Large Deviation Asymptotics for a Suspension Flow over the Natural Extension of the Continued Fraction Map

Simon Waddington

Mathematisches Institut, Bunsenstr. 3-5, 37073 Göttingen, Germany.

Abstract. We study a suspension flow over the natural extension of the continued fraction map. This flow can be regarded as an abstract model of the geodesic flow on the unit tangent bundle of the Modular Surface. We derive precise asymptotic medium and large deviation formulae with respect to the natural extension of the Gauss measure, for a class of functions satisfying a Hölder continuity condition, with a weighted Hölder constant. As corollaries we deduce dynamical analogues of the Local and Central Limit Theorems

0. Introduction

In a recent paper [W1], we proved some precise asymptotic formulae for medium and large deviation probabilities for Anosov flows on compact C^{∞} Riemannian manifolds. The prototypical example of such a flow is the geodesic flow on the unit tangent bundle of a compact negatively curved Riemannian manifold.

In this paper, we take a first step to extending our earlier results to flows on noncompact spaces. We consider a suspension flow over the natural extension of the continued fraction or Gauss transformation. More precisely, let I = [0, 1] and let S : I^{2} → I^{2} denote the natural extension of the Gauss map (see section one), and let

X = {(x, y; t) : (x, y) ∈ (0, 1] × [0, 1], 0 ≤ t < −2 log x}/ ∼

where the equivalence relation ∼ identifies the points (x, y; −2 log x) and (S(x, y); 0). Define a flow Φ on X to be is the unit speed flow under the graph of the function (0, 1] × [0, 1] → ℝ given by (x, y) ↦ −2 log x. That is, we define Φ locally by

Φ_{s}(x, y; t) = (x, y; s + t)

and extend this to the whole of X, respecting the identifications.

The flow Φ : X → X serves as a simplified abstract model of probably the best known example of a geodesic flow on the unit tangent bundle of a non-compact surface with finite area, namely the geodesic flow on the unit tangent bundle of the Modular Surface M. (In the full abstract model, one considers the suspension flow over the skew product of X with the group ℤ/2ℤ. This introduces some further technical complications.) The Modular Surface is the quotient space of the group P SL(2, ℤ) acting as Möbius transformations on the hyperbolic plane ℍ. Topologically, M is a once punctured sphere, with the puncture corresponding to a parabolic cusp.

Extensions of the results in this paper to the geodesic flow on the unit tangent bundle of the Modular Surface, as well as some geometrical and number-theoretic applications, will appear in a forthcoming article, [W3].

Let m denote the normalised, Φ-invariant measure, constructed as the product of ρ^{2} on I^{2}, where ρ is the Gauss measure, with Lebesgue measure on ℝ. This measure is the analogue of the Liouville measure for the geodesic flow. The flow Φ is ergodic (and even strong mixing) with respect to m. Thus for each F ∈ L^{1}(X; ℝ),

\frac{1}{T} ∫_{0}^{T} F (Φ_{t}y)dt → ∫ F dm as T → ∞,

for m-almost all y ∈ X. We consider the rate of convergence in probability in this ergodic theorem for a special class of functions, denoted by ℱ^{α}(X; ℝ).

For each F ∈ ℱ^{α}(X; ℝ), we show that there is a non-negative, strictly convex, real analytic function, called the entropy or rate function, denoted by H : Γ_{F} → ℝ, defined on an open interval Γ_{F} in ℝ, such that the following asymptotic medium deviation formula holds:

m\{y ∈ X : \frac{1}{T} ∫_{0}^{T} F (Φ_{t}y)dt − aT ∈ K\} ∼ C_{1}(a)(∫_{K} e^{−H′(a)t}dt)\frac{e^{−T H(a)}}{\sqrt{T}}  (0.1)

as T → ∞, for any a ∈ Γ_{F} and any compact subset K of ℝ. The constant C_{1}(a) is an explicitly computable constant, depending only on a. Moreover, the convergence in (0.1) is uniform on compact subsets of Γ_{F}. From this fact, we deduce the following general asymptotic large deviation formula:

m\{y ∈ X : \frac{1}{T} ∫_{0}^{T} F (Φ_{t}y)dt ∈ J\} ∼ C_{2}(a)\frac{e^{−T H(a)}}{\sqrt{T}}  (0.2)

as T → ∞, for any compact set J ⊂ Γ_{F} \ {∫ F dm} is compact, and

H(a) = inf_{b∈J} H(b).

The constant C_{2}(a) depends only on a. As a corollary of (0.1), we also deduce the Local and Central Limit Theorems.

Large deviations have been studied extensively in dynamical systems in recent years. However, the techniques developed in [D], [K] and [Y] for example, do not yield the more refined estimates of the form (0.1) and (0.2).

The model we consider can be related to the Modular Surface using the classical technique of coding geodesics by means of continued fractions. This was used by Artin to prove the existence of a dense geodesic on M. A detailed account of this coding can be found in [Se].

Our results are essentially based on a careful study of transfer operators associated to the continued fraction or Gauss map. The non-compactness of the space X is surmounted by truncating the singularity at a certain height, and then allow this height to increase with time, at an appropriate rate. We use these ideas to describe the analytic domain of a complex function called the complex moment generating function, which was introduced in [W1]. The asymptotic formula (0.1) then follows by applying an appropriate Tauberian theorem.

The distribution of closed geodesics on the modular surface was studied in [Po1], based on the work of Mayer [M] on the Ruelle zeta function for the Gauss map. Since the techniques used in this case use only spaces of analytic functions, they are not directly applicable to our problem.

The paper is organized as follows. Sections one and two contain basic definitions of the Gauss map and the suspended flow respectively. The main results of the paper are stated in section three.

Sections four and five contain some basic results on Perron Frobenius operators with specially weighted potentials, acting on spaces of Hölder continuous functions. In particular, we give an upper bound on the essential spectral radius of these operators. We apply these results to prove the real analycity of a thermodynamic pressure function.

Section six contains the basic definitions and properties of the entropy or rate function. In section seven, we prove that certain classes of functions defined on the unit square

are cohomologous to functions defined on the interval. This extends a well known technique of reducing invertible systems to non-invertible ones.

Section eight contains an extension of the results of sections five and six to Perron Frobenius operators acting on spaces of L^{1} functions which can be well approximated by Hölder continuous functions.

The calculation of the complex moment generating function is carried out in section nine. Essentially, we rewrite this function in terms of Perron Frobenius operators, and then apply the results of section eight. The calculations here differ from those in [W1] in that we are forced to make our estimates in the L^{1}-norm, instead of the Hölder norm.

In section ten, we complete the proof of the main theorem using an “unsmoothing” method, together with a Tauberian theorem of Katsuda and Sunada. This argument is described in detail in section five of [W1], so we only briefly sketch the main ideas, indicating an improvement to the original method.

Acknowledgement. This problem was suggested to me by Prof. Ya. G. Sinai at the conference on “Statistical Mechanics and Dynamical Systems”, held at the Hebrew University, Jerusalem, Israel, in June, 1995.

1. The Gauss Map

Let I = [0, 1] and let T : I → I be the map defined by

T x = \{ \begin{matrix}\frac{1}{x} − [\frac{1}{x}] & if x ≠ 0 \\ 0 & if x = 0. \end{matrix}

which is commonly refered to as the Gauss Map. (Here [x] denotes the integer part of the real number x.) The map T is an expanding Markov map, in fact |(T^{2})′(x)| > 4 for all x ∈ I. Indeed, if {I_{n}}_{n∈ℕ} is the partition of I whose elements are given by

I_{n} = (\frac{1}{n + 1}, \frac{1}{n}] , n ∈ ℕ

then T I_{n} = I for all n ∈ ℕ. The inverse branches T_{n} : I → I_{n} have the form

T_{n}(x) = \frac{1}{x + n}.

The Gauss map is closely related to continued fraction expansions of points in [0, 1). If x ∈ [0, 1) has an expansion of the form

x = \frac{1}{x_{0} + \frac{1}{x_{1} + \frac{1}{x_{2} + ⋯}}} x_{i} ∈ ℕ,

then we denote this by x = [x_{0}, x_{1}, x_{2}, …]. This expansion terminates after finitely many terms if and only if x is rational. If x = [x_{0}, x_{1}, x_{2}, …] is irrational, then it is easy to check that

{x} = ⋂_{n=0}^{∞} T^{−n}(I_{x_{n}}).

The map T acts as a shift operator on continued fraction transformations, that is

T ([x_{0}, x_{1}, x_{2}, …]) = ([x_{1}, x_{2}, x_{3}, …].

There is a unique T -invariant probability measure ρ on I, equivalent to Lebesgue measure l, called the Gauss measure, which is defined via the Radon-Nikodym derivative

\frac{dρ}{dl}(x) = \frac{1}{(log 2)(1 + x)}.

Let S : [0, 1]^{2} → [0, 1]^{2} be the natural extension of the Gauss map, given by

S(x, y) = (T x, T_{x_{0}}y),  (1.1)

where x ∈ I_{x_{0}}, or alternatively

S([x_{0}, x_{1}, …], [y_{−1}, y_{−2}, …]) = ([x_{1}, x_{2}, …], [x_{0}, y_{−1}, …])  (1.2)

where x = [x_{0}, x_{1}, x_{2}, …] and y = [y_{−1}, y_{−2}, y_{−3}, …], in terms of continued fraction expansions. The product measure ρ × ρ is invariant under S, and S is ergodic with respect to ρ^{2}. (The definition of S in (1.2) is not adequate for points with rational coordinates, but as the set of such points has l^{2} measure 0, we can safely neglect them.)

2. The suspended flow

In this section, we define a suspension flow over the natural extension S : I^{2} → I^{2} of the continued fraction transformation.

First define a function r : (0, 1] × [0, 1] → ℝ by r(x, y) = −2 log x. This function will be the height function for the suspended flow. Sometimes it will be convenient to regard r as a function (0, 1] → ℝ, by letting r(x) = −2 log x. Define a quotient space

X = {(z; t) ∈ (0, 1] × [0, 1] × ℝ : 0 ≤ t ≤ r(z)}/ ∼,

where the equivalence relation ∼ identifies the points (z; r(z)) and (Sz; 0). We define a flow Φ_{t} : X → X locally by

Φ_{t}(z; s) = (z; s + t)

and extend it to the whole of X, respecting the above identification.

Define a measure m on X locally by

m = \frac{ρ × ρ × l}{∫ rdρ^{2}}  (2.1)

where l is Lebesgue measure on ℝ. This means that

∫ F dm = \frac{∫ fdρ^{2}}{∫ rdρ^{2}}

where

f(z) = ∫_{0}^{r(z)} F (z; t)dt.

Since S is strong mixing, and hence ergodic with respect to ρ × ρ, we obtain the following theorem.

Theorem 1. The flow Φ is strong mixing, and therefore ergodic, with respect to the measure m.

3. Statement of results

Let Φ_{t} : X → X denote the suspension flow defined in section two, and let m denote the measure given by (2.1). Define a function r_{δ} : I^{2} → ℝ for each δ ∈ (0, 1) by

r_{δ}(x) = \{ \begin{matrix}−\frac{2}{δ}x + 2(1 − log δ) & if 0 ≤ x ≤ δ \\ r(x) & if δ < x ≤ 1, \end{matrix}  (3.1)

which can be regarded as truncations of the height function r.

We now define a special class of continuous functions on X. Some examples of functions satisfying this condition are given in [W3].

Definition 1. We say that a function F : X → ℝ lies in the set ℱ^{α}(X; ℝ), for some α ∈ (0, 1), if the following three conditions are satisfied:

(i) F ∈ C(X; ℝ). (ii) There exists a constant C_{F} > 0 such that |F (y)| ≤ C_{F} for all y ∈ X. (iii) There exists a constant K_{F} > 0 and α′/in(0, α) such that for any δ ∈ (0, 1), and for

all z_{1}, z_{2} ∈ I^{2} and t ∈ [0, 1],

|F (z_{1}; tr_{δ}(z_{1})) − F (z_{2}; tr_{δ}(z_{2}))| ≤ \frac{K_{F}}{δ^{α′}} |z_{1} − z_{2}|^{α}.

We require an additional assumption, which ensures, for example that F is nonconstant.

Definition 2. Let F ∈ ℱ^{α}(X; ℝ) be given. We say that F is non-lattice if the following condition holds: If there exist B ∈ L_{m}^{2}(X; ℂ) and λ_{1}, λ_{2} ∈ ℝ such that

B ∘ Φ_{T} = e^{i(λ_{1} ∫_{0}^{T} F ∘Φ_{t}dt+λ_{2}T)}B

for all T ∈ ℝ, then λ_{1} = 0 = λ_{2}. (Of course, if λ_{1} = 0 = λ_{2} then B is constant m-almost everywhere by the ergodicity of the flow.)

Similar conditions occur in the papers [La2], [W1]. Henceforth, we assume that F ∈ ℱ^{α}(X; ℝ), and that F is non-lattice. Throughout the rest of the paper, we will use the notation

F^{T} (y) = ∫_{0}^{T} F (Φ_{t}y)dt for T ∈ ℝ.

Our main result is the following asymptotic medium deviation result.

Theorem 2. (Medium deviations) There exists a function H : ℝ → [0, ∞] with the following properties:

(i) The set Γ_{F} = {a ∈ ℝ : H(a) < ∞} is a non-empty open interval of ℝ. (ii) The function H : Γ_{F} → [0, ∞) is real analytic, surjective and strictly convex. (iii) For a ∈ Γ_{F},

\begin{aligned}H(a) & > 0 if a ≠ ∫ F dm \\ & = 0 if a = ∫ F dm. \end{aligned}

Moreover, for every non-empty compact set K ⊂ ℝ and every a ∈ Γ_{F},

m\{y ∈ X : F^{T} (y) − T a ∈ K\} ∼ C(a) (∫_{K} e^{−H′(a)t}dt) \sqrt{\frac{H′′(a)}{2π}} \frac{e^{−T H(a)}}{\sqrt{T}}  (3.2)

as T → ∞. The constant C(a) is an explicitly computable and depends only on a. Furthermore, the convergence in (3.2) is uniform on compact subsets of Γ_{F}.

In the statement of Theorem 2, we have used the standard notation A(t) ∼ B(t) to mean

\frac{A(t)}{B(t)} → 1 as t → ∞.

The proof of Theorem 2 will be given in section 10.

The function H : ℝ → [0, ∞) given by Theorem 2 is called the entropy or rate function. Its existence is used in the following corollaries of Theorem 2.

Corollary 1. (Large deviations). Let J ⊂ Γ_{F} be a closed interval which does not contain the point ∫ F dm. Let a ∈ Γ_{F} be the unique point for which

H(a) = inf_{b∈J} H(b).

Then

m\{y ∈ X : \frac{F^{T} (y)}{T} ∈ J\} ∼ \frac{C(a)}{H′(a)}\sqrt{\frac{H′′(a)}{2π}} \frac{e^{−T H(a)}}{\sqrt{T}}

as T → ∞.

Corollary 1 can be proved by the method of [W1], section six. The following corollary is an analogue of the Local Limit Theorem from probability theory. Such results have been obtained for Anosov diffeomorphisms in [GH] and in [W1] for Anosov flows.

Corollary 2. (Local Limit Theorem). For any c ∈ ℝ and any closed interval K ⊂ ℝ of length b,

m\{y ∈ X : F^{T} (y) − T ∫ F dm − c\sqrt{T} ∈ K\} ∼ b\sqrt{\frac{H′′(∫ F dm)}{2π}}e^{−H′′(∫ F dm) \frac{c^{2}}{2}} \frac{1}{\sqrt{T}}

as T → ∞.

Proof. Let θ = ∫ F dm. Since θ is a local minimum of H, we have

H(θ) = 0 = H′(θ).

By the real analycity of H,

H (θ + \frac{c}{\sqrt{T}} ) = \frac{H′′(θ)}{2} \frac{c^{2}}{T} + O (\frac{1}{T^{3/2}} ) as T → ∞.

By making the simple observation that

m\{y ∈ X : F^{T} (y) − T θ − c\sqrt{T} ∈ K\} = m\{y ∈ X : F^{T} (y) − T (θ + \frac{c}{\sqrt{T}} ) ∈ K\},

we can apply Theorem 2 to deduce the result. (This uses the uniformity on compact subsets of Γ_{F}). ▷◁

The Central Limit Theorem now follows easily from Corollary 2. This result extends a result of Ratner [Ra] for Anosov flows.

Corollary 3. (Central Limit Theorem). For every c ∈ ℝ,

m \{y : (F^{T} (y) − T ∫ F dm)\sqrt{\frac{H′′(∫ F dm)}{T}} ≤ c\} ⟶ \frac{1}{\sqrt{2π}} ∫_{−∞}^{c} e^{− \frac{1}{2} u^{2}}du as T → ∞.

In other words, Corollary 3 states that the continuous time process (F ∘ Φ_{t})_{t≥0}, with stationary probability m, is asymptotically normally distributed, with mean ∫ F dm and variance \frac{1}{H′′(∫ F dm)}.

4. Perron Frobenius operators

In this section, we derive some elementary properties of Perron Frobenius operators for the Gauss map. For I = [0, 1], let C(I; ℂ) (respectively C(I; ℝ)) denote the Banach space of continuous complex (respectively real-valued) functions on I, equipped with the supremum norm

∥f∥_{∞} = sup_{x∈I} |f(x)|.

For α ∈ (0, 1), define the Hölder seminorm |.|_{α} by

|f|_{α} = sup \{\frac{|f(x) − f(y)|}{|x − y|^{α}} : x, y ∈ I, x ≠ y\} ,

for any f ∈ C(I; ℂ), and let

C^{α}(I; ℂ) = {f ∈ C(I; ℂ) : |f|_{α} < ∞}.

The space C^{α}(I; ℂ) is a Banach space when endowed with the norm

∥f∥_{α} = ∥f∥_{∞} + |f|_{α}.

The subspace

C^{α}(I; ℝ) = C^{α}(I; ℂ) ∩ C(I; ℝ)

is a real Banach space.

Definition 3. We say that ϕ ∈ 𝒢^{α}(I; ℝ) if the following conditions hold.

(a) ϕ : (0, 1] → ℝ is continuous.

(b) There exist constants D > 1, θ_{1} > θ_{2} > 1 such that for all x ∈ (0, 1],

\frac{1}{D} x^{θ_{1}} ≤ e^{ϕ(x)} ≤ Dx^{θ_{2}}.

(c) For all x, y ∈ I_{n},

|ϕ(x) − ϕ(y)| ≤ M_{n}|x − y|^{α}

where

M = sup_{n≥1} \{\frac{M_{n}}{n^{2α}} \} < ∞.

We say that ϕ ∈ 𝒢^{α}(I; ℂ) if ϕ : (0, 1] → ℂ is continuous and satisfies property (c) and Re ϕ satisfies (b). An example of a function in 𝒢^{α}(I; ℂ) is the map

ϕ(x) = s log x^{2}

for a fixed s ∈ ℂ with Re(s) > \frac{1}{2}.

Let B_{1}, B_{2} denote any of the function spaces C^{α}(I; ℂ), 𝒢^{α}(I; ℂ) or L_{l}^{1}(I). Two functions f, g ∈ B_{1} are said to be B_{2}-cohomologous if there exists a function w ∈ B_{2} such that

f = g + w ∘ T − w.

A function f ∈ B_{1} is called a B_{2}-coboundary if it is B_{2}-cohomologous to the function which is identically zero.

For ϕ ∈ 𝒢^{α}(I; ℂ), define the transfer operator

ℒ_{ϕ} : L_{l}^{1}(I; ℂ) → L_{l}^{1}(I; ℂ)

by

(ℒ_{ϕ}g)(x) = ∑_{n=1}^{∞} e^{ϕ(T_{n}x)}g(T_{n}x).  (4.1)

A straightforward calculation gives that

∥ℒ_{ϕ}∥_{1} ≤ D ∑_{n=1}^{∞} \frac{1}{n^{θ_{2}}} ≤ \frac{D}{θ_{2} − 1}.  (4.2)

Unless stated to the contrary, all L^{1} norms are with respect to Lebesgue measure.

Lemma 1. For ϕ ∈ 𝒢^{α}(I; ℂ), the operator ℒ_{ϕ} : C^{α}(I; ℂ) → C^{α}(I; ℂ) is well defined.

Proof. We need to show that if g ∈ C^{α}(I; ℂ) then ℒ_{ϕ}g ∈ C^{α}(I; ℂ). Firstly,

∥ℒ_{ϕ}g∥_{∞} ≤ ∥g∥_{∞} sup_{x∈I}|∑_{n=1}^{∞} e^{ϕ(T_{n}x)}|

≤ ∥g∥_{∞}D ∑_{n=1}^{∞} \frac{1}{(x + n)^{θ_{2}}}

≤ \frac{D∥g∥_{∞}}{θ_{2} − 1}

for some D > 0 and θ_{2} > 1. Secondly,

|(ℒ_{ϕ}g)(x) − (ℒ_{ϕ}g)(y)| ≤ ∑_{n=1}^{∞}|e^{ϕ(T_{n}x)} − e^{ϕ(T_{n}y)}||g(T_{n}x)| + ∑_{n=1}^{∞}|e^{ϕ(T_{n}x)}||g(T_{n}x) − g(T_{n}y)|

≤ ∥g∥_{∞} ∑_{n=1}^{∞} e^{Re ϕ(T_{n}x)}|ϕ(T_{n}x) − ϕ(T_{n}y)|e^{|ϕ(T_{n}x)−ϕ(T_{n}y)|}

+ ∑_{n=1}^{∞} e^{Re ϕ(T_{n}x)}|g(T_{n}x) − g(T_{n}y)|.

Then by applying our assumptions on ϕ, we obtain

|(ℒ_{ϕ}g)(x) − (ℒ_{ϕ}g)(y)| ≤ \frac{D}{θ_{2} − 1}(∥g∥_{∞} sup_{n∈ℕ}\{M_{n}|T_{n}x − T_{n}y|^{α}e^{M_{n}|T_{n}x−T_{n}y|^{α}}\}

+ |g|_{α} sup_{n∈ℕ}{|T_{n}x − T_{n}y|})

≤ \frac{D}{θ_{2} − 1}(∥g∥_{∞} sup_{n∈ℕ}\{\frac{M_{n}}{n^{2α}} e^{\frac{Mn}{n^{2α}}} \} + |g|_{α})|x − y|^{α}

≤ \frac{D}{θ_{2} − 1}(∥g∥_{∞}Me^{M} + |g|_{α})|x − y|^{α}

and hence that ℒ_{ϕ} ∈ B(C^{α}(I; ℂ)). ▷◁

We now give a version of the so called Ruelle-Perron-Frobenius Theorem for the transfer operator ℒ. (See [Ws], Theorems 8 and 22.) Proposition 2. Let ϕ ∈ 𝒢^{α}(I; ℝ) be given.

(i) There is a unique simple positive eigenvalue e^{P (ϕ)} of ℒ_{ϕ}, with corresponding strictly positive eigenfunction h_{ϕ} ∈ C(I; ℝ). Further, the function h_{ϕ} satisfies

h(x) ≤ h(y)e^{C_{ϕ}(x,y)}

for all x, y ∈ I, where

C_{ϕ}(x, y) = sup_{{n≥1 \atop b∈ℕ^{n}}} sup_{{x′=T_{b}x \atop y′=T_{b}y}} ∑_{i=0}^{n−1}(ϕ(T^{i}x′) − ϕ(T^{i}y′))

and

T_{b} = T_{b_{0}} ∘ T_{b_{1}} ∘ ⋯ ∘ T_{b_{n−1}}

for b = (b_{0}, b_{1}, …, b_{n−1}) ∈ ℕ^{n}.

(ii) There exists a unique Borel probability measure ν_{ϕ} on I satisfying both

ℒ_{ϕ}^{∗}ν_{ϕ} = e^{P (ϕ)}ν_{ϕ}

and ∫ h_{ϕ}dν_{ϕ} = 1. (iii) Uniformly for k ∈ C^{α}(I; ℂ),

\frac{ℒ_{ϕ}^{n}k}{e^{nP (ϕ)}} → h_{ϕ} ∫ kdν_{ϕ} as n → ∞.

(iv) The function ϕ has a unique equilibrium state μ_{ϕ}. That is, the measure μ_{ϕ} is the unique T -invariant Borel probability measure satisfying

P (ϕ) = h(μ_{ϕ}) + ∫ ϕdμ_{ϕ}.

Moreover, the measure μ_{ϕ} has no atoms and is positive on non-empty open subsets of I. Further, if ζ ∈ 𝒢^{α}(I; ℝ) then μ_{ϕ} = μ_{ζ} if and only if there exists c ∈ ℝ such that ϕ is C(I; ℝ)-cohomologous to ζ + c. The measures μ_{ϕ} and ν_{ϕ} are related by

\frac{dμ_{ϕ}}{dν_{ϕ}} = h_{ϕ}.

The following lemma is a simple consequence of Proposition 2.

Lemma 2. If ϕ ∈ 𝒢^{α}(I; ℝ) then h_{ϕ} ∈ C^{α}(I; ℝ).

Proof. For x, y ∈ I,

|C_{ϕ}(x, y)| ≤ sup_{{n≥1 \atop b∈ℕ^{n}}} sup_{{x′=T_{b}x \atop y′=T_{b}y}} ∑_{i=0}^{n−1} M_{b_{i}}|T^{i}x′ − T^{i}y′|^{α}

≤ M sup_{n≥1} sup_{b∈ℕ^{n}}(∑_{i=0}^{n−1} \frac{b_{i}^{2α}}{(b_{i}b_{i+1} … b_{n−1})^{2α}} )|x − y|^{α}

≤ K|x − y|^{α}

for a positive constant K. Hence

|h(x) − h(y)| ≤ |h(y)||e^{C_{ϕ}(x,y)} − 1|
≤ ∥h∥_{∞}|C_{ϕ}(x, y)|e^{|C_{ϕ}(x,y)|}
≤ ∥h∥_{∞}K|x − y|^{α}e^{K}. ▷◁

The next lemma contains some easy estimates which will be used frequently.

Lemma 3.

(i) For ϕ ∈ 𝒢^{α}(I; ℝ) or 𝒢^{α}(I; ℂ), there exist C > 0 and θ > 1 such that for all n ≥ 1,

∥ϕ ∘ T_{n}∥_{∞} ≤ C + θ log n.

(ii) Let ϕ ∈ 𝒢^{α}(I; ℝ), (resp. 𝒢^{α}(I; ℂ)). Then for all n ≥ 1, ϕ ∘ T_{n} ∈ C^{α}(I; ℝ), (resp. ϕ ∈ C^{α}(I; ℂ)). (iii) For any α ∈ (0, 1), we have r ∈ 𝒢^{α}(I; ℝ).

Proof. (i) This follows directly from the definition of 𝒢^{α}(I; ℝ) (resp. 𝒢^{α}(I; ℂ)). (ii) Suppose for example that ϕ ∈ 𝒢^{α}(I; ℂ). By (i), ϕ ∘ T_{n} ∈ C(I; ℂ). Further, for all n ≥ 1 and x, y ∈ I,

|ϕ(T_{n}x) − ϕ(T_{n}y)| ≤ M_{n}|T_{n}x − T_{n}y|^{α}

= M_{n}|\frac{1}{x + n} − \frac{1}{y + n}|^{α}

≤ \frac{M_{n}}{n^{2α}} |x − y|^{α}

≤ M|x − y|^{α}.

(iii) Conditions (a) and (b) in Definition 3 are easily seen to be satisfied. Let x, y ∈ I_{n} for n ≥ 1. Then

|r(x) − r(y)| = 2 |∫_{x}^{y} \frac{dt}{t}|

≤ 2(n + 1)|x − y|

≤ 2(n + 1) (\frac{1}{n(n + 1)})^{1−α} |x − y|^{α}

and

sup_{n∈ℕ} \frac{(n + 1)}{n^{2α}} (\frac{1}{n(n + 1)})^{1−α} = sup_{n∈ℕ} \frac{n + 1}{n^{2}} < ∞. ▷◁

Lemma 4. There exists a constant C > 0 such that for any n ≥ 1,

|ℒ_{ϕ}^{n}g|_{α} ≤ e^{nP (ϕ)}(C∥g∥_{∞} + \frac{1}{2^{n}} |g|_{α}).

Proof. By replacing ϕ by

ϕ − log h_{ϕ} ∘ T + log h_{ϕ} − P (ϕ),

we may assume that ϕ is normalised, that is

ℒ_{Re ϕ}1 = 1.

As in Lemma 2,

\begin{aligned}|(ℒ_{ϕ}^{2}g)(x) − (ℒ_{ϕ}^{2}g)(y)| & ≤ C_{2}∥g∥_{∞}|x − y|^{α} + |g|_{α} sup_{m,n∈ℕ}\{|T_{n,m}x − T_{n,m}y|\} \\ & ≤ (C_{2}∥g∥_{∞} + |g|_{α} \frac{1}{4^{α}} )|x − y|^{α} \end{aligned}

where we have used the fact that |(T^{2})′| > 4. Similarly, we also have

|(ℒ_{ϕ}g)(x) − (ℒ_{ϕ}g)(y)| ≤ (C_{1}∥g∥_{∞} + |g|_{α})|x − y|^{α}.

Proceeding by induction on even n and on odd n, we have

|(ℒ_{ϕ}^{2n}g)(x) − (ℒ_{ϕ}^{2n}g)(y)| ≤ (C_{2n}∥g∥_{∞} + \frac{1}{4^{nα}} |g|_{α})|x − y|^{α}

where for all n ≥ 2,

C_{2n} = C_{2} ∑_{j=1}^{n−1} \frac{1}{4^{jα}} ≤ C_{1} ∑_{j=1}^{∞} \frac{1}{4^{jα}} ,

and

|(ℒ_{ϕ}^{2n+1}g)(x) − (ℒ_{ϕ}^{2n+1}g)(y)| ≤ (C_{2n+1}∥g∥_{∞} + \frac{1}{4^{nα}} |g|_{α})|x − y|^{α},

where for all n ≥ 1,

C_{2n+1} ≤ C_{1} ∑_{j=1}^{∞} \frac{1}{4^{jα}} .

Finally, we take C = max{C_{1}, C_{2}}. ▷◁

5. The essential spectrum

In practice, we require a stronger form of Proposition 2 for the transfer operator. We will first briefly introduce the notion of the essential spectrum of a bounded linear operator on a Banach space.

Let B(V) denote the Banach algebra of bounded linear operators on a Banach space V, and let ℳ ∈ B(V). An element λ ∈ spectrum (ℳ) is said to lie in the essential spectrum, denoted by ess(ℳ), if λ is a limit point of spectrum(ℳ).

The essential spectral radius ρ_{e}(ℳ) is defined by

ρ_{e}(ℳ) = sup{|λ| : λ ∈ ess(ℳ)}.

Let ∥.∥_{e} denote the seminorm on B(V) given by

∥ℳ∥_{e} = inf{∥ℳ − K∥ : K : V → V is a compact linear operator}.

The essential spectral radius formula, due to Nussbaum, states that

ρ_{e}(ℳ) = lim_{n→∞}(∥ℳ^{n}∥_{e})^{\frac{1}{n}} .

In particular, the set ess(ℳ) does not contain any isolated eigenvalues of finite multiplicity. Further, for ε > 0, the set

{λ ∈ spectrum(ℳ) : |λ| ≥ ρ_{e}(ℳ) + ε}

consists of a finite set of eigenvalues λ_{1}, λ_{2}, …, λ_{d} say, each of which has finite multiplicity. If ρ_{e}(ℳ) < ρ(ℳ) then the operator ℳ is called quasi-compact.

Proposition 3. For any ϕ ∈ 𝒢^{α}(I; ℂ),

ρ_{e}(ℒ_{ϕ}) ≤ \frac{1}{2^{α}} e^{P (Reϕ)}.

Proof. We will assume that ϕ is normalised. By the Spectral Radius Formula and Lemma 4, we deduce immediately that

ρ(ℒ_{ϕ}) ≤ 1.

A cylinder set C(a_{0}, a_{1}, …, a_{n−1}) for a_{i} ∈ ℕ, i = 1, 2, …, n − 1, is a subset of I of the form

C(a_{0}, a_{1}, …, a_{n−1}) = {x ∈ I : x = [x_{0}, x_{1}, …] and x_{i} = a_{i} for 0 ≤ i ≤ n − 1}.

As usual, we may neglect points whose continued fraction transformations terminate after finitely many terms.

Now define a map τ_{n}^{N} : 𝒞_{n} → 𝒞_{n} on the set 𝒞_{n} of cylinders of length n, for each N ≥ 2, by

τ_{n}^{N}(C(a_{0}, a_{1}, …, a_{n−1})) = C(u_{N}(a_{0}), u_{N}(a_{1}), u_{N}(a_{2}), …, u_{N}(a_{n−1}))

where

u_{N}(a) = \{ \begin{matrix}a & if a_{i} < N \\ N & if a_{i} ≥ N. \end{matrix}

We now define expectation operators E_{n}^{N} : C^{α}(I; ℂ) → C^{α}(I; ℂ), with respect to the measure μ = μ_{Re ϕ} by

E_{n}^{N}(h) = ∑_{C∈𝒞_{n}} \frac{∫_{τ_{n}^{N} (C)} hdμ}{μ(τ_{n}^{N}(C))}χ_{C},

where χ_{C} denotes the characteristic function of the set C. For n ≥ 1, N ≥ 2, define a map t_{n}^{N} : I → I μ-almost everywhere, by

t_{n}^{N}(x) = [u_{N}(x_{0}), u_{N}(x_{1}), …, u_{N}(x_{n−1}), x_{n}, x_{n+1}, …]

where

x = [x_{0}, x_{1}, x_{2}, …].

For y, z ∈ C = C(a_{0}, a_{1}, …, a_{n−1}), we have

|h(y)−h(t_{n}^{N}(z))|

≤ max{sup{|h(y′) − h(t_{n}^{N}(z′))| : y′, z′ ∈ C, a_{i} < N for all 0 ≤ i ≤ n − 1},

sup{|h(y′) − h(t_{n}^{N}(z′))| : y′, z′ ∈ C, a_{i} > N for some 0 ≤ i ≤ n − 1}}

≤ |h|_{α} max\{\frac{1}{2^{nα}} , \frac{1}{N^{α}} \}.

In particular, it follows that

|h(y) − \frac{∫_{τ_{n}^{N} (C)} hdμ}{μ(τ_{n}^{N}(C))}| ≤ |h|_{α} max\{\frac{1}{2^{nα}} , \frac{1}{N^{α}} \}

and therefore

∥h − E_{n}^{N}(h)∥_{∞} ≤ |h|_{α} max\{\frac{1}{2^{nα}} , \frac{1}{N^{α}} \}.  (5.1)

It is also simple to show that

|h − E_{n}^{N}(h)|_{α} ≤ |h|_{α}.  (5.2)

The operator K_{n}^{N} = ℒ_{ϕ}^{n}E_{n}^{N} is compact, as the operator E_{n}^{N} has finite dimensional range, and furthermore, by Lemma 4, (5.1) and (5.2), for h ∈ C^{α}(I; ℂ),

|(ℒ_{ϕ}^{n} − K_{n}^{N})h|_{α} = |ℒ_{ϕ}^{n}(h − E_{n}^{N}h)|_{α}

= C∥h − E_{n}^{N}h∥_{∞} + \frac{1}{2^{nα}} |h − E_{n}^{N}h|_{α}

≤ |h|_{α}(C max\{\frac{1}{2^{nα}} , \frac{1}{N^{α}} \} + \frac{1}{2^{nα}} ).

Also, by (5.1) we have

∥(ℒ_{ϕ}^{n} − K_{n}^{N})h∥_{∞} ≤ |h|_{α} max\{\frac{1}{2^{nα}} , \frac{1}{N^{α}} \}

and hence

∥ℒ_{ϕ}^{n} − K_{n}^{N}∥_{α} ≤ (C + 1) max\{\frac{1}{2^{nα}} , \frac{1}{N^{α}} \} + \frac{1}{2^{nα}} .

Finally, we choose 0 < γ < \frac{1}{2} and let

N = [\frac{1}{γ^{n}} ] ,

so that

∥ℒ_{ϕ}^{n} − K_{n}^{N}∥_{α} ≤ \frac{1}{2^{nα}} (1 + (1 + C)(1 + (2γ)^{nα}))

and hence

ρ_{e}(ℒ_{ϕ}) ≤ \frac{1}{2^{α}} . ▷◁

Combining Propositions 2,3 as well as the remarks preceding Proposition 3, we obtain the following proposition.

Proposition 4. For ϕ ∈ 𝒢^{α}(I; ℝ), there exist κ ∈ (0, 1) and C > 0 such that for all k ∈ C^{α}(I; ℂ),

∥\frac{ℒ_{ϕ}^{n}k}{e^{nP (ϕ)}} − h_{ϕ} ∫ kdν_{ϕ}∥_{α} ≤ Cκ^{n}.

We now consider the regularity of the pressure function P. For this, we require the following standard perturbation lemma. (See for example Appendix V of [PP].)

Lemma 5. Let V be a Banach space. Let δ_{0} > 0 and suppose that L : B(1; δ_{0}) → B(V) is analytic. If L(1) ∈ B(V) has a simple isolated maximal eigenvalue λ(1), with corresponding eigenvector v(1), then for all ε > 0, there exists 0 < δ < δ_{0} such that if z ∈ B(1; δ), then L(z) has a simple isolated maximal eigenvalue λ(z) with corresponding eigenvector v(z) such that the following two properties are satisfied:

(i) The maps B(1; δ) → ℂ and B(1; δ) → V given by z ↦ λ(z) and z ↦ v(z) respectively, are analytic. (ii) For z ∈ B(1; δ), |λ(z) − λ(1)| < ε and

spectrum(L(z)) \ {λ(z)} ⊆ {w : |w − λ(1)| > ε}

for some ε > 0.

We now state our main result on the analycity of pressure.

Proposition 5. Let ϕ ∈ 𝒢^{α}(I; ℝ) be given. Then there exists an open neighbourhood of 1 ∈ ℂ, denoted by N(ϕ), such that the maps N(ϕ) → ℂ and N(ϕ) → C^{α}(I; ℂ) defined respectively by s ↦ e^{P (sϕ)} and s ↦ h_{sϕ}, are analytic, and furthermore,

ℒ_{sϕ}h_{sϕ} = e^{P (sϕ)}h_{sϕ}

holds for all s ∈ N(ϕ).

Before proving Proposition 5, we require two preparatory lemmas. Firstly, for any ϕ ∈ 𝒢^{α}(I; ℝ) define a map η : B(1; δ) → B(C^{α}(I; ℂ)) for any small δ > 0 by

η(s) = ℒ_{sϕ}.

Lemma 6. There exists δ > 0 such that for any k ≥ 1 and any s ∈ B(1; δ),

η^{(k)}(s) = ℒ_{sϕ} ∘ (R_{ϕ})^{k} ∈ B(C^{α}(I; ℂ))  (5.3)

where R_{ϕ} : C^{α}(I; ℂ) → 𝒢^{α}(I; ℂ) is the multiplication operator given by (R_{ϕ}g)(x) = ϕ(x)g(x), for x ∈ (0, 1].

Proof. First of all, by using the same argument as in Lemma 1, there is a δ_{3} > 0 such that

ℒ_{sϕ} ∘ (R_{ϕ})^{k} ∈ B(C^{α}(I; ℂ))

for any k ≥ 1 and s ∈ B(1; δ_{3}). We will prove the identity (5.3) only in the case k = 1. The argument can then be completed by induction on k.

For s, w ∈ B(1; δ_{3}), define

Υ_{w,s} = \frac{η(w) − η(s)}{w − s} − γ(s)

where

γ(s) = ℒ_{sϕ} ∘ R_{ϕ}.

Let ε > 0 be arbitrary. Let

ξ = inf{t ∈ ℝ : ∥ℒ_{tϕ}∥_{α} < ∞}

and by the hypothesis on ϕ, we have ξ < 1. For s ∈ B(1; δ_{3}) let δ_{0} = \frac{1}{2}|Re(s) − ξ|. Then we may choose a constant K_{1} > 0 such that

sup_{x∈I} {∑_{n=1}^{∞} e^{−Re(w)|ϕ(T_{n}x)|}ϕ(T_{n}x)^{2}} ≤ K_{1}

for any w ∈ \overline{B(s; δ_{0})}. Now choose

δ_{1} = min\{δ_{0}, \frac{ε}{2K_{1}} \}.

Using the elementary inequality

|e^{z} − 1 − z| ≤ |z|^{2}e^{|z|} for z ∈ ℂ,

if |s − w| < δ_{1}, then

|(Υ_{w,s}g)(x)| ≤ ∥g∥_{∞} ∑_{n=1}^{∞} e^{Re(s)ϕ(T_{n}x)}|\frac{e^{(w−s)ϕ(T_{n}x)} − 1 − (w − s)ϕ(T_{n}x)}{w − s}|

≤ ∥g∥_{∞}|w − s| ∑_{n=1}^{∞} e^{(−|w−s|+Re(s))|ϕ(T_{n}x)|}ϕ(T_{n}x)^{2}

≤ |w − s|K_{1}∥g∥_{∞}

< \frac{ε}{2}∥g∥_{∞},

and hence

∥Υ_{w,s}g∥_{∞} < \frac{ε}{2}∥g∥_{∞}.

Now let s, δ_{0}, K be as above and choose

δ_{2} = min{δ_{0}, K_{1}M(2 + (2δ_{0} + 1)e^{(1+2δ_{0})M})}.

Then by proceeding as above, we have that for any g ∈ C^{α}(I; ℂ) with ≢ 0, and |s−w| < δ_{2},

|(Υ_{w,s}g)(x) − (Υ_{w,s}g)(y)|

≤ |w − s||∑_{n=1}^{∞} e^{−(δ_{0}+Re(s))|ϕ(T_{n}x)|}ϕ(T_{n}x)^{2}g(T_{n}x)

− ∑_{n=1}^{∞} e^{−(δ_{0}+Re(s))|ϕ(T_{n}y)|}ϕ(T_{n}y)^{2}g(T_{n}y)|

≤ |w − s||∑_{n=1}^{∞} e^{−(δ_{0}+Re(s))|ϕ(T_{n}x)|}(ϕ(T_{n}x)^{2}g(T_{n}x) − ϕ(T_{n}y)^{2}g(T_{n}y))|

+ |w − s||∑_{n=1}^{∞} e^{−(δ_{0}+Re(s))|ϕ(T_{n}x)|}ϕ(T_{n}y)^{2}g(T_{n}y)(1 − e^{(Re(s)+δ_{0})(|ϕ(T_{n}x)|−|ϕ(T_{n}y)|)})|

After a simple rearrangement we then obtain

|(Υ_{w,s}g)(x)−(Υ_{w,s}g)(y)|

≤ |w − s| ∑_{n=1}^{∞} e^{−(δ_{0}+Re(s))|ϕ(T_{n}x)|}(|ϕ(T_{n}x)^{2} − ϕ(T_{n}y)^{2}||g(T_{n}x)|

+ ϕ(T_{n}x)^{2}|g(T_{n}x) − g(T_{n}(y)|

+ ϕ(T_{n}y)^{2}|g(T_{n}y)||1 − e^{(Re(s)+δ_{0})(|ϕ(T_{n}x)|−|ϕ(T_{n}y)|)}|)

Finally, by applying our assumptions on ϕ and g, we obtain

|(Υ_{w,s}g)(x)−(Υ_{w,s}g)(y)|

≤ |w − s|K_{1}M(2∥g∥_{∞} + |g|_{α} + (1 + 2δ_{0})∥g∥_{∞}e^{(1+2δ_{0})M})|x − y|^{α} ≤ |w − s|K_{1}M∥g∥_{α}(2 + (1 + 2δ_{0})e^{(1+2δ_{0})M})|x − y|^{α}

< \frac{ε}{2}∥g∥_{α}

and hence

∥Υ_{w,s}g∥_{α} < ε∥g∥_{α}

provided that |s − w| < min{δ_{1}, δ_{2}}. ▷◁

We continue with the definition of δ_{0} as given in Lemma 6. Lemma 7. For any s ∈ B(1; δ_{0}), there exists ε > 0 and K_{2} > 0 such that for all k ≥ 1,

∥η^{(k)}(s)∥_{α} ≤ \frac{K_{2}k!}{ε^{k}}.

Proof. For g ∈ C^{α}(I; ℂ) with g ≢ 0, we have

\begin{matrix}|(η^{(k)}(s)g)(x)− & (η^{(k)}(s)g)(y)| \\ ≤ |∑_{n=1}^{∞} e^{−sϕ(T_{n}x)}g(T_{n}x)(ϕ(T_{n}x)^{k} − ϕ(T_{n}y)^{k})| \\ + |∑_{n=1}^{∞}(e^{−sϕ(T_{n}x)}g(T_{n}x) − e^{−sϕ(T_{n}y)}g(T_{n}y))ϕ(T_{n}y)^{k}|. \end{matrix} (5.4)

We consider the two terms in (5.4) separately. Firstly by Lemma 3(i),

|∑_{n=1}^{∞}e^{−sϕ(T_{n}x)}g(T_{n}x)(ϕ(T_{n}x)^{k} − ϕ(T_{n}y)^{k})|

≤ M|x − y|^{α}∥g∥_{∞} ∑_{n=1}^{∞} e^{−Re(s)ϕ(T_{n}x)}|∑_{j=1}^{k−1} ϕ(T_{n}x)^{k−j}ϕ(T_{n}y)^{j}|

≤ M|x − y|^{α}∥g∥_{∞}K_{3}k ∑_{n=1}^{∞} \frac{(C + θ log n)^{k}}{n^{θRe(s)}}

for constants K_{3} > 0, C > 0 and θ > 1. Using the estimate

∑_{n=1}^{∞} \frac{(C + θ log n)^{k}}{n^{θRe(s)}} ≤ K_{3}γ^{k} ∑_{n=1}^{∞} \frac{(log n)^{k}}{n^{θRe(s)}}

≤ K_{3}γ^{k} ∫_{1}^{∞} \frac{(log u)^{k}}{u^{θRe(s)}} du

≤ \frac{K_{4}γ^{k}}{(Re(s) − θ^{−1})^{k}}

for a constant K_{4} > 0 independent of s, k, and a constant γ > 1. Thus we have

\begin{matrix}|∑_{n=1}^{∞} e^{−sϕ(T_{n}x)}g(T_{n}x) & (ϕ(T_{n}x)^{k} − ϕ(T_{n}y)^{k})| \\ ≤ K_{4}M∥g∥_{∞} \frac{kγ^{k}}{(Re(s) − θ^{−1})^{k}} |x − y|^{α}. \end{matrix}

For the second term in (5.4), we have

|∑_{n=1}^{∞}(e^{−sϕ(T_{n}x)}g(T_{n}x) − e^{−sϕ(T_{n}y)}g(T_{n}y))ϕ(T_{n}y)^{k}|

≤ ∑_{n=1}^{∞} e^{−Re(s)ϕ(T_{n}x)}(|g(T_{n}x) − g(T_{n}y)|

+|g(T_{n}x)||e^{−Re(s)(ϕ(T_{n}x)−ϕ(T_{n}y))} − 1|)(C + θ log n)^{k}

≤ (|g|_{α} + ∥g∥_{∞}M|Re(s)|e^{M|Re(s)|})|x − y|^{α} ∑_{n=1}^{∞} e^{−Re(s)ϕ(T_{n}x)}(C + θ log n)^{k}

≤ ∥g∥_{α}K_{6} \frac{kγ^{k}}{(Re(s) − θ^{−1})^{k}} |x − y|^{α}

by proceeding as before. Here K_{6} > 0 does not depend on s or k. Thus, we conclude that

|η^{(k)}(s)g|_{α} ≤ ∥g∥_{α}K_{6} \frac{kγ^{k}}{(Re(s) − θ^{−1})^{k}} .  (5.5)

A similar estimate shows that

∥η^{(k)}(s)g∥_{∞} ≤ ∥g∥_{∞} \frac{kγ^{k}}{(Re(s) − θ^{−1})^{k}} .  (5.6)

Thus by combining (5.5) and (5.6), we have

\begin{aligned}∥η^{(k)}(s)∥_{α} & ≤ \frac{K_{7}kγ^{k}}{(Re(s) − θ^{−1})^{k}} \\ & ≤ \frac{K_{7}k!}{ε^{k}} \end{aligned}

where K_{7} = 2K_{6} and

ε = \frac{(Re(s) − θ^{−1})}{γ} > 0.

This completes the proof of the Lemma. ▷◁

Proof of Proposition 5. Let ϕ ∈ 𝒢^{α}(I; ℝ) be given. By Lemmas 6,7, there exists δ_{0} > 0 such that the map s ↦ η(s) is analytic for s ∈ B(1; δ_{0}). The conclusions of the proposition now follow directly from Lemma 5. ▷◁

Let ϕ ∈ 𝒢^{α}(I; ℝ) be given. Using Proposition 5, we may define P (sϕ) to be the principal branch of log(e^{P (sϕ)}) for each s ∈ N(ϕ). Then s ↦ P (sϕ) is analytic as a map N(ϕ) → ℂ.

Let M(I) denote the space of Borel probability measures on I. Using standard perturbation theory, we may define a weak* analytic map N(ϕ) → M(I) by s ↦ ν_{sϕ}, such that

ℒ_{sϕ}^{∗}ν_{sϕ} = e^{P (sϕ)}ν_{sϕ}

and

∫ h_{sϕ}dν_{sϕ} = 1

for all s ∈ N(ϕ). (The map s ↦ ν_{sϕ} is weak* analytic means that for each g ∈ C^{α}(I; ℂ), the map N(ϕ) → ℂ given by

s ↦ ∫ gdν_{sϕ}

is analytic.)

Proposition 6. Let ϕ ∈ 𝒢^{α}(I; ℂ) be given.

(i) For a ∈ [0, 2π) and g ∈ L_{l}^{2}(I; ℂ), the following statements are equivalent.

(a) ℒ_{ϕ}g = e^{P (Re ϕ)+ia}g.

(b) g ∘ T = e^{−i(a+(Im ϕ))}g.

(ii) ρ(ℒ_{ϕ}) ≤ e^{P (Re ϕ)}, and if ℒ_{ϕ} has no eigenvalues of modulus e^{P (Re ϕ)} then additionally it follows that ρ(ℒ_{ϕ}) < e^{P (Re ϕ)}.

(iii) If there g ∈ L_{l}^{2}(I; ℂ) is an eigenfunction of ℒ_{ϕ}, then it can be chosen in C^{α}(I; ℂ).

The proof of the equivalence of statements (a) and (b) follows exactly as in Proposition 6.2 of [PP], by applying the convexity result given in Lemma 8. Part (ii) follows immediately from the quasi-compactness of ℒ_{ϕ}. Part (iii) is again proved by applying the argument in Proposition 6.2 of [PP].

Lemma 8. Let (a_{n})_{n=1}^{∞} be a sequence of points in (0, 1) satisfying

∑_{n=1}^{∞} a_{n} = 1.

Let (x_{n})_{n=1}^{∞} be a sequence of points on the unit circle S^{1}, and suppose that

∑_{n=1}^{∞} a_{n}x_{n} = y

for some y ∈ S^{1}. Then x_{n} = y for all n ≥ 1.

Proof. For each n ≥ 1, we may write

y = a_{n}x_{n} + (1 − a_{n}) ∑_{{k=1 \atop k≠n}}^{∞} \frac{a_{k}}{(1 − a_{n})}x_{k}  (5.5)

where

|∑_{{k=1 \atop k≠n}}^{∞}(\frac{a_{k}}{1 − a_{n}} )x_{k}| ≤ 1.

Thus (5.5) expresses y ∈ S^{1} as a strict convex combination of points in the closed unit disc. We conclude that

(1 − a_{n})x_{n} = ∑_{{k=1 \atop k≠n}}^{∞} a_{k}x_{k}

or equivalently , y = x_{n}. ▷◁

Motivated by Proposition 6, now define a notion of independence for functions in 𝒢^{α}(I; ℂ).

Definition 4. (i) Let ϕ_{1}, ϕ_{2} ∈ 𝒢^{α}(I; ℂ), with Im(ϕ_{i}) ≢ 0 for i = 1, 2, be given. Then ϕ_{1}, ϕ_{2} are said to be T -independent if whenever there exist γ ∈ L_{l}^{2}(I; ℂ) and λ_{1}, λ_{2} ∈ ℝ such that

γ ∘ T = e^{i(λ_{1}Im(ϕ_{1})+λ_{2}Im(ϕ_{2}))}γ

then λ_{1} = 0 = λ_{2}. (ii) A function ϕ ∈ 𝒢^{α}(I; ℂ) with Im(ϕ) ≢ 0 is called T -non-lattice if whenever there exist λ ∈ ℝ, γ ∈ L_{l}^{2}(I; ℂ) and c ∈ [0, 2π) such that

γ ∘ T = e^{i(λIm(ϕ)+c)}γ.

then λ = 0 = c. (In other words, a function is non-lattice if it is independent of constant functions.) Let Λ^{α}(I; ℂ) denote the set of lattice functions (i.e. those functions which are not non-lattice).

We can extend the foregoing definition of pressure for real valued functions to Λ^{α}(I; ℂ) by defining

P (ϕ) = P (Re ϕ) + c

whenever

γ ∘ T = e^{i(Im(ϕ)+c)}γ

for γ ∈ L_{l}^{2}(I; ℂ) and c ∈ [0, 2π). Similarly, we can define h_{ϕ} = h_{Re(ϕ)}, ν_{ϕ} = ν_{Re(ϕ)} and N(ϕ) = N(Re ϕ). In particular, the map N(ϕ) → ℂ given by s ↦ P (sϕ) is analytic for ϕ ∈ Λ^{α}(I; ℂ).

The following proposition is an extension of Propositions 4,6, and can be proved by using standard arguments in Perturbation Theory. Analogous results have been obtained for subshifts of finite type in [La2], and the same arguments are applicable in this setting.

Proposition 7. Let B ⊂ C^{α}(I; ℂ) be compact.

(i) Let ϕ ∈ Λ^{α}(I; ℂ) and let K_{1} ⊂ N(ϕ) ⊂ ℂ be compact. Then there exists δ_{1} > 0 such that

(1 + δ_{1})^{n} ∥\frac{ℒ_{sϕ}^{n}k}{e^{nP (sϕ)}} − h_{sϕ} ∫ kdν_{sϕ}∥_{α} ⟶ 0 as n → ∞,

uniformly for s ∈ K_{1}, k ∈ B. (ii) Let ϕ ∈ 𝒢^{α}(I; ℝ) and let K_{2} be a compact subset of ℂ \ ℝ such that sϕ ∉ Λ^{α}(I; ℂ) for all s ∈ K_{2}. Then there exists δ_{2} > 0 such that

(1 + δ_{2})^{n} \frac{∥ℒ_{sϕ}^{n}k∥_{α}}{e^{nP (Re(s)ϕ)}} ⟶ 0 as n → ∞,

uniformly for all s ∈ K_{2}, k ∈ B.

6. The entropy function

In this section we define the entropy or rate function. First we define a new function space ℋ^{α}(I; ℝ). This is identical to the definition of 𝒢^{α}(I; ℝ), except for the restriction on θ_{1} and θ_{2}.

Definition 5. We say that ϕ ∈ ℋ^{α}(I; ℝ) if the following properties are satisfied.

(i) ϕ : (0, 1] → ℝ is continuous. (ii) There exist constants D > 1, and θ_{1}, θ_{2} ∈ ℝ with θ_{1} > θ_{2}, such that

\frac{1}{D} x^{θ_{1}} ≤ e^{ϕ(x)} ≤ Dx^{θ_{2}}

for all x ∈ (0, 1]. (iii) For all x, y ∈ I_{n},

|ϕ(x) − ϕ(y)| ≤ M_{n}|x − y|^{α}

where

M = sup_{n≥1} \{\frac{M_{n}}{n^{2α}} \} < ∞.

Throughout this the remainder to this section, we assume that ϕ ∈ ℋ^{α}(I; ℝ) is given. First we require an elementary lemma.

Lemma 9. There exists a unique real number t such that

P (ϕ − tr) = 0.

Proof. By Lemma 3(iii), for all t ∈ ℝ sufficiently large, we have ϕ − tr ∈ 𝒢^{α}(I; ℝ). Since r > 0 l-almost everywhere, by the variational principle,

P (ϕ − tr) = h(μ_{ϕ−tr}) + ∫ (ϕ − tr)dμ_{ϕ−tr} → −∞ as t → ∞.

Further, by Proposition 2(iii),

P (ϕ − tr) = lim_{k→∞} \frac{1}{k} log ℒ_{ϕ−tr}^{k}1.  (6.1)

Then there exist θ ∈ ℝ and a constant C > 0 such that for any x ∈ I,

\begin{aligned}(ℒ_{ϕ−tr}1)(x) & = ∑_{n=1}^{∞} exp\{(ϕ − tr)(\frac{1}{x + n})\} \\ & ≥ C^{2} ∑_{n=1}^{∞} \frac{1}{(1 + n)^{θ+2t}} \end{aligned}

the latter term tending to infinity as t tends to \frac{1}{2}(1−θ) from above. By (6.1), we therefore have

(ℒ_{ϕ−tr}^{k}1)(x) ≥ (C^{2} ∑_{n=1}^{∞} \frac{1}{(1 + n)^{θ+2t}} )^{k}

for all x ∈ I, all k ≥ 1 and hence

P (ϕ − tr) ≥ C^{2} ∑_{n=1}^{∞} \frac{1}{(1 + n)^{θ+2t}} .

By the continuity of pressure, there exists t such that P (ϕ − tr) = 0. Also, since r > 0 l-almost everywhere, the map t ↦ P (ϕ − tr) is strictly monotonic decreasing, and hence the solution is unique. ▷◁

By Lemma 9, we may thus define a function β : ℝ → ℝ implicitly by

P (tϕ − β(t)r) = 0

for all t ∈ ℝ. By the Implicit Function Theorem and Proposition 5, the function β is real analytic. Letting μ_{t} denote the measure μ_{tϕ−β(t)r}, standard calculations ([Ru1] or [PP] section five), give

β′(t) = \frac{∫ ϕdμ_{t}}{∫ rdμ_{t}}

and

β′′(t) = \frac{1}{∫ rdμ_{t}} ∑_{n=0}^{∞} ∫ (ϕ − μ_{t}(ϕ))(x)(ϕ − μ_{t}(ϕ))(T^{n}x)dμ_{t}(x).

Lemma 10. Suppose that ϕ satisfies the following condition: if there exist λ_{1}, λ_{2} ∈ ℝ and γ ∈ L_{l}^{2}(I; ℂ) such that

γ ∘ T = e^{i(λ_{1}ϕ+λ_{2}r)}γ

then λ_{1} = 0 = λ_{2}. Then β′′(t) > 0 for all t ∈ ℝ.

Proof. If ϕ satisfies the hypothesis of the lemma, then for all t ∈ ℝ, tϕ − β(t)r is not cohomologous to zero. Then the result follows by a standard argument. See for example [PP], Proposition 4.12. ▷◁

We now proceed under the additional assumption that ϕ satisfies the hypothesis in Lemma 10. By Lemma 10, the map ℝ → ℝ given by t ↦ β′(t) is strictly increasing. Let Γ_{ϕ} ⊂ ℝ be defined by

Γ_{ϕ} = {β′(t) : t ∈ ℝ}.

Then for each a ∈ Γ_{ϕ}, there exists a unique b(a) ∈ ℝ such that β′(b(a)) = a. The function b : Γ_{ϕ} → ℝ is strictly increasing, surjective and real analytic. We define a function H : ℝ → ℝ by

H(a) = sup{at − β(t) : t ∈ ℝ}

where the supremum may take the value ∞. Standard properties of the Legendre transform then give that

H(a) = \{ \begin{matrix}ab(a) − β(b(a)) & for a ∈ Γ_{ϕ} \\ ∞ & for a ∉ Γ_{ϕ}. \end{matrix}

By the Inverse Function Theorem,

b′(a) = \frac{1}{β′′(b(a))}.

Thus we have that

\begin{aligned}H′(a) & = ab′(a) + a − β′(b(a))b′(a) \\ & = b(a) \end{aligned}

and in particular, H′(a) = 0 if and only if a = β′(0) = ^{\frac{∫ ϕdρ}{∫ rdρ}}. Furthermore, H′′(a) > 0

since b is strictly increasing. We conclude that H is a strictly convex, non-negative function with a unique minimum at a = β′(0).

7. Cohomology

Let F ∈ ℱ^{α}(X; ℝ) be given, and define functions f ∈ L_{l^{2}}^{2}(I^{2}; ℝ) and f_{δ} ∈ C(I^{2}; ℝ) by

f(z) = ∫_{0}^{r(z)} F (z; t)dt and f_{δ}(z) = ∫_{0}^{r_{δ}(z)} F (z; t)dt

respectively, where δ ∈ (0, 1). Recall that r_{δ} : I → ℝ was defined in (3.1). We remark that r_{δ} ∈ C^{α}(I; ℝ), and that the following elementary estimates hold.

∥r_{δ}∥_{∞} = 2(1 − log δ)  (7.1)
|r_{δ}|_{α} ≤ \frac{2}{δ}  (7.2)
∥r − r_{δ}∥_{1} = δ  (7.3)
∥r − r_{δ}∥_{2} ≤ Cδ^{\frac{1}{3}}  (7.4)

for a constant C > 0.

Let C^{α}(I^{2}; ℂ) and C^{α}(I^{2} : ℝ) denote respectively the Banach spaces of complex and real valued α-Hölder continuous functions on I^{2}. The following result extends a theorem of Sinai ([Si]).

Proposition 8. For δ ∈ (0, 1), there exist functions g_{δ} ∈ C^{\frac{α}{2}} (I^{2}; ℝ) and k_{δ} ∈ C^{\frac{α}{2}} (I^{2}; ℝ) such that g_{δ} depends only on the first coordinate (i.e. g_{δ}(x, y_{1}) = g_{δ}(x, y_{2}) for all x ∈ I, y_{1}, y_{2} ∈ I), and

f_{δ} = g_{δ} + k_{δ} ∘ S − k_{δ}.

There also exist functions k ∈ L_{l^{2}}^{2}(I^{2}; ℝ), g ∈ L_{l^{2}}^{2}(I^{2}; ℝ) and positive constants K_{1}, K_{2} such that

∥g − g_{δ}∥_{2} ≤ K_{1}δ^{\frac{1}{3}} , ∥k − k_{δ}∥_{2} ≤ K_{2}δ^{\frac{1}{3}}

and

f = g + k ∘ S − k.

Moreover, g = g(x, y) depends only on the first coordinate and g ∈ ℋ^{α/2}(I; ℝ). The first step in the proof of Proposition 8 is to convert our assumption on F to a condition on f. Lemma 12. There exist constants C_{1}, C_{2}, C_{3} > 0 such that for all δ ∈ (0, 1),

(i) |f_{δ}|_{α} ≤ \frac{C_{1}}{δ} (ii) ∥f − f_{δ}∥_{2} ≤ C_{2}δ^{\frac{1}{3}} . (iii) ∥f − f_{δ}∥_{1} ≤ C_{3}δ. (iv) ∥f_{δ}∥_{∞} ≤ 2C_{F} (1 − log δ). Proof. (i) For any z_{1}, z_{2} ∈ I^{2},

|f_{δ}(z_{1}) − f_{δ}(z_{2})|

= |∫_{0}^{r_{δ}(z_{1})} F (z_{1}; t)dt − ∫_{0}^{r_{δ}(z_{2})} F (z_{2}; t)dt|

= |r_{δ}(z_{1}) ∫_{0}^{1} F (z_{1}; tr_{δ}(z_{1}))dt − r_{δ}(z_{2}) ∫_{0}^{1} F (z_{2}; tr_{δ}(z_{2}))dt|

≤ |r_{δ}(z_{1}) − r_{δ}(z_{2})||∫_{0}^{1} F (z_{1}; tr_{δ}(z_{1}))dt|

+ |r_{δ}(z_{2})||∫_{0}^{1} F (z_{1}; tr_{δ}(z_{1}))dt − ∫_{0}^{1} F (z_{2}; tr_{δ}(z_{2}))dt|

≤ |r_{δ}|_{α}(sup_{y∈X} |F (y)|)d_{E}(z_{1}, z_{2})^{α} + ∥r_{δ}∥_{∞} max_{t∈I}|F (z_{1}; tr_{δ}(z_{1})) − F (z_{1}; tr_{δ}(z_{2}))|.

where d_{E} denotes the Euclidean metric on I^{2}. But by Definition 1,

max_{t∈[0,1]}|F (z_{1}; tr_{δ}(z_{1})) − F (z_{2}; tr_{δ}(z_{2}))| ≤ \frac{K_{F}}{δ^{α′}} d_{E}(z_{1}, z_{2})^{α}.

Combining the two previous estimates, we have that

|f_{δ}(z_{1}) − f_{δ}(z_{2})| ≤ (|r_{δ}|_{α}( sup_{y∈T_{1}M} |F (y)|) + \frac{C}{δ^{α′}} ∥r_{δ}∥_{∞})d_{E}(z_{1}, z_{2})^{α}

≤ (\frac{2}{δ} sup_{y∈T_{1}M} |F (y)| + 2\frac{C}{δ^{α′}} (1 − log δ))d_{E}(z_{1}, z_{2})^{α}

≤ \frac{2C}{δ} d_{E}(z_{1}, z_{2})^{α}.

(ii) We use the estimate

|f(z) − f_{δ}(z)| = |∫_{0}^{r(z)} F (z; t)dt − ∫_{0}^{r_{δ}(z)} F (z; t)dt|

= |∫_{r_{δ}(z)}^{r(z)} F (z; t)dt|

≤ (sup_{y∈X} |F (y)|)|r(z) − r_{δ}(z)|

for l^{2}- almost all z ∈ I^{2}. Hence

\begin{aligned}∥f − f_{δ}∥_{2} & ≤ C_{F} ∥r − r_{δ}∥_{2} \\ & ≤ Cδ^{\frac{1}{3}} \end{aligned}

as required. (iii) This is proved using a similar argument to (ii). (iv) This follows from (7.1). ▷◁

Proof of Proposition 8. By a change of variables, we may rewrite f_{δ} as

f_{δ}(x, y) = r_{δ}(x) ∫_{0}^{1} F (x, y; tr_{δ}(x))dt.

By Definition 1(iii), for all z_{1}, z_{2} ∈ I^{2},

sup_{t∈[0,1]} |F (z_{1}; tr_{δ}(z_{1})) − F (z_{2}; tr_{δ}(z_{2}))| ≤ \frac{K_{F}}{δ^{α′}} d_{E}(z_{1}, z_{2})^{α}.

By using the argument of Proposition 1.2 in [PP], there exist bounded continuous functions G, K, independent of δ, such that for all t ∈ [0, 1] and all z ∈ I^{2},

F (z; tr_{δ}(z)) = G(z; tr_{δ}(z)) + K(S(z); tr_{δ}(Sz)) − K(z; tr_{δ}(z)).

Again by Proposition 1.2 in [PP], the Hölder norms of G(.; tr_{δ}(.)) and K(.; tr_{δ}(.)) can be estimated from above by the Hölder norm of F (.; r_{δ}(.)). Thus there exist constants L, M > 0 such that for all z_{1}, z_{2} ∈ I^{2},

sup_{t∈[0,1]} |G(z_{1}; tr_{δ}(z_{1})) − G(z_{2}; tr_{δ}(z_{2}))| ≤ \frac{L}{δ^{α′}} d_{E}(z_{1}, z_{2})^{α/2}

and

sup_{t∈[0,1]} |K(z_{1}; tr_{δ}(z_{1})) − K(z_{2}; tr_{δ}(z_{2}))| ≤ \frac{M}{δ^{α′}} d_{E}(z_{1}, z_{2})^{α/2}

where

G(x, y_{1}; tr_{δ}(x)) = G(x, y_{2}; tr_{δ}(x))

for all x, y_{1}, y_{2} ∈ I and t ∈ [0, 1]. By integration, we obtain the identity

f_{δ} = g_{δ} + k_{δ} ∘ S − k_{δ}  (7.5)

where

g_{δ}(z) = r_{δ}(z) ∫_{0}^{1} G(z; tr_{δ}(z))dt

and

k_{δ}(z) = r_{δ}(z) ∫_{0}^{1} K(z; tr_{δ}(z))dt.

Moreover, g_{δ} depends only on the first coordinate, and g_{δ}, k_{δ} ∈ C^{α/2}(I^{2}; ℝ). Next note that for δ_{1}, δ_{2} ∈ (0, 1),

\begin{aligned}|g_{δ_{1}}(z) − g_{δ_{2}}(z)| & = | ∫_{0}^{r_{δ_{1}} (z)} G(z; t)dt − ∫_{0}^{r_{δ_{2}} (z)} G(z; t)dt| \\ & ≤ C_{F} |r_{δ_{1}}(z) − r_{δ_{2}}(z)| \end{aligned}

and so by using (7.4), we have that

∥g_{δ_{1}} − g_{δ_{2}}∥_{2} ≤ C|δ_{1} − δ_{2}|^{1/3}.

Hence (g_{δ}) is a Cauchy sequence in L_{l^{2}}^{2}(I^{2}; ℝ), and there exists g ∈ L_{l^{2}}^{2}(I^{2}; ℝ) such that

∥g − g_{δ}∥_{2} ≤ Cδ^{1/3}.

A similar argument yields k ∈ L_{l^{2}}^{2}(I^{2}; ℝ) such that

∥k − k_{δ}∥_{2} ≤ Cδ^{1/3}.

By letting δ → 0 in (7.5), we obtain

f = g + k ∘ S − k

as required.

Finally, we can rewrite g_{δ} as

g_{δ}(x) = ∫_{0}^{r_{δ}(x)} G(x, y; t)dt

for a fixed y ∈ I. Letting δ → 0 gives in the L^{2} sense

g(x) = ∫_{0}^{r(x)} G(x, y; t)dt.

Since r_{δ} = r in the interval [δ, 1] for any δ ∈ (0, 1), we conclude that g is continuous on (0, 1].

Now we check conditions (i)-(iii) in Definition 5 in order to show that g ∈ ℋ^{α/2}(I; ℝ). Since G is uniformly bounded,

(inf G)r(x) ≤ g(x) ≤ (sup G)r(x)

and hence

x^{θ_{1}} ≤ e^{g(x)} ≤ x^{θ_{2}}

for all x ∈ (0, 1], where θ_{1} = inf G and θ_{2} = sup G, which verifies (ii).

To check condition (iii), let x_{1}, x_{2} ∈ I_{n} and fix y ∈ I. Note that if x_{1}, x_{2} ∈ I_{n} then

|x_{1} − x_{2}| ≤ \frac{1}{n(n + 1)}.

Therefore

|g(x_{1}) − g(x_{2})| = |∫_{0}^{r(x_{1})} G(x_{1}, y; t)dt − ∫_{0}^{r(x_{2})} G(x_{2}, y; t)dt|

≤ |∫_{r(x_{1})}^{r(x_{2})} G(x_{1}, y; t)dt| + |∫_{0}^{r(x_{2})}(G(x_{1}, y; t) − G(x_{2}, y; t))dt|

≤ |r(x_{1}) − r(x_{2})|(sup G) + ( sup_{x∈I_{n}} |r(x)|) sup_{t∈[0,1]}|G(x_{1}, y; tr_{δ}(x_{1})) − G(x_{2}, y; tr_{δ}(x_{2}))|

where δ = \frac{1}{(n+1)}, and hence

|g(x_{1}) − g(x_{2})| ≤ 2 ((sup G)(n + 1)|x_{1} − x_{2}|^{1− \frac{α}{2}} + (L log n + 1)(n + 1)^{α′}) |x_{1} − x_{2}|^{α/2}
≤ 2 ((sup G)(n + 1)^{1−α} + L(n + 1)^{α}) |x − y|^{α/2}

and moreover,

\frac{1}{n^{α}} 2((sup G)\frac{1}{n^{1−α}} + L(n + 1)^{α}) < ∞

for all n ∈ ℕ. ▷◁

8. A modified transfer operator theorem

The main result of this section is an analogue of Proposition 7 for families for L_{l}^{1}(I; ℂ) functions, which can be uniformly well approximated by Hölder continuous functions.

Proposition 9. Let B ⊂ ℝ^{n} be compact and let K = {k^{b} ∈ L_{l}^{1}(I; ℂ) : b ∈ B}. Suppose there exist constants C_{1}, C_{2} such that for each δ > 0, there is a family {k_{δ}^{b} ∈ C^{α}(I; ℂ) : b ∈ B} satisfying

\begin{aligned}∥k^{b} − k_{δ}^{b}∥_{1} & ≤ C_{1}δ \\ ∥k_{δ}^{b}∥_{α} & ≤ \frac{C_{2}}{δ} \end{aligned}  (8.1)

for all b ∈ B.

(i) Let ϕ ∈ 𝒢^{α}(I; ℂ) be a lattice function and let K_{1} ⊂ N(ϕ) ⊂ ℂ be compact. Then there exists δ_{1} > 0 such that for any s ∈ K_{1} and b ∈ B,

(1 + δ_{1})^{n} ∥\frac{ℒ_{sϕ}^{n}k^{b}}{e^{nP (sϕ)}} − h_{sϕ} ∫ k^{b}dν_{sϕ}∥_{1} → 0  (8.2)

as n → ∞. (ii) Let K_{2} be a compact subset of ℂ\ℝ, let ϕ ∈ 𝒢^{α}(I; ℝ), and suppose that sϕ is non-lattice, for all s ∈ K_{2}. Then there exists δ_{2} > 0 such that

(1 + δ_{2})^{n} ∥\frac{ℒ_{sϕ}^{n}k^{b}}{e^{nP (Re(s)ϕ)}} ∥_{1} → 0  (8.3)

as n → ∞.

Proof. (i) Let {k^{b} : b ∈ B} be a set of functions satisfying the hypotheses of the proposition. We will use the inequality

\begin{matrix}∥\frac{ℒ_{sϕ}^{n}k^{b}}{e^{nP (sϕ)}} − & h_{sϕ} ∫ k^{b}dν_{sϕ}∥_{1} ≤ ∥\frac{ℒ_{sϕ}^{n}(k^{b} − k_{δ}^{b})}{e^{nP (sϕ)}} ∥_{1} \\ + ∥\frac{ℒ_{sϕ}^{n}k_{δ}^{b}}{e^{nP (sϕ)}} − h_{sϕ} ∫ k_{δ}^{b}dν_{sϕ}∥_{1} + ∥h_{sϕ}∥_{1} |∫ (k_{δ}^{b} − k^{b})dν_{sϕ}| \end{matrix}

and estimate each of the three terms separately. We take δ = γ^{n} for some γ ∈ (0, 1) to be chosen later. Then for the first term, we have

∥\frac{ℒ_{sϕ}^{n}(k^{b} − k_{δ}^{b})}{e^{nP (sϕ)}} ∥_{1} ≤ \frac{∥ℒ_{sϕ}^{n}1∥_{1}}{|e^{nP (sϕ)}|}∥k^{b} − k_{δ}^{b}∥_{1}

≤ \frac{∥ℒ_{sϕ}^{n}1∥_{α}}{|e^{nP (sϕ)}|}∥k^{b} − k_{δ}^{b}∥_{1}

≤ 2C_{1}δ∥h_{sϕ}∥_{α}

≤ C_{4}γ^{n}

for a constant C_{4} > 0, uniformly for s ∈ K_{1}, b ∈ B.

Secondly, by applying Proposition 7, there exist κ ∈ (0, 1) and constants C_{5}, C_{6} > 0 such that for all s ∈ K_{1} and b ∈ B,

∥\frac{ℒ_{sϕ}^{n}k_{δ}^{b}}{e^{nP (sϕ)}} − h_{sϕ} ∫ k_{δ}^{b}dν_{sϕ}∥_{1} ≤ C_{5}∥\frac{ℒ_{sϕ}^{n}k_{δ}^{b}}{e^{nP (sϕ)}} − h_{sϕ} ∫ k_{δ}^{b}dν_{sϕ}∥_{α}

≤ C_{6}κ^{n}∥k_{δ}^{b}∥_{α}

≤ C_{7} \frac{κ^{n}}{δ}

= C_{7} (\frac{κ}{γ})^{n} .

For the third term, note that for s ∈ K_{1} and b ∈ B,

|∫ (k^{b} − k_{δ}^{b})dν_{sϕ}| = ∥k^{b} − k_{δ}^{b}∥_{1,ν_{sϕ}}
≤ C_{3}δ
= C_{3}γ^{n}

for a constant C_{3} > 0. Thus by taking γ ∈ (κ, 1) and

0 < δ_{1} < (max\{\frac{κ}{γ}, γ\})^{−1} − 1,

we have that

(1 + δ_{1})^{n} ∥\frac{ℒ_{sϕ}^{n}k^{b}}{e^{nP (sϕ)}} − h_{sϕ} ∫ k^{b}dν_{sϕ}∥_{1} → 0

as n → ∞, uniformly for s ∈ K_{1} and b ∈ B.

In this case, we again let δ = γ^{n}, to give

∥\frac{ℒ_{sϕ}^{n}k^{b}}{e^{nP (Re(s)ϕ)}} ∥_{1} ≤ \frac{∥ℒ_{sϕ}^{n}(k^{b} − k_{δ}^{b})∥_{1}}{e^{nP (Re(sϕ))}} + \frac{∥ℒ_{sϕ}^{n}k_{δ}^{b}∥_{1}}{e^{nP (Re(sϕ))}}

≤ \frac{∥ℒ_{sϕ}^{n}1∥_{α}}{e^{nP (Re(sϕ))}} ∥k^{b} − k_{δ}^{b}∥_{1} + \frac{∥ℒ_{sϕ}^{n}k_{δ}^{b}∥_{α}}{e^{nP (Re(sϕ))}}

≤ C_{8}δ + κ^{n}∥k_{δ}^{b}∥_{α}

≤ C_{8}γ^{n} + C_{2} (\frac{κ}{γ})^{n}

by applying Proposition 7(ii). The result then follows by choosing γ ∈ (κ, 1) and

0 < δ_{2} < (max\{\frac{κ}{γ}, γ\})^{−1} − 1.

This completes the proof. ▷◁

9. The moment generating function

For any K ∈ C(X; ℂ), define the moment generating function Z_{K} for Φ_{t} : X → X by

Z_{K}(s) = ∫_{0}^{∞} e^{−sT} (∫ e^{K^{T} (y)}dm(y))dT  (9.1)

where s ∈ ℂ, provided the integral converges.

For F ∈ ℱ^{α}(X; ℝ), we also define a function f ∈ L_{l^{2}}^{2}(I^{2}; ℝ) by

f(z) = ∫_{0}^{r(z)} F (z; t)dt.

Lemma 12. If F ∈ ℱ^{α}(X; ℝ) is non-lattice, then f satisfies the following condition: if there exist λ_{1}, λ_{2} ∈ ℝ and γ ∈ L_{l^{2}}^{2}(I^{2}; ℂ) such that

γ ∘ S = e^{i(λ_{1}f+λ_{2}r)}γ

then λ_{1} = 0 = λ_{2}.

Proof. Suppose that there exist γ ∈ L_{l}^{2}(I^{2}; ℂ) and λ_{1}, λ_{2} ∈ ℝ such that

γ ∘ S = e^{i(λ_{1}f+λ_{2}r)}γ.  (9.2)

Then define a function B : X → ℂ by

B(z; t) = γ(z)e^{i(λ_{1}F^{t}(z;0)+λ_{2}t)}.

By (9.2), B is well defined and lies in L_{m}^{2}(X; ℂ), and so, by construction,

B ∘ Φ_{T} = e^{i(λ_{1}F^{T} +λ_{2}T)}B.

If F is non-lattice, then λ_{1} = 0 = λ_{2} as required. ▷◁

By Proposition 8, there exist g ∈ ℋ^{α/2}(I; ℝ) and k ∈ L_{l^{2}}^{2}(I^{2}; ℝ) such that
f = g + k ∘ S − k.

Taking ϕ = g in section six, and defining Γ_{F} = Γ_{g}, we have a well defined entropy function H : Γ_{F} → ℝ. Note that by Lemma 12, g satisfies the hypothesis of Lemma 10, so that H has the properties stated in Theorem 2.

Define

R = ∫_{I^{2}} rdρ^{2}

. By a straightforward integration,

R = \frac{2π^{2}}{3 log 2}.

Substituting (−iw + H′(a))(F − a) for K in (9.1), where F ∈ ℱ^{α}(X; ℝ) and a ∈ Γ_{F}, we let

Z(s, w, a) = Z_{(−iw+H′(a))(F −a)}(s − H(a) − 1)

where w ∈ ℝ, wherever this is well defined. The following result describes in detail the analytic behaviour of the function Z(s, w, a).

Proposition 10. Suppose that F ∈ ℱ^{α}(X; ℝ) is non-lattice. Let a ∈ Γ_{F} be arbitrary. Then the following statements hold.

(i) Z(s, w, a) is analytic for (s, w) ∈ {s : Re(s) > 1} × ℝ. (ii) There exists an open neighbourhood U of (1, 0) in ℂ^{2} such that, for each (s, w) ∈ U,

Z(s, w, a) = \frac{C(a)}{s − s(w, a)} + J_{1}(s, w, a)

where C(a) ≠ 0 depends only on a, and J_{1}(s, w, a) is analytic for all (s, w) ∈ U. The function s(w, a) is real analytic in w for |w| < ε ( for some ε > 0 sufficiently small), and s(w, a) is the unique simple pole of Z(s, w, a) in U. Further,

(a) \frac{∂}{∂w} Re s(w, a)|_{w=0} = 0,

(b) \frac{∂}{∂w} Im s(w, a)|_{w=0} = 0,

(c) \frac{∂^{2}}{∂w^{2}} Re s(w, a)|_{w=0} = −\frac{1}{H′′(a)} < 0,

(d) \frac{∂^{2}}{∂w^{2}} Im s(w, a)|_{w=0} = 0,

(e) s(w, a) is an even function of w. (iii) Z(s, w, a) is analytic for (s, w) in an open neighbourhood V of {s : Re(s) = 1, s ≠ 1} × {0}. (iv) For ω ∈ ℝ \ {0}, Z(s, w, a) is analytic in an open neighbourhood W of {s : Re(s) = 1} × {ω}.

Moreover, if J ⊂ Γ_{F} is a non-empty, compact set then U, V, W and ε can be chosen to depend only on J.

Proof. We assume that F ∈ ℱ^{α}(X; ℝ) is non-lattice.

Let Δ denote the Dirac Delta Function. We will use a modified version of a technical device employed in [Po2] and [W1], namely the identity

e^{F^{T +θ}(z;0)} = ∑_{n=0}^{∞} (∫_{0}^{r(S^{n}z)} e^{F^{v+r^{n}(z)}(z;0)}Δ(θ + T − v − r^{n}(z))dv) .  (9.3)

This identity can be interpreted rigorously, as described in [W1]. Note that only one term in the summation in (9.3) can be non-zero. Using the decomposition of m in (2.1), we have

\begin{aligned}Z_{F} (s) & = \frac{1}{2R} ∫_{0}^{∞} e^{−sT} (∫_{I^{2}} e^{F^{T} (z;θ)}dθdρ^{2}(z))dT \\ & = \frac{1}{2R} ∫_{0}^{∞} e^{−sT} (∫_{I^{2}} ∫_{0}^{r(z)} e^{F^{T +θ}(z;0)−F^{θ}(z;0)}dθdρ^{2}(z))dT. \end{aligned}

Substituting in (9.3) gives

\begin{aligned}Z_{F} (s) & = \frac{1}{2R} ∑_{n=0}^{∞}(∫_{I^{2}} ∫_{0}^{r(z)} e^{−s(r^{n}(z)−θ)}\{∫_{0}^{r(S^{n}z)} e^{−sv+F^{v+r^{n}(z)}(z;0)}dv\} \\ & e^{−F^{θ}(z;0)}dθdρ^{2}(z)) \\ & = \frac{1}{2R} ∑_{n=0}^{∞}(∫_{I^{2}} e^{−sr^{n}(z)}\{∫_{0}^{r(S^{n}z)} e^{−sv+F^{v+r^{n}(z)}(z;0)}dv\} \\ & \{∫_{0}^{r(z)} e^{sθ−F^{θ}(z;0)}dθ\}dρ^{2}(z)) \\ & = \frac{1}{2R} ∑_{n=0}^{∞}(∫_{I^{2}} e^{−sr^{n}(x)+f^{n}(x)}B_{1}(s, S^{n}z)D_{1}(s, z)dρ^{2}(z)) \end{aligned}  (9.4)

where

B_{1}(s, z) = ∫_{0}^{r(z)} e^{−sv+F^{v}(z;0)}dv

and

D_{1}(s, z) = ∫_{0}^{r(z)} e^{sθ−F^{θ}(z;0)}dθ.

By Proposition 8, there exist functions g, k ∈ L_{l^{2}}(I^{2}) such that

f = g + k ∘ S − k,

where g depends on future coordinates and g ∈ ℋ^{α/2}(I; ℝ). We may replace α by α/2 without loss of generality. Then we have

Z_{F} (s) = \frac{1}{2R} ∑_{n=0}^{∞} ∫_{I^{2}} e^{−sr^{n}(x)+g^{n}(x)}B_{2}(s, S^{n}z)D_{2}(s, z)dρ^{2}(x, y).

where z = (x, y), and

B_{2}(s, z) = e^{−k(z)}B_{1}(s, z),

and

D_{2}(s, z) = e^{k(z)}D_{1}(s, z).

Using an approximation argument used to study the correlation functional in [Ru2], which we adapted to the complex moment generating function in [W2], we may assume that the functions B_{2}(s, z) and D_{2}(s, z) depend only on the first coordinate in the z variable. Note that the functions B_{2}(s, .) and D_{2}(s, .) are well approximable by Hölder continuous functions, in the sense of Proposition 9 and in particular equation (8.1). Thus we may rewrite Z_{F} (s) in the following form

Z_{F} (s) = \frac{1}{2R} ∑_{n=0}^{∞}(∫_{0}^{1} e^{−sr^{n}+g^{n}}B_{2}(s, T^{n}.)D_{2}(s, .)\}dρ).

This equality should be understood as being modulo the addition of a function analytic in Re(s) > 1 − ε for some ε > 0. Using the facts that ℒ_{−r}^{∗}l = l and dρ(x) = h(x)dx, (where we regard ℒ_{−r} as an operator on L_{l}^{1}(I; ℂ)), it follows that

Z_{F} (s) = \frac{1}{2R} ∑_{n=0}^{∞} ∫_{0}^{1} ℒ_{−r}^{n}\{e^{−sr^{n}+g^{n}}hB_{2}(s, T^{n}.)D_{2}(s, .)\}(x)dx

and hence, by a simple rearrangement,

Z_{F} (s) = \frac{1}{2R} ∑_{n=0}^{∞} ∫_{0}^{1} B_{2}(s, x, e)ℒ_{−(s+1)r+g}^{n}\{hD_{2}(s, .)\}(x)dx.

Making our substitution for F, we let

\begin{aligned}Z(s, w, a) & = Z_{(−iw+H′(a))(F −a)}(s − H(a) − 1) \\ & = \frac{1}{2R} ∑_{n=0}^{∞}(∫_{0}^{1} C(s, w, a, x)ℒ_{b(s,w,a)}^{n}\{E(s, w, a, .)\}dx) \end{aligned}  (9.5)

where

b(s, w, a) = (−iw + H′(a))(g − ar) − (s − H(a))r

and C(s, w, a, .), E(s, w, a, .) ∈ L_{l}^{1}(I; ℂ) for each s, w, a. We also let

K(s, w, a) = ∥C(s, w, a, .)∥_{1}∥E(s, w, a, .)∥_{1}.

Proof of (i). By Propositions 1,2, we have that for each (s, w, a), there is a β ∈ (0, 1) depending on s, a, such that

|Z(s, w, a)| ≤ K(s, w, a) ∑_{n=0}^{∞}∥ℒ_{b(Re(s),0,a)}^{n}1∥_{1}

≤ K(s, w, a) ∑_{n=0}^{∞}∥ℒ_{b(Re(s),0,a)}^{n}1∥_{α}

≤ K(s, w, a) ∑_{n=0}^{∞} e^{nP (b(Re(s),0,a))}(1 + β^{n})

< ∞,

provided P (b(Re(s), 0, a)) < 0, that is

P (ρ(a)(g − ar) − (Re(s) − H(a))r) < 0.

By the strict monotonicity of pressure and the definition of H, this is true if Re(s) > 1. Thus Z(s, w, a) is analytic for (s, w) ∈ {s : Re(s) > 1} × ℝ, for any a ∈ Γ_{F}.

Proof of (ii). By Proposition 9(i), provided ε is sufficiently small, and U is a sufficiently small neighbourhood of 1 ∈ ℂ, there exists δ_{1} > 0 such that for all s ∈ U, and w ∈ (−ε, ε),

(1 + δ_{1})^{n} ∥\frac{ℒ_{b(s,w,a)}^{n}(E(s, w, a, .))}{e^{P (b(s,w,a))}} − h_{b(s,w,a)} ∫ E(s, w, a, .)dν_{b(s,w,a)}∥_{1} ⟶ 0  (9.6)

as n → ∞. From (9.5) and (9.6), we therefore have that

\begin{aligned}Z & (s, w, a) \\ & = \frac{1}{2R} ∑_{n=0}^{∞}(∫_{0}^{1} C(s, w, a, x)h_{b(s,w,a)}dx)(∫_{0}^{1} E(s, w, a, .)dν_{b(s,w,a)})e^{nP (b(s,w,a))} \\ & + \frac{1}{2R} ∑_{n=0}^{∞} e^{nP (b(s,w,a))}(∫_{0}^{1} C(s, w, a, x)\{\frac{ℒ_{b(s,w,a)}^{n}(E(s, w, a, .))}{e^{nP (b(s,w,a))}} \\ & −h_{b(s,w,a)} ∫_{0}^{1} E(s, w, a, .)dν_{b(s,w,a)}\}dx) \end{aligned} (9.7)

where

|∑_{n=0}^{∞} e^{nP (b(s,w,a))} ∫_{0}^{1} C(s, w, a, x)\{\frac{ℒ_{b(s,w,a)}^{n}(E(s, w, a, .))}{e^{nP (b(s,w,a))}}

− h_{b(s,w,a)} ∫_{0}^{1} E(s, w, a, .)dν_{b(s,w,a)}}dx|

≤ K(s, w, a) ∑_{n=0}^{∞} \frac{e^{nReP (b(s,w,a))}}{(1 + δ_{1})^{n}}

≤ K(s, w, a) ∑_{n=0}^{∞} \frac{1}{(1 + δ_{1})^{n}}

by Proposition 9(i), which implies that the second term in (9.7) is analytic in an open neighbourhood U of (s, w) = (1, 0). Thus we have that

Z(s, w, a) = \frac{A(a)}{1 − e^{P (b(s,w,a))}} + J_{1}(s, w, a)

where

A(a) = \frac{1}{2R}(∫_{0}^{1} C(1, 0, a, x)h_{b(1,0,a)}dx)(∫_{0}^{1} E(1, 0, a, x)dν_{b(1,0,a)})

and J_{1}(s, w, a) is analytic for (s, w) ∈ U. In particular, U can be chosen uniformly on compact subsets of Γ_{F}, and A(a) > 0 for all a ∈ Γ_{F}.

Properties (a)-(e) follow exactly as in Lemma 3 of [W1], using standard formulae for the derivatives of the pressure function.

Proof of (iii). Now suppose that t ≠ 0 and note that

Im(b(1 + it, 0, a)) = −tr.

In particular, b(1 + it, 0, a) is T -non-lattice for any t ∈ ℝ \ {0}. By Proposition 6, there exists η_{1} ∈ (0, 1) such that

ρ(ℒ_{b(1,0,a)}) ≤ 1 − η_{1},

where ρ(ℒ) denotes the spectral radius of the operator ℒ. By the upper semicontinuity of the map t ↦ ρ(ℒ_{tψ}) for ψ ∈ 𝒢^{α}(I; ℂ), given η_{2} ∈ (η_{1}, 1), we can find ε_{1} > 0 and ξ_{1} > 0 such that, for each pair (λ, w) in the compact set

{(λ, w) : |λ| ≤ ξ_{1}, |w| ≤ ε_{1}}

we have

ρ(ℒ_{b(1+λ+it,w,a)}) ≤ 1 − η_{2}.

Therefore by Proposition 9(ii), there exists δ_{2} > 0 such that for s = 1 + λ + it,

(1 + δ_{2})^{n}∥\frac{ℒ_{b(s,w,a)}^{n}\{E(s, w, a, .)\}}{e^{nP (b(s,w,a))}} ∥_{1} ⟶ 0

as n → ∞, for all |λ| < ξ_{1}, w ∈ (−ε_{1}, ε_{1}).

Thus by (9.3) again, Z(s, w, a) is analytic for all

(s, w) ∈ (1 + i(t − ξ_{1}), 1 + (t + ξ_{1})) × (−ε_{1}, ε_{1}).

Hence we see that, for each a ∈ Γ_{F}, the functions Z(s, w, a) is analytic in an open neighbourhood V of {s : Re(s) = 1, s ≠ 1} × {0} as required. It is also easy to see by using Proposition 9(ii), that V can be chosen uniformly on compact subsets of Γ_{F}. This completes the proof of (iii).

Proof of (iv). Let ω ∈ ℝ \ {0} and let t ∈ ℝ be fixed. This time there are two cases to consider. Firstly, suppose that

b(1 + it, ω, a) ∈ Λ^{α}(I; ℂ),

say

γ ∘ T = e^{i(Im b(1+it,ω,a)+c)}γ

for some γ ∈ L_{l}^{2}(I; ℂ) and c ∈ [0, 2π). By Proposition 6 and Proposition 9(i), there exists an open neighbourhood U_{3} of (1 + it, ω) in ℂ^{2} such that

Z(s, w, a) = \frac{N(s, w, a)}{1 − e^{P (b(s,w,a))}} + J_{2}(s, w, a)

for (s, w) ∈ U_{3}, where N(s, w, a) and J_{2}(s, w, a) are analytic for (s, w) ∈ U_{3}. Further, by Lemma 12, c ≠ 0. Thus

P (b(s, w, a)) ≠ 0

for all (s, w) ∈ U_{3}, and so Z(s, w, a) is analytic for (s, w) ∈ U_{3}.

Secondly, suppose that b(1 + it, ω, a) is T -non-lattice. By duplicating the analysis for case (iii), we can find ε_{2}, ξ_{2} > 0 such that, for any (λ, w) in the compact set

{(λ, w) : |λ| ≤ ξ_{2}, |w − ω| ≤ ε_{2}},

we have that

ρ(ℒ_{b(1+λ+it,w,a)}) ≤ 1 − η_{2}

for some η_{2} ∈ (0, 1). Thus by Proposition 9(ii), there exists δ_{3} > 0 such that for s = 1 + λ + it,

(1 + δ_{3})^{n}∥\frac{ℒ_{b(s,w,a)}^{n}\{E(s, w, a, .)\}}{e^{nP (b(s,w,a))}} ∥_{1} ⟶ 0

as n → ∞, for |λ| < ξ_{2}, |w − ω| < ε_{2}. From (9.5), Z(s, w, a) is analytic in an open neighbourhood W of {s : Re(s) = 1} × {ω}, for each ω ∈ ℝ \ {0} as required. As before, W may be chosen uniformly on compact subsets of Γ_{F}. This completes the proof of part (iv). ▷◁

10. Proof of Theorem 2

The proof of Theorem 2 can be carried out as in section five of [W1]. For the benefit of the reader, we give a brief account of the method.

Let Δ denote the space of C^{∞} test functions on ℝ, i.e.

Δ = {u ∈ C^{∞}(ℝ) : sup_{t∈ℝ}{(1 + |t|^{m})|u^{(n)}(t)|} < ∞, for all m, n ≥ 0}.

For u ∈ Δ, let û denote the Fourier-Laplace transform of u, defined by

û(z) = ∫_{−∞}^{∞} e^{zt}u(t)dt  (10.1)

for z ∈ ℂ, wherever the integral converges. If u ∈ Δ, then the function ℝ → ℝ given by w ↦ û(iw) is also in Δ.

Let (k_{N})_{N=1}^{∞} denote the approximate identity whose Fourier transform is given by

k̂_{N}(iw) = \{ \begin{matrix}exp \{−\frac{w^{2}}{N^{2}−w^{2}} \} & if |w| ≤ N, \\ 0 & otherwise. \end{matrix}

(One can obtain an explicit formula for k_{N} itself by Fourier inversion). We remark that the map w ↦ k̂_{N}(iw) is in Δ, and that k̂_{N}(iw) → 1 as N → ∞, uniformly for w in any compact subset of ℝ.

Let J ⊂ Γ_{F} be a fixed non-empty, compact set. To prove Theorem 2, it suffices, by standard smoothing arguments, to show that for any non-negative u ∈ Δ with compact support,

E_{m}u(F^{T} − aT) ∼ C(a)\sqrt{\frac{H′′(a)}{2π}}û(−H′(a))\frac{e^{−T H(a)}}{\sqrt{T}}  as T → ∞,  (10.2)

uniformly for all a ∈ J.

For F ∈ ℱ^{α}(X; ℝ) and a ∈ J, let

Θ_{T} (t) = m{y ∈ X : F^{T} (y) − aT ≤ t},

for all T ≥ 0, t ∈ ℝ. We note that Θ_{T} has Fourier transform

\begin{aligned}Θ̂_{T} (z) & = ∫ e^{z(F^{T} −aT)(y)}dm(y) \\ & = E_{m}(e^{z(F^{T} −aT)}). \end{aligned}  (10.3)

By using Parseval’s identity and the Riemann-Lebesgue Lemma, we have

\begin{aligned}E_{m}u(F^{T} − aT) & = ∫_{−∞}^{∞} u(t)e^{−H′(a)t}d (Θ_{T} (t)e^{H′(a)t}) \\ & = C(a)\sqrt{\frac{H′′(a)}{2π}} \frac{e^{−T H(a)}}{\sqrt{T}} ∫_{−∞}^{∞} u(t)e^{−H′(a)t}dq_{H′(a)}^{(T)}(t) \end{aligned}  (10.4)

where q_{H′(a)}^{(T)} is the positive Borel measure on ℝ given by

dq_{H′(a)}^{(T)}(t) = \frac{1}{C(a)}\sqrt{\frac{2π}{H′′(a)}} \frac{\sqrt{T}}{e^{−T H(a)}} e^{H′(a)t}dH_{T} (t).  (10.5)

To prove (10.2), it therefore suffices to show that

∫_{−∞}^{∞} u(t)e^{−H′(a)t}dq_{H′(a)}^{(T)}(t) → ∫_{−∞}^{∞} u(t)e^{−H′(a)t}dt = û(−H′(a))  (10.6)

as T → ∞, uniformly for all a ∈ J.

Now let

Z(s; a) = ∫_{0}^{∞} e^{−(s−H(a)−1)T} E_{m}u(F^{T} − aT)dT  (10.7)

wherever this integral is well defined. By Parseval’s identity, we have

Z(s; a) = \frac{1}{2π} ∫_{−∞}^{∞} û(−iw + H′(a))Z(s, w, a)dw.  (10.8)

By using an “unsmoothing” lemma stated in [W1], Lemma 3, it suffices to prove (10.6) for convolutions of functions of the form k_{N} ∗ v, where v ∈ Δ is non-negative and N ≥ 1.

Let a ∈ Γ_{F} and v ∈ Δ be non-negative. Define

Z_{N,v}(s; a) = ∫_{0}^{∞} e^{−(s−H(a)−1)T} E_{m}(v_{N,a}(F^{T} − aT))dT  (10.9)

where v_{N,a} = (k_{N}e_{H′(a)}) ∗ (ve_{ρ(a)}) and e_{c} : ℝ → ℝ is given by e_{c}(t) = e^{ct} for any c ∈ ℝ. Then (10.8) and Proposition 10(i), (iii), (iv),

Z_{N,v}(s, a) = \frac{1}{2π} ∫_{−ε}^{ε} k̂_{N}(iw)v̂(iw)Z(s, w, a)dw + J_{1}(s)  (10.10)

where J_{1}(s) is analytic in U_{1}, where U_{2} is an open neighbourhood of {s : Re(s) ≥ 1}, depending only on J.

By applying Proposition 10 to Z_{N,v}(s, a) and following a calculation in [KS], we obtain the following proposition.

Proposition 11. Let a ∈ Γ_{F} be arbitrary. Then the limit

lim_{σ→1} (Z_{N,v}(σ + it, a) − v̂(0)C(a)\sqrt{\frac{H′′(a)}{2}} \frac{1}{(σ + it − 1)^{1/2}} )  (10.11)

exists for almost every t ∈ ℝ, and is in the space W_{loc}^{1,1}(ℝ) of locally integrable functions with locally integrable first derivatives. Further, there exists a locally integrable function h(t) such that

|Z_{N,v}(σ + it, a) − v̂(0)C(a)\sqrt{\frac{H′′(a)}{2}} \frac{1}{(σ + it − 1)^{1/2}} | ≤ h(t)

for σ > 1.

If a ∈ J ⊂ Γ_{F}, with J compact, then the convergence in (10.11) is uniform in J, and h can be chosen depending only on J.

By an integration by parts, we may rewrite Z_{N,v}(s, a) as

\begin{aligned}Z_{N,v}(s, a) & = ∫_{0}^{∞} e^{−(s−H(a)−1)T} E_{m}v_{N,a}(F^{T} − aT )dt \\ & = ∫_{0}^{∞} e^{−sT} dα_{N,v,a}(T) \end{aligned}  (10.12)

where

α_{N,v,a}(T) = ∫_{0}^{T} e^{(H(a)+1)t}E_{m}v_{N,a}(F^{t} − at)dt.

We now require the following Tauberian Theorem.

Proposition 12. Let J ⊂ ℝ be compact. Let α : J × ℝ → ℝ be continuous and write α_{a}(T) = α(a, T). Let α_{a}(0) = 0 and suppose that α_{a}(T) is monotonic non-decreasing, for all a ∈ J. Let f_{a}(s) be a family of functions, depending continuously on a ∈ J, such that f_{a}(s) is analytic for all Re(s) > 1 and all a ∈ J. Let a ↦ A_{a} be a continuous map J → ℝ \ {0}. Suppose that the identity

f_{a}(s) = \frac{A_{a}}{(s − 1)^{1/2}} − ∫_{0}^{∞} e^{−sT} dα_{a}(T)  (10.13)

holds, for all Re(s) > 1 and all a ∈ J. Further suppose that

lim_{δ→0} f_{a}(1 + δ + it)  (10.14)

exists for almost every t ∈ ℝ, and is in W_{loc}^{1,1}(ℝ), independent of a ∈ J. Further suppose that there is a locally integrable function h(t), depending only on J, such that

|f(1 + δ + it)| ≤ h(t)  (10.15)

for δ > 0. Then

α_{a}(T) ∼ \frac{A_{a}}{\sqrt{π}} \frac{e^{T}}{\sqrt{T}}  as T → ∞,  (10.16)

uniformly for a ∈ J.

Applying Propositions 11, 12 to the function Z_{N,v}, as expressed in (10.12), we obtain

α_{N,v,a}(T) ∼ v̂(0)C(a)\sqrt{\frac{H′′(a)}{2π}} \frac{e^{T}}{\sqrt{T}}  as T → ∞.  (10.17)

uniformly for a ∈ J.

After a further integration by parts, we then obtain

E_{m}((k_{N}e_{H′(a)}) ∗ (ve_{H′(a)})(F^{T} − aT)) ∼ v̂(0)C(a)\sqrt{\frac{H′′(a)}{2π}} \frac{e^{−T H(a)}}{\sqrt{T}}  (10.18)

as T → ∞, uniformly for a ∈ J. Thus from (10.4), we deduce that

∫_{−∞}^{∞}(k_{N} ∗ v)(t)dq_{H′(a)}^{(T)}(t) ⟶ v̂(0)  as T → ∞,  (10.19)

uniformly for a ∈ J, as required.

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