemannian manifold M and a basic set Λ for φt we consider the Ruelle transfer operator Lf−sτ+zg, where f and g are real-valued Hölder functions on Λ, τ is the roof function and s, z ∈ C are complex parameters. Under some assumptions about φt we establish estimates for the iterations of this Ruelle operator in the spirit of the estimates for operators with one complex parameter (see [4], [21], [22]). Two cases are covered: (i) for arbitrary Hölder f, g when | Im z| ≤ B| Im s| for some constants B > 0, 0 < μ < 1 (μ = 1 for Lipschitz f, g), (ii) for Lipschitz f, g when | Im s| ≤ B1| Im z| for some constant B > 0 . Applying these estimates, we obtain a non zero analytic extension of the zeta function ζ(s, z) for Pf − < Re(s) < Pf and |z| small enough with simple pole at s = s(z). Two other applications are considered as well: the first concerns the Hannay-Ozorio de Almeida sum formula, while the second deals with the asymptotic of the counting function πF (T ) for weighted primitive periods of the flow φt.


Introduction
Let M be a C 2 complete (not necessarily compact) Riemannian manifold, φ t : M −→ M (t ∈ R) a C 2 flow on M and let ϕ t : M −→ M be a C 2 weak mixing Axiom A flow ( [2], [11]). Let Λ be a basic set for φ t , i.e. Λ is a compact locally maximal invariant subset of M and φ t is hyperbolic and transitive on Λ.
Given a Hölder continuous function F : Λ −→ R and a primitive periodic orbit γ of φ t , denote by λ(γ) the least period of γ. The weighted period of γ is defined by λ F (γ) = λ(γ) 0 F (φ t (x γ ))dt, where x γ ∈ γ. The weighted version of the dynamical zeta function (see Sect. 9 in [11]) is given by For F = 0 we obtain the classical Ruelle dynamical zeta function.
It is well known (see for instance Chapter 6 in [11]) that the analysis of the dynamical zeta function can be reduced to that of a Dirichlet series by using a symbolic coding of Λ given by a fixed Markov family {R i } k i=1 . For our analysis it is convenient to consider a Markov family of pseudo-rectangles R i = [U i , S i ] = {[x, y] : x ∈ U i , y ∈ S i } (see section 2 for the notation and more details). Let P : R = ∪ k i=1 R i −→ R be the related Poincaré map, let τ (x) > 0 be the first return time function on R, and let σ : U = ∪ k i=1 U i −→ U be the shift map given by σ = π (U ) • P, where π (U ) : R −→ U is the projection along stable leaves. The flow φ t on Λ is naturally related to the suspension flow σ τ t on the suspension space R τ (see section 2 for details). There exists a natural semi-conjugacy projection π(x, t) : R τ −→ Λ which is one-to-one on a residual set (see [2]). Then following the results in [2], [3], a closed σ-orbit {x, σx, ..., σ n−1 x} is projected to a closed orbit γ in Λ with a least period λ(γ) = τ n (x) := τ (x) + τ (σ(x)) + ... + τ (σ n−1 (x)).
Passing to the symbolic model (see [2], Chapter 6 in [11]), the analysis of ζ ϕ (s, F ) is reduced to that of the Dirichlet series η(s) = F (π(x, t))dt : R −→ R. To deal with certain problems (see Chapter 9 in [11] and [17]) it is necessary to study a more general series η g (s) = The precise definition of the Ruelle operator acting on spaces of Hölder functions is given in section 4. Thus, the strategy for the proof of the analytic continuation of the dynamical zeta function comprises two majors steps: (I) Prove that suitable "contraction" estimates for the iterations of the Ruelle operator L n f −sτ +zg imply the convergence by packets of the Dirichlet series which yields an analytic continuation of the corresponding zeta function.
(II) Establish suitable "contraction" estimates for the iterations.
This strategy has been used for zeta functions depending on one complex parameter and related spectral estimates, called Dolgopyat estimates, have been proved in many cases ( [4], [20], [21], [22]) under some conditions on φ t . The most general case of such estimates known so far for Ruelle operators with one complex parameter is that described by the Standing Assumptions in section 4 below (see [21], [22]).
In this paper we study both problems (I) and (II) for zeta functions and Ruelle operators depending on two parameters s, z ∈ C. These problems are motivated by particular important applications in mind, however we believe they are also of an independent interest.
1.1. Results. Under some hypothesis on the flow φ t (see section 4 for our standing assumptions) we prove spectral estimates for the iterations of Ruelle operator L n f −sτ +zg with two complex parameters s, z ∈ C. These estimates are in the spirit of those obtained in [4], [20], [21], [22] for Ruelle operators with one complex parameter s ∈ C. It should be emphasized that the transition from one to two complex parameters is highly non-trivial, and so far there have been no results of this kind in the literature. In particular, in the treatment of this case completely new difficulties appear when | Im s| → ∞ and | Im z| → ∞.
In what follows, first in Theorem 5 we prove spectral estimates in the case of arbitrary Hölder continuous functions f, g when there exist constants B > 0 and 0 < µ < 1 such that | Im z| ≤ B| Im s| µ and | Im s| ≥ b 0 > 0. When f, g are Lipschitz one can take µ = 1. This covers completely the case when |z| is bounded and the estimates have the same form as those for operators with one complex parameter. Moreover, these estimates are sufficient for the applications in [11] and [18] when |z| runs in a small neighbourhood of 0 (see sections 7 and 8). Notice that in the special case of a geodesic flow on a surface with negative curvature in the proof of Lemma 3.5 in [18] it was mentioned that one can obtain a non-vanishing extension of ζ(s, z) for sufficiently small |z|. However no proof of this result was given, and indeed one needs some of the results in this paper to obtain this -in particular, the generalisation of Ruelle's lemma to the case of two complex parameters (see section 3) and the estimates of the corresponding Ruelle operator established in sections 5 and 6 below. In fact, in section 6 we deal with the more difficult situation when f, g are Lipschitz and there exists a constant B 1 > 0 such that | Im s| ≤ B 1 | Im z| (see Theorem 6).
To study the analytic continuation of ζ(s, z) for P f − η 0 < Re s < P f , we need a generalisation of Ruelle's lemma mentioned above which yields a link between the convergence by packets of a Dirichlet series like (1.3) below and log ζ(s, z) and the estimates of the iterations of the corresponding Ruelle operator. The reader may consult [24] for a precise result in this direction completing some points the previous works ( [19], [16], [9]), treating this question. For our needs in this paper we prove in section 3 an analogue of this lemma for Dirichlet series with two complex parameters following the approach in [24]. Combining Theorem 4 with the estimates in Theorem 5 (b), we obtain the following Theorem 1. Assume the standing assumptions in section 4 fulfilled for a basic set Λ. Then for any Hölder continuous functions F, G : Λ −→ R there exists η 0 > 0 such that the function ζ(s, z) admits a non vanishing analytic continuation for (s, z) ∈ {(s, z) ∈ C 2 : P f − η 0 ≤ Re s, s = s(z), |z| ≤ η 0 } with a simple pole at s(z). The pole s(z) is determined as the root of the equation P r(f −sτ +zg) = 0 with respect to s for |z| ≤ η 0 .
Applying the results of sections 5 and 6, we study also the analytic continuation of ζ(s, iw) for P f − η 0 < Re s and w ∈ R, |w| ≥ η 0 , in the case when F, G : Λ −→ R are Lipschitz functions (see Theorem 7). Here both complex parameters s, w may go to infinity, the analysis of this case is more complicated and we study the situation when z = iw. Our investigation was motivated by the necessity to have an analytic continuation of the zeta functions appearing in the arguments in [23], [7]. This analytic continuation combined with the arguments in [23] opens some perspectives for investigations on sharp large deviations for Anosov flows with exponentially shrinking intervals in the spirit of [12]. Some other applications are also possible, in particular we expect to obtain the result of Theorem 7 for arbitrary Hölder functions F, G : Λ −→ R, which for now is an open problem.
Our first application concerns the so called Hannay-Ozorio de Almeida sum formula (see [5], [10], [18]). Let φ t : M −→ M be the geodesic flow on the unit-tangent bundle over a compact negatively curved surface M . In [18] it was proved that there exists > 0 such that if (δ(T )) −1 = O(e T ), then for every Hölder continuous function G : where γ runs over the set of primitive periodic orbits of the flow in M , )|, while µ is the unique φ t -invariant probability measure which is absolutely continuous with respect to the volume measure on M . The measure µ is called SRB (Sinai-Ruelle-Bowen) measure (see [3]). Notice that in the above case the Anosov flow φ t is weak mixing and M is an attractor. Applying Theorem 1 and the arguments in [18], we prove the following Theorem 2. Let Λ be an attractor, that is there exists an open neighborhood V of Λ such that Λ = ∩ t≥0 φ t (V ). Assume the standing assumptions of section 4 fulfilled for the basic set Λ. Then there exists > 0 such that if (δ(T )) −1 = O(e T ), then for every Hölder function G : Λ −→ R the formula (1.2) holds with the SRB measure µ for φ t .
Our second application concerns the counting function where γ is a primitive period orbit for φ t : Λ −→ Λ, λ(γ) is the least period and λ F (γ) = λ(γ) 0 F (φ t (x γ ))dt, x γ ∈ γ. For F = 0 we obtain the counting function π 0 (T ) = #{γ : λ(γ) ≤ T }. These counting functions have been studied in many works (see [16] for references concerning π 0 (T ) and [11], [15] for the function π F (T )). The study of π F (T ) is based on the analytic continuation of the function which is just the function ζ(s, 0) defined above. We prove the following Theorem 3. Let Λ be a basic set and let F : Λ −→ R be a Hölder function. Assume the standing assumptions of section 4 fulfilled for Λ. Then there exists > 0 such that In the case when φ t : T 1 (M ) −→ T 1 (M ) is the geodesic flow on the unit tangent bundle T 1 (M ) of a compact C 2 manifold M with negative section curvatures which are 1 4 -pinching the above result has been established in [15]. It follows from [21] and [22] that the special case of a geodesic flow in [15] is covered by Theorem 3.
The proof of Theorem 5 for Hölder functions f and g ≡ 0 implies some new result even for the Ruelle operator with one complex parameter under the standing assumptions. For example, we have to study quite precisely the approximations of f by smooth functions and estimate the Lipschitz constants of the corresponding eigenfunctions related to maximal eigenvalues. This particular result is given in Lemma 4 and appears to be of an independent interest.
The results of our work for contact Anosov flows satisfying some pinching conditions, called in section 4 simplifying assumptions, have been announced in [13]. Here we treat a more general case and present detailed proofs of the results.
Then P r(0) = h, where h > 0 is the topological entropy of φ t and ζ φ (s) is absolutely convergent for Re s > h (see Chapter 6 in [11]).

Example 2. Consider the expansion function
Introduce the function λ u (γ) = λ E (γ) and we define f : R −→ R by Then we have −λ u (γ) = f n (x) , f is Hölder continuous and P r(f ) = 0 (see [3]). Consequently, the series is absolutely convergent for Re s > 0 and nowhere zero and analytic for Re s ≥ 0 except for a simple pole at Re s = 0 (see Theorem 9.2 in [11]). The roof functions τ (x) is constant on stable leaves of rectangles R i of the Markov family, so we can assume that τ (x) depends only on x ∈ U. By a standard argument (see [11]) we can replace f by a Hölder functionf (x) which depends only on x ∈ U so that f ∼f . Thus the series (1.3) can be written by functionsf , τ depending on only x ∈ U . We keep the notation f below assuming that f depends only on x ∈ U. The analysis of the analytic continuation of (1.3) is based on spectral estimates for the iterations of the Ruelle operator (see [4], [16], [21], [22], [24] for more details).
Example 3. Let f, τ be real-valued Hölder functions and let P f > 0 be the unique real number such that P r(f − P f τ ) = 0. Let g(x) = τ (x) 0 G(π(x, t))dt, where G : Λ −→ R is a Hölder function. Then if the suspended flow σ τ t is weak-mixing, the function (1.1) is nowhere zero analytic function for Re s > P f and z in a neighborhood of 0 (depending on s) with nowhere zero analytic extension to Re s = P f (s = P f ) for small |z|. This statement is just Theorem 6.4 in [11]. To examine the analytic continuation of ζ(s, z) for P f −η 0 ≤ Re s and small |z|, it is necessary to have some spectral estimates for the iterations of the Ruelle operator (1.4) The analytic continuation of ζ(s, z) for small |z| and that of η g (s) play a crucial role in the argument in [18] concerning the Hannay-Ozorio de Almeida sum formula for the geodesic flow on compact negatively curved surfaces. We deal with the same question for Axiom A flows on basic sets in section 8.
Example 4. In [7] for Anosov flows the authors examine the spectral properties of the Ruelle operator (1.4) with f = 0 and z = iw, w ∈ R and the analyticity of the corresponding L-function L(s, z). The properties of the Ruelle operator L n f −(P f +a+ib)τ +iw , w ∈ R, n ∈ N, are also rather important in the paper [23] dealing with the large deviations for Anosov flows. Here as above P f ∈ R is such that P r(f − P f τ ) = 0. However, it is important to note that in [7] and [23] the analysis of the Ruelle operators covers mainly the domain Re s ≥ P f and there are no results treating the spectral properties for P f − η 0 ≤ Re s < P f and z = iw, w ∈ R. To our best knowledge the analytic continuation of the function ζ(s, z) for these values of s and z has not been investigated in the literature so far which makes it quite difficult to obtain sharper results.

Preliminaries
As in section 1, let φ t : M −→ M be a C 2 Axiom A flow on a Riemannian manifold M , and let Λ be a basic set for φ t . The restriction of the flow on Λ is a hyperbolic flow [11]. For any x ∈ M let W s (x), W u (x) be the local stable and unstable manifolds through x, respectively (see [2], [6], [11]). When M is compact and M itself is a basic set, φ t is called an Anosov flow. It follows from the hyperbolicity of Λ that if 0 > 0 is sufficiently small, there exists 1 > 0 such that if x, y ∈ Λ and d(x, y) < 1 , then W s 0 (x) and φ [− 0 , 0 ] (W u 0 (y)) intersect at exactly one point [x, y] ∈ Λ (cf. [6]). That is, there exists a unique t ∈ [− 0 , 0 ] such that φ t ([x, y]) ∈ W u 0 (y). Setting ∆(x, y) = t, defines the so called temporal distance function.
We will use the set-up and some arguments from [21]. As in [21], fix a (pseudo-) Markov family The function τ is the so called first return time associated with R. Let σ : U −→ U be the shift map given by σ = π (U ) • P, where π (U ) : R −→ U is the projection along stable leaves. Let U be the set of those points x ∈ U such that P m (x) is not a boundary point of a rectangle for any integer m. In a similar way define R. Clearly in general τ is not continuous on U , however under the assumption that the holonomy maps are Lipschitz (see section 4) τ is essentially Lipschitz on U in the sense that there exists a constant L > 0 such that if x, y ∈ U i ∩ σ −1 (U j ) for some i, j, then |τ (x) − τ (y)| ≤ L d(x, y). The same applies to σ : U −→ U .
It is possible to construct a Markov family R so that A is irreducible and aperiodic (see [2]). Consider the suspension space where by ∼ we identify the points (x, τ (x)) and (σx, 0). The corresponding suspension flow is defined by σ τ t (x, s) = (x, s+t) on R τ taking into account the identification ∼ . For a Hölder continuous function f on R, the topological pressure Pr(f ) with respect to σ is defined as where M σ denotes the space of all σ-invariant Borel probability measures and h(σ, m) is the entropy of σ with respect to m. We say that f and g are cohomologous and we denote this by f ∼ g if there exists a continuous function w such that

Ruelle's lemma with two complex parameters
Let B( U ) be the space of bounded functions q : U −→ C with its standard norm q 0 = sup x∈ U |g(x)|. Given a function q ∈ B( U ), the Ruelle transfer operator L q : is Lipschitz on U with respect to the Riemann metric, then L q preserves the space C Lip ( U ) of Lipschitz functions q : U −→ C. Similarly, if q is ν-Hölder for some ν > 0, the operator L q preserves the space C ν ( U ) of ν-Hölder functions on U . In this section we assume that g, τ and f are real-valued ν−Hölder continuous functions onÛ . Then we can extend these functions as Hölder continuous on U .
We define the Ruelle operator L g−sr+zf : Next, for ν > 0 define the ν-norm on a set B ⊂ U by Let w ν = w ∞ + |w| ν , and denote by . ν be the corresponding norm for operators. Let χ i (x) be the characteristic function of U i . Introduce the sum Our purpose is to prove the following statement which can be considered as Ruelle's lemma with two complex parameters.
We have the following The proof is a repetition of that of Lemma 2.5 in [24] and we leave the details to the reader.
Proposition 1. Let m ≥ 1 and let w be a function which is ν-Hölder continuous on all cylinder of length m + 1. Then for the transfer operator L f −sτ +zg we have Proof. Let w be ν-Hölder on U iα for all i such that A(i, α 0 ) = 1. Let x, y ∈ Int U α and let Repeating this argument, we get and we conclude that Now, as in [24], we will choose in every cylinder U α a point x α ∈ U α . For the reader's convenience we recall the choice of x α .
(1) If U α has an n-periodic point, then we take x α ∈ U α so that σ n x α = x α .
(2) If U α has no n-periodic point and n > 1 we choose Let χ α be the characteristic function of U α . Then Lemma 3.4 and Lemma 3.5 in [24] are applied without any change and we get The proof is elementary by using the fact that Now we repeat the argument in [24] without any change and conclude that Thus, the proof of (3.1) is reduced to an estimate of the difference Observe that x α and xᾱ are on the same cylinder Uᾱ. According to Proposition 1, the function L n f −sτ +zg χ α is ν-Hölder continuous on Uᾱ. Consequently, for every n ≥ 2 we obtain where . µ denotes the operator norm derived from the ν-Hölder norm. Going back to (3.2), we deduce This it makes possible to apply (2.1) and to conclude that To finish the proof we have to estimate the term L m g−sr+zf χ β ν . Given a word α of length n > 1 and The proof is a repetition of that of Lemma 3.7 in [24] and it is based on the definition of σ −1 α above and the fact that For every admissible word β with |β| = m, we fix a point y β ∈ σ(U β m−1 ) which will be chosen as in [24]. Define z β = σ −1 β (y β ). Lemma 3. There exist constants B 0 > 0, B 1 > 0, B 2 > 0 such that we have the estimate Proof. We will follow the proof of Lemma 3.8 in [24]. Let x and y be in the same Markov leaf.
In the case when x / ∈ σ(U β m−1 ), we repeat the same argument. So we will consider the case when both x and y are in σ(U β m−1 ).
We have . On the other hand, applying Lemma 1 with w = τ , we get The same argument works for the terms involving f m and g m , applying Lemma 1 with w = f, g, respectively. Thus we obtain

and this implies an estimate for
For the product involving zg m we have the same estimate with B 2 , |z|, c 0 and c in the place of B 1 , |s|, a 0 and a. A similar estimate holds for the term containing f m with a constant B 3 in the place of B 1 . Taking the product of these estimates we obtain a bound for |L m f −sτ +zg (χ β )(x) − L m f −sτ +zg (χ β )(y)|, this implies the desired estimate for the µ-Hölder norm of L f −msτ +zg (χ β ). This completes the proof. Now the proof of Theorem 4 is reduced to the estimate of |β|=m e (f m −aτ m +cg m )(z β ) . Introduce the real-valued function h = f −aτ +cg. Then we have to estimate |β|=m e h m (z β ) . For this purpose we repeat the argument on pages 232-234 in [24] and deduce with some constant d 0 > 0 depending only on the matrix A and every > 0 the bound Combing this with the previous estimates, we get (3.1) which completes the proof of Theorem 4.

Ruelle operators -definitions and assumptions
For a contact Anosov flows φ t with Lipschitz local stable holonomy maps it is proved in section 6 in [21] that the following local non-integrability condition holds: (LNIC): There exist z 0 ∈ Λ, 0 > 0 and θ 0 > 0 such that for any ∈ (0, 0 ], anyẑ ∈ Λ∩W u (z 0 ) and any tangent vector We will say that φ t has a regular distortion along unstable manifolds over the basic set Λ if there exists a constant 0 > 0 with the following properties: ) for any z ∈ Λ and any T > 0.
(b) For any ∈ (0, 0 ] and any ρ ∈ (0, 1) there exists δ ∈ (0, ] such that for any z ∈ Λ and any A large class of flows on basic sets having regular distortion along unstable manifolds is described in [22].
In this paper we work under the following Standing Assumptions: (A) φ t has Lipschitz local holonomy maps over Λ, (B) the local non-integrability condition (LNIC) holds for φ t on Λ, (C) φ t has a regular distortion along unstable manifolds over the basic set Λ.
A rather large class of examples satisfying the above conditions is provided by imposing the following pinching condition: We should note that (P) holds for geodesic flows on manifolds of strictly negative sectional curvature satisfying the so called 1 4 -pinching condition. (P) always holds when dim(M ) = 3. Simplifying Assumptions: φ t is a C 2 contact Anosov flow satisfying the condition (P).
As shown in [22] the pinching condition (P) implies that φ t has Lipschitz local holonomy maps and regular distortion along unstable manifolds. Combining this with Proposition 6.1 in [21], shows that the Simplifying Assumptions imply the Standing Assumptions.

As in section 2 consider a fixed Markov family
The Standing Assumptions imply the existence of constants c 0 ∈ (0, 1] and γ 1 > γ 0 > 1 such that (2.1) hold.
In what follows we will assume that f and g are fixed real-valued functions in C α ( U ) for some fixed α > 0. Let P = P f be the unique real number so that Pr(f − P τ ) = 0, where Pr(h) is the topological pressure of h with respect to the shift map σ defined in Section 2. Given t ∈ R with t ≥ 1, following [4], denote by f t the average of f over balls in U of radius 1/t. To be more precise, first one has to fix an arbitrary extension f ∈ C α (V ) (with the same Hölder constant), where V is an open neighborhood of U in M , and then take the averages in question. Then f t ∈ C ∞ (V ), so its restriction to U is Lipschitz (with respect to the Riemann metric) and: ( In the special case f ∈ C Lip (U ) we set f t = f for all t ≥ 1. Similarly for g. Let λ 0 > 0 be the largest eigenvalue of L f −P τ , and letν 0 be the (unique) probability measure on U with , and L f 0 1 = 1.
Given real numbers a and t (with |a| + 1 |t| small), denote by λ at the largest eigenvalue of L ft−(P +a)τ on C Lip (U ) and by h at the corresponding (positive) eigenfunction such that U h at dν at = 1, where ν at is the unique probability measure on U with L * ft−(P +a)τ ν at = ν at . As is well-known the shift map σ : U −→ U is naturally isomorphic to an one-sided subshift of finite type. Given θ ∈ (0, 1), a natural metric associated by this isomorphism is defined (for x = y) by d θ (x, y) = θ m , where m is the largest integer such that x, y belong to the same cylinder of length m. There exist θ = θ(α) ∈ (0, 1) and β ∈ (0, α) such that (d(x, y)) α ≤ Const d θ (x, y) and d θ (x, y) ≤ Const (d(x, y)) β for all x, y ∈ U . One can then apply the Ruelle-Perron-Frobenius theorem to the sub-shift of fine type and deduce that h at ∈ C β ( U ). However this is not enough for our purposes -in Lemma 4 below we get a bit more.
Consider an arbitrary β ∈ (0, α). It follows from properties (a) and (c) above that there exists a constant C 0 > 0, depending on f and α but independent of β, such that for all |a| ≤ 1 and t ≥ 1. Since Pr(f − P τ ) = 0, it follows from the analyticity of pressure and the eigenfunction projection corresponding to the maximal eigenvalue λ at = e Pr(ft−(P +a)τ ) of the Ruelle operator L ft−(P +a)τ on C β (U ) (cf. e.g. Ch. 3 in [11]) that there exists a constant a 0 > 0 such that, taking C 0 > 0 sufficiently large, we have for |a| ≤ a 0 and 1/t ≤ a 0 . We may assume C 0 > 0 and a 0 > 0 are taken so that 1 Given real numbers a and t with |a|, 1/t ≤ a 0 consider the functions One checks that M at 1 = 1.
Taking the constant C 0 > 0 sufficiently large, we may assume that We will now prove a simple uniform estimate for Lip(h at ). With respect to the usual metrics on symbol spaces this a consequence of general facts (see e.g. Sect. 1.7 in [1] or Ch. 3 in [11]), however here we need it with respect to the Riemann metric.
The proof of the following lemma is given in the Appendix.
Lemma 4. Taking the constant a 0 > 0 sufficiently small, there exists a constant T > 0 such that for all a, t ∈ R with |a| ≤ a 0 and t ≥ 1/a 0 we have h at ∈ C Lip ( U ) and Lip(h at ) ≤ T t.

Ruelle operators depending on two parameters -the case when b is the leading parameter
Throughout this section we work under the Standing Assumptions made in section 4 and with fixed real-valued functions f, g ∈ C α ( U ) as in section 4. Throughout 0 < β < α are fixed numbers.
We will study Ruelle operators of the form L f −(P f +a+ib)τ +zg , where z = c + iw, a, b, c, w ∈ R, and |a|, |c| ≤ a 0 for some constant a 0 > 0. Such operators will be approximates by operators of the form In fact, since f at − ibτ + zg t is Lipschitz, the operators L abtz preserves each of the spaces For |b| ≥ 1, as in [4], consider the norm .
Our aim in this section is to prove the following be a Markov family for φ t over Λ as in section 2. Then for any real-valued functions f, g ∈ C α ( U ) we have: A 0 log |b| ν and |w| ≤ B |b| ν .
We will first prove part (a) of the above theorem and then derive part (b) by a simple approximation procedure. To prove part (a) we will use the main steps in section 5 in [21] with necessary modifications. The proof of part (c) is just a much simpler version of the proof of (b).
Define a new metric D on U by D(x, y) = min{diam(C) : x, y ∈ C , C a cylinder contained in U i } if x, y ∈ U i for some i = 1, . . . , k, and D(x, y) = 1 otherwise. Rescaling the metric on M if necessary, we will assume that diam(U i ) < 1 for all i. As shown in [20], D is a metric on U with d(x, y) ≤ D(x, y) for x, y ∈ U i for some i, and for any cylinder C in U the characteristic function χ C of C on U is Lipschitz with respect to D and Lip D (χ C ) ≤ 1/diam(C).
We will denote by C ≤ A D(u, u ) for all u, u ∈ U that belong to the same U i for some i = 1, . . . , k. Notice We begin with a lemma of Lasota-Yorke type, which necessarily has a more complicated form due to the more complex situation considered. It involves the operators L abtz , and also operators of the form . Fix arbitrary constants ν ∈ (0, 1) andγ with 1 <γ < γ 0 .
Lemma 5. Assuming a 0 > 0 is chosen sufficiently small, there exists a constant A 0 > 0 such that for all a, c, t ∈ R with |a|, |c| ≤ a 0 and t ≥ 1 the following hold: for all m ≥ 1 and all u, u ∈ U i , i = 1, . . . , k. whenever u, u ∈ U i for some i = 1, . . . , k. In particular, if for some constant B > 0, then for some constant A 1 > 0.
A proof of this lemma is given in the Appendix. From now on we will assume that a 0 , η 0 and A 0 are fixed with the properties in Lemma 5 above and a, b, c, w, t ∈ R are such that |a| ≤ a 0 , c ≤ η 0 , |b|, t, |w| ≥ 1 and (5.1) hold. As before, set z = c + id.
We will use the entire set-up and notation from section 4 in [21]. In what follows we recall the main part of it.
Next, fix an arbitrary orthonormal basis e 1 , . . . , e n in E u (z 0 ) and a C 1 parametrization r(s) = exp u z 0 (s), s ∈ V 0 , of a small neighborhood W 0 of z 0 in W u 0 (z 0 ) such that V 0 is a convex compact neighborhood of 0 in R n ≈ span(e 1 , . . . , e n ) = E u (z 0 ). Then r(0) = z 0 and ∂ ∂s i r(s) |s=0 = e i for all i = 1, . . . , n. Set U 0 = W 0 ∩ Λ. Shrinking W 0 (and therefore V 0 as well) if necessary, we may assume that U 0 ⊂ Int Λ (U 1 ) and ∂r ∂s i (s), ∂r ∂s j (s) − δ ij is uniformly small for all i, j = 1, . . . , n and s ∈ V 0 , so that 21]): (a) For a cylinder C ⊂ U 0 and a unit vector ξ ∈ E u (z 0 ) we will say that a separation by a ξ-plane occurs in Let S ξ be the family of all cylinders C contained in U 0 such that a separation by an ξ-plane occurs in C.
Fix U 0 and U with the properties described in Lemma 1; then U = U . Setδ = min 1≤ ≤ 0δ j , n 0 = max 1≤ ≤ 0 m , and fix an arbitrary pointẑ 0 ∈ U Fix integers 1 ≤ n 1 ≤ N 0 and 0 ≥ 1, unit vectors η 1 , η 2 , . . . , η 0 ∈ E u (z 0 ) and a non-empty open subset U 0 of W 0 with the properties described in Lemma 6. By the choice of U 0 , σ n 1 : U 0 −→ U is one-to-one and has an inverse map ψ : U −→ U 0 , which is Lipschitz. Set , where A 0 ≥ 1 is the constant from Lemma 5.4, and fix an integer N ≥ N 0 such that Then fix maps v Since U 0 is a finite union of open cylinders, it follows from Lemma 6(d) that there exist a constant δ = δ (U 0 ) > 0 such that Fix δ with this property. Set and let b ∈ R be such that |b| ≥ 1 and 1 |b| ≤ δ .
Next, let D 1 , . . . , D q be the list of all closed cylinders contained in U 0 that are subcylinders of co-length p 0 q 0 of some C m (1 ≤ m ≤ p).
Denote by J = J(a, b) the set of all dense subsets J of Ξ. Although the operator N here is different, the proof of the following lemma is very similar to that of Lemma 5.8 in [21]. Lemma 8. Given the number N , there exist ρ 2 = ρ 2 (N ) ∈ (0, 1) and a 0 = a 0 (N ) > 0 such that In what follows we assume that h, H ∈ C and , Definitions. We will say that the cylinders D j and D j are adjacent if they are subcylinders of the same C m for some m. If D j and D j are contained in C m for some m and for some = 1, . . . , 0 there exist u ∈ D j and v ∈ D j such that d(u, v) ≥ 1 2 diam(C m ) and r −1 (v)−r −1 (u) r −1 (v)−r −1 (u) , η ≥ θ 1 , we will say that D j and D j are η -separable in C m .
As a consequence of Lemma 6(b) one gets the following. for all u ∈ Z j and u ∈ Z j , where c 2 =δ ρ 16 .
The following lemma is the analogue of Lemma 5.10 in [21] and represents the main step in proving Theorem 1. To prove this we need the following lemma which coincides with Lemma 14 in [4] and its proof is almost the same.
Sketch of proof of Lemma 10. We use a modification of the proof of Lemma 5.10 in [21].
Given j = 1, . . . , q, let m = 1, . . . , p be such that D j ⊂ C m . As in [21] we find j , j = 1, . . . , q such that D j , D j ⊂ C m and D j and D j are η -separable in C m .
Fix , j and j with the above properties, and set Z = Z j ∪ Z j ∪ Z j . If there exist t ∈ {j, j , j } and i = 1, 2 such that the first alternative in Lemma 11(b) holds for Z t , and i, then µ ≤ 1/4 implies χ (i) (u) ≤ 1 for any u ∈ Z t .
Assume that for every t ∈ {j, j , j } and every i = 1, 2 the second alternative in Lemma 11(b) holds for Z t , and i, i.e. |h(v ) is a cylinder with diam(C ) ≤ 1 c 0 γ N −n 1 |b| . Thus, the estimate (9.3) in the Appendix below implies Using the above assumption, (5.1), (5.2) and (4.4), and assuming e.g.
The proof of Theorem 5(b) can be derived using an approximation procedure as in [4] -see the Appendix below for some details.

Spectral estimates when w is the leading parameter
Here we try to repeat the arguments from the previous section however changing the roles of the parameters b and w. We continue to use the assumptions made at the beginning of section 5, however now we suppose that f ∈ C Lip ( U ). We will consider the case |b| ≤ B |w| (6.1) for an arbitrarily large (but fixed) constant B > 0.

Assume that G : Λ −→ R is a Lipschitz functions which is constant on stable leaves of
where without loss of generality we may assume that D ≥ 1. We will also assume that The function is constant on stable leaves of R, so it can be regarded as a function on U . Clearly g ∈ C Lip ( U ).
Remark. Notice that if we replace G by G + d for some constant d > 0, then Choose and fix d > 0 so that Lip(G) G 0 +d ≤μ. Then for G = G + d and g = g + dτ we have Lip(G ) min G ≤μ, and the operator L fa−i bτ +i wg = L fa−i b τ +i wg , where b = b + dw. Thus, without loss of generality we may assume that Lip(G) min G ≤μ, which is equivalent to (6.2). As in [12], this will imply a nonintegrability property for g (see Lemma 10 below). In other words, dealing with an initial function G one has to first change it to arrange (6.2), and then with the new parameters b and w that appear in front of iτ and ig consider the cases |w| ≤ B|b| (as in Theorem 5(c)) and |b| ≤ B|w|, which is considered in this section.
As in section 5, we will use the set-up and some arguments from [21]. Let R = {R i } k i=1 be a Markov family for φ t over Λ as in section 2.
Here we prove the following analogue of Theorem 5(c).
Theorem 6. Let φ t : M −→ M be a C 2 flow satisfying the Standing Assumptions over the basic set Λ. Assume in addition that (6.2) holds. Then for any real-valued functions f, g ∈ C Lip ( U ), any constants > 0 and B > 0 there exist constants 0 < ρ < 1, a 0 > 0, w 0 ≥ 1 and C = C(B, ) > 0 such that if a, c ∈ R satisfy |a|, |c| ≤ a 0 , then for all integers m ≥ 1 and all b, w ∈ R with |w| ≥ w 0 and |b| ≤ B |w|.
Recall the definitions of λ 0 > 0,ν 0 , h 0 , f 0 from section 4; now we have h 0 , f 0 ∈ C Lip ( U ). Fix a small a 0 > 0. Given a real number a with |a| ≤ a 0 , denote by λ a the largest eigenvalue of L f −(P +a)τ on C Lip (U ) and by h a the corresponding (positive) eigenfunction such that U h a dν a = 1, where ν a is the unique probability measure on U with L * f −(P +a)τ ν a = ν a . Given real numbers a, b, c, w with |a|, |c| ≤ a 0 consider the functioñ f a = f − (P + a)τ + ln h a − ln(h a • σ) − ln λ a and the operators Notice that Lf a 1 = 1. Taking the constant C 0 > 0 sufficiently large, we may assume that Thus, ssuming a 0 > 0 is chosen sufficiently small, there exists a constant T > 0 (depending on f and a 0 ) such that for |a| ≤ a 0 . As before, we will assume that T ≥ max{ τ 0 , Lip(τ | U ) }, and also that Lip(g) ≤ T and g 0 ≤ T .
Essentially in what follows we will repeat (a simplified version of) the proof of Theorem 5, so we will use the set-up in section 5 -see the text after Lemma 6, up to and including the definition of 1 .
Let a, b, c, w ∈ R be so that |a|, |c| ≤ a 0 , |w| ≥ w 0 , where w 0 is a sufficiently large constant defined as b 0 in section 5, and |b| ≤ B|w|. Set z = c + iw.
Let C m (1 ≤ m ≤ p) be the family of maximal closed cylinders contained in U 0 with diam(C m ) ≤ 1 |w| such that U 0 ⊂ ∪ p j=m C m and U 0 = ∪ p m=1 C m . As before we have Fix an integer q 0 ≥ 1 as in Sect. 5, and let D 1 , . . . , D q be the list of all closed cylinders contained in U 0 that are subcylinders of co-length p 0 q 0 of some C m (1 ≤ m ≤ p).
Next, define the cylinders Z j = σ n 1 ( D j ) and X Lemma 13. Assuming a 0 > 0 is chosen sufficiently small, there exists a constant A 0 > 0 such that for all a, c ∈ R with |a|, |c| ≤ a 0 the following hold: for all m ≥ 1 and all u, u ∈ U i , i = 1, . . . , k.
(b) If the functions h and H on U and E > 0 are such that H > 0 on U and |h . . , k, then for any integer m ≥ 1 and any b, w ∈ R with |b|, |w| ≥ 1, for z = c + iw we have (u, u ). whenever u, u ∈ U i for some i = 1, . . . , k.
The proof is a simplified version of that of Lemma 5 and we omit it.
Next, changing appropriately the definition of a dense subset J of Ξ, Lemma 8 holds again replacing K E|b| ( U ) by K E|w| ( U ). and Define the functions χ (i) : U −→ C by , The crucial step in this section is to prove the following analogue of Lemma 9: Lemma 14. Let j, j ∈ {1, 2, . . . , q} be such that D j and D j are contained in C m and are ηseparable in C m for some m = 1, . . . , p and = 1, . . . , 0 . Then |γ (u) − γ (u )| ≥ c 3 1 for all u ∈ Z j and u ∈ Z j , where c 3 = Aδ ρ 32 .
To prove the above we need the following.
Next, we need to prove the analogue of Lemma 10.
Sketch of proof of Lemma 16. We will use Lemma 11 which holds again with (5.3)-(5.4) replaced by (6.6)-(6.7). Given j = 1, . . . , q, let m = 1, . . . , p be such that D j ⊂ C m . As in [21] we find j , j = 1, . . . , q such that D j , D j ⊂ C m and D j and D j are η -separable in C m .
Fix , j and j with the above properties, and set Z = Z j ∪ Z j ∪ Z j . If there exist t ∈ {j, j , j } and i = 1, 2 such that the first alternative in Lemma 11(b) holds for Z t , and i, then µ ≤ 1/4 implies χ (i) (u) ≤ 1 for any u ∈ Z t .
Proof of Theorem 6. This is now the same as the proof of Theorem 5(a).
7. Analytic continuation of the function ζ(s, z) Consider the function ζ(s, z) introduced in section 1. Recall that s = a + ib, z = c + iw with real a, b, c, w ∈ R. First, we assume that f and g are functions in C α (Λ) with some 0 < α < 1. Passing to the symbolic model defined by the Markov family R we obtain functions 2 in C α (R) which we denote again by f and g. We assume that P r(f − P f τ ) = 0 and we set s = P f + a + ib. The functions f , g depend on x ∈ R. A second reduction is to replace f and g by functionsf ,ĝ ∈ C α/2 (U ) depending only on x ∈ U so that f =f + h 1 − h 1 • σ, g =ĝ + h 2 − h 2 • σ (see Proposition 1.2 in [11]). Since for periodic points with σ n x = x we have f n (x) =f n (x), g n (x) =ĝ n (x), we obtain the representation In this section we will prove under the standing assumptions that there exists > 0 and 0 > 0 such that the function ζ(s, z) has a non-vanishing zero analytic continuation for − ≤ a ≤ 0 and |z| ≤ 0 with a simple pole at s = s(z), s(0) = P f . Here s(z) is determined from the equation P r(f − sτ + zg) = 0. For simplicity of the notation we denote belowf andĝ again by f , g.
To obtain a representation of the function η g (s) = ∂ log ζ(s,z) ∂z z=0 for s sufficiently close to P f , notice that for such values of s we have where m is the equilibrium state of f − P f τ , µ F is the equilibrium state of F and A 0 (s) and A 1 (s) are analytic in a neighborhood of P f (see Chapter 6 in [11]). More precisely, µ F is a σ τ t invariant probability measure on R τ such that where h(σ τ 1 , µ F ) is the metric entropy of σ τ 1 with respect to µ F (see Chapter 6 in [11]). Taking η 0 small enough, for |z| ≤ η 0 , | Re s − P f | ≤ η 0 and | Im s| ≥ η 0 from the estimates for Z n (f − (P f + a + ib)τ + zg) above, we deduce | log ζ(s, z)| ≤ C max 1, | Im s| 1+ .
To estimate η g (s), as in [17], we apply the Cauchy theorem for the derivative with δ > 0 sufficiently small. Thus we obtain a O max 1, | Im s| 1+ bound for the function Decreasing η 0 and applying Phragmén-Lindelöf theorem, by a standard argument we obtain a bound O max 1, | Im s| α with 0 < α < 1. Consequently, we have the following Proposition 3. Under the assumptions of Theorem 1 there exist η 0 > 0 and 0 < α < 1 such that for Re s > P f − η 0 we have with an analytic function A(s) satisfying the estimate |A(s)| ≤ C max 1, | Im s| α .
is Lipschitz continuous on R. Moreover, if the representative of G in the suspension space R τ is constant on stable leaves, the function g(x) depends only on x ∈ U. Now we introduce two definitions of independence. Definition 1. Two functions f 1 , f 2 : U → R are called σ− independent if whenever there are constants t 1 , t 2 ∈ R such that t 1 f 1 + t 2 f 2 is co homologous to a function in C(U : 2πZ), we have t 1 = t 2 = 0.
For a function G ∈ F τ (R) consider the skew product flow S G t on S 1 × R τ by S G t (e 2πiα , y) = e 2πi(α+G t (y)) , σ τ t (y) .

Definition 2 ([8]
). Let G ∈ F τ (R). Then G and σ τ t are flow independent if the following condition is satisfied. If t 0 , t 1 ∈ R are constants such that the skew product flow S H t with H = t 0 + t 1 G is not topologically mixing, then t 0 = t 1 = 0.
Notice that if G and σ τ t are flow independent, then the flow σ τ t is topologically weak mixing and the function G is not co homologous to a constant function. On the other hand, if G and σ τ t are flow independent, then g(x) = τ (x) 0 G(π(x, t))dt and τ are σ− independent. Below we assume that g and τ are σ− independent and we suppose that F, G is a Lipschitz functions Λ having representative in R τ which are constant on stable leaves. Thus we obtain functions f , g which are in C Lip ( U ). We will now obtain an analytic continuation of ζ(s, z) for P f −η 0 < Re s < P f and z = iw. Set r(s, w) = f − (P f + a + ib)τ + iwg. We choose M > 0 large enough so that we can apply Theorem 6 for |w| ≥ M. We consider two cases. Case 1. η 0 ≤ |w| ≤ M. We consider two sub cases: 1a) | Im s| ≤ M 1 , 1b) | Im s| ≥ M 1 . Here M 1 > 0 is chosen large enough so that Theorem 5 (b) holds with | Im s| ≥ M 1 .
Let | Im s| ≤ M 1 . Assume first that Im r(s 0 , w 0 ) is cohomologous to c + 2πQ with an integervalued function Q ∈ C(U ; Z) and a constant c ∈ [0, 2π). If c = 0, since g and τ are σ− independent, from the fact that bτ + wg is co homologous to a function in C(U ; 2πZ), we deduce b = w = 0 which is impossible because b = Im s = 0. Thus we have c = 0. Consequently, the operator L f −s 0 τ +iwg has an eigenvalue e ic . Then there exists a neighborhood U 1 ⊂ C × R of (s 0 , w 0 ) such that for (s, w) ∈ U 1 we have P r(r(s, w)) = 0 and for (s, w) ∈ U 2 we have an analytic extension of log ζ(s, w) given by log ζ(s, w) = K 1 (s, w) 1 − e P r(r(s,w)) + J 1 (s, w) with functions K 1 (s, w), J 1 (s, w) analytic with respect to s for (s, w) ∈ U 1 . Second, let Im r(s 0 , w 0 ) be not cohomologous to c + 2πQ. Then the spectral radius of L f −s 0 τ +iwg is strictly less than 1 and this will be the case for (s, w) is a small neighborhood U 2 ⊂ C × R of (s 0 , w 0 ). Applying Theorem 4, this implies easily that log ζ(s, iw) has an analytic continuation with respect to s.
Case 2. |w| ≥ M . We consider two-sub cases: 2a) | Im s| ≥ B|w|, 2b) | Im s| ≤ B|w|, B = M 1 M . If we have 2a), we apply Theorem 5(c). In the case 2b) we use the argument of section 6 replacing g(x) by g (x) = g(x) + dτ (x), where the constant d > 0 is chosen so that for the function G = G + d we have Lip G min G ≤μ, whereμ > 0 is the constant introduced in section 5. Next we write L f −(P f +a+ib)τ +iwg = L f −(P f +a+i(b+dw)τ +iwg .
For the Ruelle operator involving g we can apply Theorem 6 since |b + dw| ≤ (B + d)|w|, |w| ≥ M and g is a Lipschitz function. An application of Theorem 4 implies the analytic continuation of log ζ(s, iw) for P f − η 0 ≤ Re s ≤ P f and |w| ≥ M. From the above analysis we deduce the following Theorem 7. Assume the standing assumptions fulfilled for the basic set Λ. Let F, G : Λ −→ R be Lipschitz functions having representatives in R τ which are constant on stable leaves. Assume that g and τ are σ-independent. Then there exists η 0 > 0 such that ζ(s, iw) admits a non zero analytic continuation with respect to s for P f − η 0 ≤ Re s, w ∈ R and |w| ≥ η 0 .

Applications
8.1. Hannay-Ozorio de Almeida sum formula. The proof of (1.2) in [18] is based on the analytic continuation of the Dirichlet series λ G (γ)e m(−λ u (γ)−(s−1)λ(γ)) , s ∈ C for 1 − η 0 ≤ Re s < 1. For this purpose the authors examine the analytic continuation of the symbolic function η g (s) with g(x) = τ (x) 0 G(π(x, t))dt defined in section 1 and they use the fact that the difference η(s) − η g (s) is analytic in a region Re s > 1 − , > 0. Next for the geodesic flow on surfaces with negative curvature they establish Proposition 3 with P f = 1. Since M is an attractor, the equilibrium state of the function −E(x) is just the SRB measure µ of φ t (see [3]) and the residuum in (7.2) becomes Gdµ.
For the proof of Proposition 3 in [18] the authors exploit the link between the analytic continuation of ζ(s, z) and the spectral estimates of the Ruelle operator obtained by Dolgopyat [4]. However, in [18] Ruelle's lemma in [16] was used whose proof is rather sketchy and contains some steps which are not done in detail (see [24] for more information and comments concerning these steps and the gaps in their proofs). On the other hand, the estimates of Dolgopyat [4] are established only for Ruelle operators with one complex parameter, and to take into account the second parameter z some complementary analysis is necessary.
We should mention that [24] contains a correct and complete proof of Ruelle's lemma in the case of one complex parameter and a Hölder function τ (x). A version of this lemma with two complex parameters is given in section 3 above. Next, in Theorem 5 the spectral estimates for the Ruelle operator with two complex parameters are established for Axiom A flows on a basic set Λ of arbitrary dimension under the standing assumptions. If Λ is an attractor, according to [3], the equilibrium state of −E(x) coincides with the SRB measure µ of φ t . Thus we can apply Proposition 3 to obtain a representation of η g (s) with residue Gdµ. Using (7.2) and repeating the argument of section 4 in [18], we obtain Theorem 2.

8.2.
Asymptotic of the counting function for period orbits. As we mentioned in section 1, the analysis of π F (T ) is based on the analytic continuation of the function ζ(s, 0) defined in section 1. From the arguments in section 7 with z = 0 and the proof of Proposition 3 we get the following admits an analytic continuation for P r(F )−η 0 ≤ Re s with a simple pole at s = P r(F ) with residue 1. Moreover, there exists 0 < α < 1 such that for | Im s| ≥ 1 we have To obtain an asymptotic of π F (T ), we examine the functions By a standard argument (see [16] and [15]) we obtain the representation where in the second equality the estimate (8.1) is used. This implies an asymptotic for Ψ(T ) and repeating the argument in [16], [15], one obtains Theorem 3.

Appendix: Proofs of some lemmas
Proof of Lemma 4. Denote by F θ ( U ) the space of all functions h : U −→ R that are Lipschitz with respect to d θ . Let g ∈ C Lip ( U ), and let θ = θ α ∈ (0, 1) be as in section 4. Then g ∈ F θ ( U ). Let λ > 0 be the maximal positive eigenvalue of L g on F θ ( U ) and let h > 0 be a corresponding normalized eigenfunction. By the Ruelle-Perron-Frobenius theorem, we have that 1 λ m L m g 1 converges uniformly to h. We will show that there exists a constant C > 0 such that 1 λ m Lip(L m g 1) ≤ C for all m; this would then imply immediately that h ∈ C Lip ( U ) and Lip(h) ≤ C.
Take an arbitrary constant K > 0 such that 1/K ≤ h(x) ≤ K for all x ∈ U . Given u, u ∈ U i for some i = 1, . . . , k and an integer m ≥ 1 for any v ∈ U with σ m (v) = u, denote by v = v (v) the unique v ∈ U in the cylinder of length m containing v such that σ m (v ) = u . By (2.1) we have for some constant C > 0. Thus, = e C Lip(g) C KLip(g) d(u, u ) (L m g h)(u) = e C Lip(g) C KLip(g) d(u, u ) λ m h(u) ≤ λ m C K 2 e C Lip(g) Lip(g) d(u, u ).
Thus, for every integer m the function 1 λ m L m g 1 ∈ C Lip ( U ) and 1 λ m Lip(L m g 1) ≤ C K 2 e C Lip(g) Lip(g). As mentioned above this proves that the eigenfunction h ∈ C Lip ( U ).
Using this with g = f t proves that h at ∈ C Lip ( U ) for all |a| ≤ a 0 and t ≥ 1/a 0 . However the above estimate for Lip(h at ) would be of the form ≤ C e C t t for some constant C > 0, which is not good enough.
We will now show that, taking a 0 > 0 sufficiently small, we have Lip(h at ) ≤ Ct for some constant C > 0 independent of a and t.
Using (4.2) and choosing a 0 > 0 sufficiently small, we have λ at γ >γ for all |a| ≤ a 0 and t > 1/a 0 . Fix an integer m 0 ≥ 1 so large that C 2 0 c 0γ m < 1 2 for m ≥ m 0 . There exists a constant d 0 > 0 depending on m 0 such that for any u, u belonging to the same U i but not to the same cylinder of length m 0 we have d(u, u ) ≥ d 0 . For such u, u we have So, to estimate Lip(h at ) it is enough to consider pairs u, u that belong to the same cylinder of length m 0 . Fix for a moment a, t with |a| ≤ a 0 and t ≥ 1/a 0 . Set L = sup u =u |hat(u)−hat(u )| d (u,u ) , where the supremum is taken over all pairs u = u that belong to the same cylinder of length m 0 . If L < Lip(h at ), then the above implies Lip(h at ) ≤ 2C 0 d 0 ≤ 2C 0 d 0 t. Assume that L = Lip(h at ). Then there exist u, u belonging to the same cylinder of length m 0 such that 3L 4 < |h at (u) − h at (u )| d(u, u ) .
At the same time, by property (i), f t 0 ≤ T for some constant T > 0, so Using the constants C 1 , C 2 > 0 from the proof of part (a), (9.5) and e C 2 a 0 < γ/γ we get As in [4] and [21] we need the following lemma whose proof is omitted here, since it is very similar to the proof of Lemma 5 given above.
We will now approximate L f −(P +a+ib)τ +zg by L ft−(P +a+ib)τ +gt in two steps. First, using the above it follows that L n f −(P +a+ib)τ +cg+iwgt h β,b = L n ft−(P +a+ib)τ +zgt e (f n −f n