Pressure function and limit theorems for almost Anosov flows

We obtain limit theorems (Stable Laws and Central Limit Theorems, both Gaussian and non-Gaussian) and thermodynamic properties for a class of non-uniformly hyperbolic flows: almost Anosov flows, constructed here. The proofs of the limit theorems for these flows are applications of corresponding theorems in an abstract functional analytic framework, which may be applicable to other classes of non-uniformly hyperbolic flows.


Introduction and summary of the main results
This paper is a contribution to the theory of limit theorems and thermodynamic formalism in the context of non-uniformly hyperbolic flows on manifolds. In particular we introduce a new class of 'almost Anosov flows', a natural analogue of almost Anosov diffeomorphisms introduced in [HY95], and prove limit theorems for natural observables ψ with respect to the SRB measure, also giving the form of the associated pressure function. This example is presented in the context of a general framework, which may be applicable to a range of other non-uniformly hyperbolic flows. We recall that various statistical properties for several classes of Anosov flows are known [D98, D03, L04], but none of these results apply to the class of 'almost Anosov flows' considered here.
Given a flow (Φ t ) t with a real-valued potential φ, we suppose that there is an equilibrium state µ φ . A standard way of studying the behaviour of averages of a real-valued observable ψ is to consider a Poincaré section Y and study the induced first return map F : Y → Y , the induced potentialsφ andψ, with the induced measure µφ. In the discrete time case, in fact in the setting where all the dynamical systems are countable Markov shifts, [S06, Section 2] showed a one-to-one relation between limit laws for ψ and the asymptotic form of the pressure function P(φ + sψ), as s → 0. This function experiences a phase transition at s = 0, and its precise form determines the type of limit laws (in particular, the index of the stable law), see [MTo04,Z07] and [S06,Theorem 7]. Furthermore, under certain conditions on ψ, [S06,Theorem 8] gave an asymptotically linear relation between the induced pressure P(φ + sψ) and P(φ + sψ). On the limit theorems side, [MTo04,Z07] and [S06,Theorem 7] show how one can go between results on the induced and on the original system: the first of these also applies in the flow setting. In these cases, the tail of the roof function/return time to the Poincaré section (both for the flow and the map) plays a major role in determining the form of the results. Here we will follow this paradigm in the setting of a new class of flows. The proofs of the main theorems are facilitated by corresponding theorems in an abstract functional analytic framework. Applying this to the considered example requires precise estimates on the tails of the roof function, which we prove for our main example.
Our central example is an almost Anosov flow, which is a flow having a continuous flowinvariant splitting of the tangent bundle T M = E u ⊕ E c ⊕ E s (where E c q is the one-dimension flow direction) such that we have exponential expansion/contraction in the directions E u q , E s q , except for a finite number of periodic orbits (in our case a single orbit). We can think of these as perturbed Anosov flows, where the perturbation is local around these periodic orbits, making them neutral. A precise description is given in Section 2. Almost Anosov diffeomorphisms have been introduced in [HY95] and sufficiently precise estimates on the tail of the return function to a 'good' set have been obtained in [BT17]. (These estimates are for both finite and infinite measure preserving almost Anosov diffeomorphisms.) We emphasise that the roof function is not constant on the stable direction, which is a main source of difficulty. When considering these examples we choose φ so that µ φ is the SRB measure. We build on the construction in [BT17] to obtain the asymptotic of the tail behaviour of the roof functions for almost Anosov flows (when viewed as suspension flows). The main technical results for these systems are Propositions 2.2 and 2.4. These are then used to prove the main result Theorem 2.5.
For a major part of the statements of the abstract theorems in this paper (in contexts not restricted to almost Anosov flows) we do not require any Markov structure for our system: it is only when we want to see our results in terms of the pressure function (making the connection between the leading eigenvalue of the twisted transfer operator of the base map and pressure as in [S99,Theorem 4]) that this is needed. Our setup requires good functional analytic properties of the Poincaré map in terms of abstract Banach spaces of distributions. Using the rather mild abstract functional assumptions described in Section 4, in Sections 5 we obtain stable laws, standard and non-standard CLT; this is the content of Proposition 5.1. In Section 6 we recall [S06, Theorem 7] and [MTo04, Theorem 1.3] to lift Proposition 5.1 to the flow, which allow us to prove Proposition 6.1.
In Section 7 we do exploit the assumption of the Markov structure to relate the definition of the pressure P(φ + sψ), s ≥ 0, with that of the family of eigenvalues of the family of twisted transfer operators of the Poincaré map; the twist is in terms of the roof function of the suspension flow and the potential ψ. Using this type of identification, in Theorem 7.1 we relate the induced pressure P(φ + sψ) with the original pressure. Using the main result in Section 7, in Section 8 we summarise the results for the abstract framework in the concluding Theorem 8.2, which gives the equivalence between the asymptotic behaviour of the pressure function P (φ + sψ) and limit theorems. It is this summarising result that can be viewed as a version of Theorems 2-4 and 2 The setup for almost Anosov flows

Description via ODEs
Let M be an odd-dimensional compact connected manifold without boundary. An almost Anosov flow is a flow having continuous flow-invariant splitting of the tangent bundle T M = E u ⊕ E c ⊕ E s (where E c q is the one-dimension flow direction) such that we have exponential expansion/contraction in the direction of E u q , E s q , except for a finite number of periodic orbits (in our case a single orbit Γ). After restricting to an irreducible component if necessary, Anosov flows are automatically transitive, cf. [BS02,5.10.3]. As noted in the introduction, we can think of almost Anosov flows as perturbed Anosov flows, where the perturbation is local around Γ, making Γ neutral.
Let Φ t : M × R → M be an almost Anosov flow on a 3-dimensional manifold. We assume that the flow in local Cartesian coordinates near the neutral periodic orbit Γ := (0, 0) ×T is determined by a vector field X : M → T M defined as:  ẋ ẏ z x(a 0 x 2 + a 2 y 2 ) −y(b 0 x 2 + b 2 y 2 ) 1 + w(x, y) where 1 a 0 , a 2 , b 0 , b 2 ≥ 0 with ∆ = a 2 b 0 − a 0 b 2 = 0, and w is a homogeneous function with exponent ρ ≥ 0 as leading term, cf. Remark 2.6. For the Limit Theorem 2.5, we additionally require a 2 > b 2 (so that β > 1, the finite measure case). We will use a Poincaré section Σ which is {z = 0} in local coordinates near Γ, but which is thought to be defined across the whole manifold so that the Poincaré maps is defined everywhere on Σ. The Poincaré map of the Anosov flow (i.e., before the perturbation rendering Γ neutral) is Anosov itself, and hence has a Markov partition, for instance based on arcs in W s (p) ∪ W u (p) for p = (0, 0), but not W s loc (p) ∪ W u loc (p) because we want p in the interior of a partition element. As the perturbation is local around Γ and leaves local stable and unstable manifolds of Γ unchanged, the same Markov partition can be used for the almost Anosov flow.
Let us call the horizontal (i.e., (x, y)-component) of the flow Φ hor t . This is the vector field of [BT17,Equation (4)], with κ = 2 and the vertical component is added as a skew product. Therefore we can take some crucial estimates from the estimates of Φ hor t in [BT17, Proposition 2.1].
The flow Φ t has a periodic orbit Γ = {p} × T of period 1 (which is neutral because DX is zero on Γ), and it has local stable/unstable manifolds W s loc (Γ) = {0} × (−ǫ, ǫ) × T 1 and W u loc (Γ) = (−ǫ, ǫ) × {0} × T 1 . It is an equilibrium point of neutral saddle type if we only consider Φ hor t . The time-1 mapf of Φ hor t is an almost Anosov map, with Markov partition {P i } i≥0 , where we assume that p is an interior point ofP 0 .
Given q ∈f −1 (P 0 ), defineτ (q) := min{t > 0 : Φ hor t (q) ∈Ŵ s }, (2.2) whereŴ s is the stable boundary leaf ofP 0 , see Figure 1. LetŴ u (y) denote the unstable leaf off intersecting (0, y) ∈Ŵ s (p). The functionτ is strictly monotone onŴ u (y). For y > 0 and T ≥ 1, let ξ(y, T ) denote the distance between (0, y) and the (unique) point q ∈Ŵ u (y) withτ (q) = T . The crucial information from [BT17, Proposition 2.1] is The notation and absence of mixed terms a 1 xy and b 1 xy goes back to [H00], who could not treat mixed terms. In [BT17], the mixed terms are absent too, in order to make the explicit form of the first integrals possible. Importantly, (2.1) ensure that the two vertical coordinate planes are local stable and unstable manifold of Γ. The only linear transformation that preserves this are trivial scalings in the coordinate directions. Such scalings reduce b 2 /a 2 and a 0 /b 0 to single parameters, but we didn't do this to maintain the possibility to compare proofs with [BT17, H00] more easily. It is possible to treat mixed terms to some extent, see [B19], but the additional required technicalities are beyond the purpose of this paper.
x y y 1 Figure 1: The first quadrant of the rectangleP 0 , with stable and unstable foliations of time-1 map f = Φ hor 1 drawn vertically and horizontally, respectively. Also the integral curve of q is drawn.
Remark 2.1 In fact, for the most general vector fields treated in [BT17,Equation (3)] (i.e., vector fields with higher order terms), we cannot do better than "big tails" estimates: The next proposition gives an estimate of integrals along such curves. This allows us to estimate the tail of the roof function (when viewing Φ t as a suspension flow) in Proposition 2.4 and also, the tail of induced potentials (see Remark 2.6). In the first instance we use it to estimate the vertical component of the flow Φ t (compared to t).
Proposition 2.2 Let θ : R 2 → R be a homogeneous function with exponent ρ ∈ R such that θ(x, 0) ≡ 0 ≡ θ(0, y) for x = 0 = y. Then there is a constant C ρ > 0 (given explicitly in the proof ) such that for every q withτ (q) = T as in (2.2), (2.5) For the proof of Proposition 2.2 we recall some notation and the form of the first integral L(x, y) from [BT17, Lemma 2.2], replacing κ from that paper by 2. Recall that ∆ := a 2 b 0 − a 0 b 2 = 0. Let u, v ∈ R be the solutions of the linear equation that is: (2.6) Note that u, v and ∆ all have the same sign. Define: is a first integral of (i.e., preserved by) Φ hor t (and therefore of Φ t ). For the proof of Proposition 2.2, we follow the proof of [BT17, Proposition 2.1]. In comparison, we have κ from [BT17, Proposition 2.1] equal to 2, our current integrand is more complicated, but we only need first order error terms.
The case ρ > 2: The value of Θ(T ) based on the leading terms (2.16) of the integrand only is Next, by inserting the asymptotics of G(T ) from (2.14), the factor T ρ 2 −1 cancels: The case ρ = 2: The factor G(T ) ρ 2 −1 in (2.15) now disappears and the leading terms (2.16) are θ 0 c ρ 0 0 M −1 and θ ∞ c ρ 0 2 M −1 respectively. This gives Inserting again the values of ξ and ω from [BT17, Proposition 2.1] gives This completes the proof.

Description via suspension flows, tail estimates of the roof function
The 3-dimensional time-1 map Φ 1 preserves no 2-dimensional submanifold of M. Yet in order to model Φ t as a suspension flow over a 2-dimensional map, we need a genuine Poincaré map. For this we choose a section Σ transversal to Γ and containing a neighbourhood U of p. As an example, Σ could be T 2 × {0}, and the Poincaré map to T 2 × {0} could be (a local perturbation of) Arnol'd's cat map; in this case (and most cases) M is not homeomorphic to T 3 because the homology is more complicated, see [BF13,N76]. Let h : Σ → R + , h(q) = min{t > 0 : Φ t (q) ∈ Σ} be the first return time. Assuming that sup Σ |w(x, y)| < 1, the first return time h is bounded and bounded away from zero, i.e., 0 < inf Σ h < sup Σ h. There is no loss of generality in assuming that inf Σ h ≥ 1.
The Poincaré map f := Φ h : Σ → Σ has a neutral saddle point p at the origin. Its local stable/unstable manifolds are W s loc (p) = {0} × (−ǫ, ǫ) and W u loc (p) = (−ǫ, ǫ) × {0}. Because the flow Φ t is a perturbation of an Anosov flow, and f is a Poincaré map, it has a finite Markov partition {P i } i≥0 and we can assume that p is in the interior of P 0 . In the sequel, let U be a neighbourhood of p that is small enough that (2.1) is valid on U × [0, 1] but also that f (U) ⊃P 0 ∪ P 0 .
In order to regain the hyperbolicity lacking in f , let be the first return time to Y := Σ \ P 0 . Then the Poincaré map is the corresponding first return time.
| be the potential for the SRB measure of the flow. Let µφ be the F -invariant equilibrium measure of the potentialφ : soφ is the logarithm of the derivative in the unstable direction of the Poincaré map F . This is at the same time the SRB-measure for F and thus is absolutely continuous conditioned to unstable leaves.
x y y 1 Figure 3: The first quadrant of the rectangle P 0 , with stable and unstable foliations of Poincaré map f = Φ h drawn vertically and horizontally, respectively. Also one of the integral curves is drawn.
Proof The function τ is defined on Σ \ P 0 and The set Y {r≥2} is a rectangle with boundaries consisting of two stable and two unstable leaves of the Poincaré map f . Let W u (y) denote the unstable leaf of f inside Y {r≥2} with (0, y) as (left) boundary point. Let y 1 < y 2 be such that W u (y 1 ) and W u (y 2 ) are the unstable boundary leaves of Y {r≥2} . The unstable foliation off = Φ hor 1 does not entirely coincide with the unstable foliation of f . LetŴ u (y) denote the unstable leaf off with (0, y) as (left) boundary point. BothŴ u (y) and W u (y) are C 1 curves emanating from (0, y); let γ(y) denote the angle between them. Then the lengths as t → ∞, where the last equality and the notation ξ 0 (y) and β = (a 2 + b 2 )/(2b 2 ) come from (2.3).
We decompose the measure µφ on Y {r≥2} as The conditional measures µ W u (y) on W u (y) are equivalent to Lebesgue m W u (y) with density h u 0 = dµ W u (y) dm W u (y) tending to a constant h * (y) at the boundary point (0, y). Therefore, as t → ∞, for C * = y 2 y 1 h * (y)| cos γ(y)| ξ 0 (y) dν u (y), and using (2.3) in the third line. This proves the result.

Main results for the Poincaré map F and flow Φ t
Throughout this section we assume the setup and notation of Subsection 2.2. We emphasise that we are in the finite measure setting, so We recall that the natural potential associated to the SRB-measure for F isφ = − log |DF | W u , which is the induced version of φ = lim t→0 − log |DΦt| W u t . Our main result can be viewed as a version of the results in [S06] for the flow Φ t ; it gives gives the link between limit theorems for (Φ t , ψ) for (unbounded from below) potentials ψ on M and pressure function P(φ + sψ), s ≥ 0. We let φ + sψ, s ≥ 0 be the family of induced potentials and denote the associate pressure function by P(φ + sψ). Some background on equilibrium states and pressure function (along with their induced versions) is recalled in Section 3.
Proof of Theorems 2.5 This follow directly from applying Theorem 8.2 (Φ t , φ), which uses a special case of Proposition 5.1. Appendix B verifies all the abstract hypotheses of Proposition 5.1 and Theorem 8.2 for the flow Φ t with base map F and roof function τ introduced in Section 2 and potential ψ defined in the statement of Theorem 2.5. The extra assumptions (including coboundary assumptions) that ensure σ = 0 in Proposition 5.1(ii) and Theorem 8.2 (b) are verified in Appendix B.3.

Background on equilibrium states for suspension flows
Here we outline the standard general theory of thermodynamic formalism for suspension flows. We start with a discrete time dynamical system F : Y → Y . Defining M F to be the set of F -invariant probability measures, for a potential ψ : Y → [−∞, ∞] we define the pressure as Given a roof function τ : computed modulo identifications. We will suppose throughout that inf τ > 0, but note that the case inf τ = 0 can also be handled, see for example [Sav98,IJT15]. Barreira and Iommi [BI06] define pressure as has summable variations. This also makes sense for general suspension flows, so we will take (3.1) as our definition. In [AK42], there is a bijection between F t -invariant probability measures µ, and the corresponding F -invariant probability measuresμ given by the identification where m is Lebesgue measure. That is, whenever there is such a µ there is such aμ, and vice versa. Moreover, Abramov's formula for flows [Ab59b] gives the following characterisation of entropy: and clearly One consequence of these formulas is that the pressure in (3.1) is independent of the choice of cross section Y . This follows essentially from the fact that if we choose a subset of Y and reinduce there, then Abramov's formula gives the same value for pressure (here we can use the discrete version of Abramov's formula [Ab59a]). Note that we say 'Abramov's formula' in both the continuous and discrete time cases as the formulas are analogous [Ab59a,Ab59b], and similarly for integrals, where the formula holds by the ergodic and ratio ergodic theorems. We say that µ φ is an equilibrium state for φ if h(µ φ ) + φ dµ φ = P(φ). The same notion extends to the induced system (Y, F,φ). In that setting we may also have a conformal measure mφ forφ. This means that mφ(F (A)) = A e −φ dmφ for any measurable set A on which F is injective. Later on, in order to link our limit theorem results with pressure, we will assume that F : Y → Y is Markov. In that setting our assumptions on φ will be equivalent to assuming P(φ) = 0, in which case we will have a equilibrium states and conformal measures µφ and mφ.

Abstract setup
We start with a flow f t : M → M, where M is a manifold. Let Y be a co-dimension 1 section of M and define τ : Y → R + to be a roof function. We think of τ as a first return of f t to Y and define F : as described in Section 3. Throughout, we assume that τ is bounded from below.
Given the potential φ : M → R and its induced versionφ : Y → R defined in (3.2), we assume that there is a conformal measure mφ for (F,φ). In the rest of this section we recall the abstract framework and hypotheses in [LT16] as relevant for studying limit theorems.

Banach spaces and equilibrium measures for
We assume that there exist two Banach spaces of distributions B, B w supported on Y such that for some α, α 1 > 0 (iii) The transfer operator R 0 associated with F maps continuously from C α to B and R 0 admits a continuous extension to an operator from B to B, which we still call R 0 .
(iv) The operator R 0 : B → B has a simple eigenvalue at 1 and the rest of the spectrum is contained in a disc of radius less than 1.
A few comments on (H1) are in order and here we just recall the similar ones in [LT16]. We note that (H1)(i) should be understood in terms of the following usual convention (see, for instance, [GL06,DL08]): there exist continuous injective linear maps π i such that π 1 (C α ) ⊂ B, π 2 (B) ⊂ B w and π 3 (B w ) ⊂ (C α 1 ) ′ . Throughout, we leave such maps implicit, but recall their meaning here. In particular, we assume that π = π 3 • π 2 • π 1 is the usual embedding, i.e., for all h ∈ C α and g ∈ C α 1 Via the above identification, the conformal measure mφ can be identified with the constant function 1 both in (C α 1 ) ′ and in B (i.e., π(1) = mφ). Also, by (H1)(i), B ′ ⊂ (C α ) ′ , hence the dual space can be naturally viewed as a space of distributions. Next, note that B ′ ⊃ (C α 1 ) ′′ ⊃ C α 1 ∋ 1, thus we have mφ ∈ B ′ as well. Moreover, since mφ ∈ B and 1, g 0 = g, 1 0 = g dµ s φ , mφ can be viewed as the element 1 of both spaces B and (C α 1 ) ′ .

4.2
Transfer operator R defined w.r.t. the equilibrium measure µφ Given the equilibrium measure µφ ∈ B, we let R : L 1 (µφ) → L 1 (µφ) be the transfer operator for the first return map F : Y → Y w.r.t. the invariant measure µφ given by Y Rv w dµφ = Y v w •F dµφ for every bounded measurable w.
Under (H1)(i)-(iv), we further assume The transfer operator R maps continuously from C α to B and R admits a continuous extension to an operator from B to B, which we still call R.
By (H1)(iv) and H1(v), the operator R : B → B has a simple eigenvalue at 1 and the rest of the spectrum is contained in a disc of radius less than 1. While the spectra of R 0 and R are the same, the spectral projection P = lim n→∞ R n acts differently on B, B w . In particular, P 1 = 1, while P 0 1 = µφ, P 0 1 = h 0 . Throughout the rest of this paper, for any g ∈ C α , we let g, 1 := g, P 0 1 0 = P 0 1, g 0 = Y g dµφ and note that P 0 g = h g, 1 0 and P g = g, 1 .

4.3
Further assumptions on the transfer operator R Given R as defined in Subsection 4.2, for u ≥ 0 and τ : Y → R + we define the perturbed transfer operatorR (u)v := R(e −uτ v).
By [BMT,Proposition 4.1], a general proposition on twisted transfer operators that holds independently of the specific properties of F , we can write for sufficiently small positive u, where ω : R → [0, 1] is an integrable function with supp ω ⊂ [−1, 1] and r 0 is analytic such that r 0 (0) = 1.
For most results we require that (H2) There exists a function ω satisfying (4.1) and C ω > 0 such that Further, we assume the usual Doeblin-Fortet inequality: (H3) There exist σ 0 > 1 and C 0 , C 1 > 0 such that for all h ∈ B, for all n ∈ N and for all u > 0,

Refined assumptions on τ
For the purpose of obtaining limit laws for F (and in the end f t ) we assume that (H4) One of the following holds as t → ∞, for some slowly varying function ℓ and β ∈ (1, 2]. When β = 2, we assume τ / ∈ L 2 (µφ) and ℓ is such that the functionl(t) = t 1 ℓ(x) x dx is unbounded and slowly varying.
5 Limit laws for (τ, F ) In this section we obtain limit theorems for the Birkhoff sum τ n = n−1 j=0 τ •F j . Under the abstract assumptions formulated in Section 4, we obtain the asymptotics of the leading eigenvalue ofR(u) (as in Subsections 5.1 and 5.2 below). In turn, as clarified in Subsection 5.3, the asymptotics of this (family of) eigenvalues give the asymptotics of the Laplace transform E µφ (e −ub −1 n τn ), for suitable normalising sequences b n , proving the claimed limit theorems. Our result reads as follows.
. Note here that the slowly varying function ℓ from the general theory later on reduces to ℓ(n) = C * , because the tail estimates (2.3) have this form.
Remark 5.2 The above result holds for all piecewise C 1 (Y ) observables v in the space B 0 (Y ) defined in Appendix B.3. For item (i), we need to assume that v is in the domain of a stable law with index β ∈ (1, 2] with v / ∈ L 2 (µφ) and adjusted C * = C * (v), while for item (ii) we assume v ∈ L 2 (µφ) and a non-coboundary condition precisely formulated in Proposition 6.1 (ii).
The rest of this section is devoted to the proof of the above result.
By (H1)(iv), 1 is an eigenvalue forR(0). By (H3), there exists a family of eigenvalues λ(u) well-defined for u ∈ [0, δ 0 ), for some δ 0 > 0. Also, for u ∈ [0, δ 0 ),R(u) has a spectral decomposition for all n ∈ N, some fixed C > 0 and some σ 0 < 1. Here, P (u) is a family of rank one projectors and we write Equivalently, we normalise such that v(u), 1 = 1 and write The result below gives the continuity of the families P (u) and v(u) (in B w ).

Estimates required for the CLT under (H4)(ii)
For the CLT case we need the following Proposition 5.7 Assume (H1), (H2) and (H4)(ii). Suppose that τ = h • F − h, for any h ∈ B. Then there exists σ = 0 such that 1 − λ(u) = τ * u + σ 2 2 u 2 (1 + o(1)). We need to ensure that under the assumptions of Proposition 5.7, which do not require that τ ∈ B, there exists the required σ = 0. The argument goes by and large as [G04b, Proof of Theorem 3.7] (which works with a different Banach space) with the exception of estimating the second derivative of the eigenvalueλ defined below. The argument in [G04b, Proof of Theorem 3.7] for estimating such a derivative does not directly apply to our setup due to: i) the two Banach spaces B, B w at our disposal are not embedded in L p , p > 1; ii) the presence of log q(u) in Lemma 5.4. Our estimates below rely heavily on (H2) and (H4)(ii).
As usual, we can can reduce the proof to the case of mean zero observables. Letτ = τ − τ * . We recall that under (H4)(ii), R(τ h) ∈ B, for every in h ∈ B and the same holds forτ . By H1(v), DefineR(u) = R(e −uτ ). Clearly,R(u) has the same continuity properties asR(u). Letλ(u) be the associated family of eigenvalues. Recall that λ(u) is the family of eigenvalues associated witĥ R(u). Hence, λ(u) = e uτ * λ (u). As in the previous sections, let v(u) be the family of associated eigenvectors normalised such that v(u), 1 = 1. The next three results are technical tools required in the proof of Proposition 5.7; the third, which has a longer proof, is postponed to Subsection 5.2.1.
Lemma 5.9 Assume the setup of Proposition 5.7. Then where D(u) = o(1) as u → 0, and χ is defined as in (5.4).

Proof of Lemma 5.9
Proof of Lemma 5.9 Although d duR (u) is bounded in B w , a priori d du v(u) is not; this a consequence of Lemma A.1 (i). However, we argue that due to (H4)(ii), d du v(u) is well-defined at 0 and as such, we get the claimed formula for d , we have that for any h ∈ B, Recall that P (u) is the eigenprojection forR(u), so forR(u) as well.

Proof of Proposition 5.1
We start with part (i), i.e., the stable and non standard Gaussian laws. Let τ n and b n be as in Proposition 5.1. Using (5.2) and Corollary 5.6 (based on [AD01b]) we obtain that the Laplace transform E µφ (e −ub −1 n τn ) converges (as n → ∞) to the Laplace transform of either the stable law or of the N (0, 1) law. The Proposition 6.1 Assume (H1). Let g : M → R and let g * = Yḡ dµφ. Suppose that the twisted operatorR g (u)v = R(e −uḡ v) satisfies (H2) (with τ replaced byḡ). Then the following hold as T → ∞, w.r.t. µφ (or any probability measure ν ≪ µφ).
Proof We use that f t : M → M can be represented as a suspension flow F t : Y τ → Y τ . Under the present assumptions,ḡ satisfies all the assumptions of Proposition 5.1 (with τ replaced byḡ).
Letḡ n = n−1 j=0ḡ • F j and recall τ n = n−1 j=0 τ • F j . Proposition 5.1 (i) applies toḡ n ; the argument goes word for word as in the proof of Proposition 5.1 (i) for τ n . Under the present assumptions onḡ, item (ii) of Proposition 5.1 applies toḡ n with the argument used in the proof of Proposition 5.7 applying word for word withḡ instead of τ .
Item (i) follows from this together with Lemma 6.3 below (correspondence between stable laws/non standard Gaussian for the base map F and suspension flow. Item (ii) follows in the same way using [MTo04, Theorem 1.3] instead of Lemma 6.3.
The next result is a version of [S06, Theorem 7] (generalising [MTo04, Theorem 1.3]) for suspension flows which holds in a very general setup; in particular, it is totally independent of method used to prove limit theorems for the base map. Lemma 6.3 Assume Y τ dµφ < ∞ and let g ∈ L q (µ φ ), for q > 1. Suppose that there exists a sequence b n = n −ρ ℓ(n) for ρ ∈ (1, 2] and ℓ a slowly varying function such that b −1 n τ n − n Y τ dµφ is tight on (Y, µφ). Then the following are equivalent: Recall that g : Y τ → R and defineĝ T (y, u) = g T (y). Recallḡ(y) = τ (y) 0 g(y, u) du. By assumption, b −1 nḡ n converges in distribution on (Y, µ φ ). Hence, (as in [MTo04, Lemma 3.1]), b −1 nĝ n converges in distribution on (Y τ , µ τ φ ); this is a consequence of [E04,Theorem 4]. In what follows we adopt the convention b T = b(T ). Write Since ℓ is slowly varying, Step 2 of the the proof of Theorem 7] (a generalization of [MTo04, Lemma 3.4]) applies and thus, 1 , we have that τ ∈ L 1/ρ (µ φ ). By assumption, g ∈ L q (µ τ φ ), for q > 1. Hence, the assumptions of [MTo04, Lemma 2.1] are satisfied with a = p = 1/ρ and any b := q > 1 (with a, b, p as there). By [MTo04,Lemma 2 . To conclude, note thatĝ n[x,T ] (y, u) = g n[x,T ] (y). By [S06, Step 3 of the the proof of Theorem 7 Asymptotics of P(φ + sψ) for the flow f t In this section and the next we shall assume that F : Y → Y is Markov, which allows us to express our results in terms of pressure. We will also assume that P (φ) = 0 for all the potential functions φ involved. [S06,Theorem 8] gives a link between the shape of the pressure of a given discrete time finite measure dynamical system and an induced version (this was extended in [BTT18] to some infinite measure settings). Here we give a version of this result in the abstract setup of Section 4 along with suitable assumptions on a second potential ψ. Note that our assumptions are not directly comparable with those in [S06,Theorem 8].
Given ψ : M → R and its induced versionψ on Y , we define the 'doubly perturbed' operator R(u, s)v := R(e −uτ e sψ v).
Under (H5), we require the following extended version of (H3).
Lemma 7.2 Assume (H1), (H2), (H5) and (H7). Then for all s ∈ (0, δ 1 ), there exists c > 0 such that Proof Using (4.1), writê By (H7), (e sψ − 1)1 a ∈ B for any element a in the (Markov) partition A. Without loss of generality we assume that supp ω(t − τ ) is a subset of a finite union ∪ a∈At a of elements in A. Thus, τ and t are of the same order of magnitude for a ∈ A t and (e sψ − 1)1 supp ω(t−τ ) Bw ≪ max a∈At (e sψ − 1)1 a Bw .
Lemmas 5.3 and 7.2 ensure that (u, s) →R(u, s) is analytic in u and continuous in s, for all s ∈ [0, δ 0 ), when viewed as an operator from B to B w with where q(u) is as defined in Lemma 5.3.
For a further technical result exploiting (H7) we introduce the following notation. For u, s ≥ 0, set g = sψ −uτ , soR(u, s) = R(e g ). For N > 1 and x ∈ [a], define the 'lower' and 'upper' flattened versions of g as if τ (y) ≤ N for all y ∈ a, inf y∈[a] g(y) otherwise, if τ (y) ≤ N for all y ∈ a, sup y∈[a] g(y) otherwise.
Lemma 7.5 Assume (H2), (H5), (H6) and (H7). Then for fixed u, s ∈ [0, δ 0 ), The reason to introduce g ± N is that, unlike g, the potentials g ± N have summable variation, and therefore, P (φ + g ± N ) = log λ ± N (u, s) by [S99, Lemma 6]. Here [S99, Theorem 3] is used to equate Gurevich pressure in [S99, Lemma 6] and variational pressure in this paper, and then [S99, Theorem 4] together with the existence of the eigenfunctions v ± N (u, s) and eigenmeasures m ± N (u, s) of R ± N (u, s) and its dual, to conclude thatφ + g ± N are positively recurrent. From this, together with (7.4) and Lemma 7.5, we conclude P(φ + sψ − u) = log λ(u, s). (7.5) We note that this connection between the eigenvalues of a perturbed transfer operator and the pressure can be seen in similar contexts in [S06].
Proof of Lemma 7.5 Assume u = 0 and/or s = 0. Proceeding similarly to the argument of Lemma 7.2, we writeR Without loss of generality we assume that supp ω(t − τ ) is a subset of a finite union ∪ b∈Bt b of elements b ∈ A. Note that for any v ∈ C α (Y ), We only have to consider those b ∈ B t with b ∩ {τ > N} = ∅, otherwise (e g − e g ± N )1 b = 0. This together with (H7) gives Recall γ < 1 for κ > 1. By (H2) and the previous displayed equation, for any ǫ ∈ (0, 1 − γ), By equation (7.3), ∞ 0 t κ−ǫ M(t) B→Bw dt < ∞. Using (7.7) and (7.3), This together with the argument recalled in the proof of Lemma 5.4 gives where for fixed u, s ∈ [0, δ 0 ), v(u, s) and v ± N (u, s) are the eigenfunctions associated with λ(u, s) and λ ± N (u, s), respectively. Thus, The conclusion follows from these estimates together with the argument used in Lemma 5.5 and the first part of the proof of Lemma 7.4.
Proof Set u 0 = P(φ + sψ) which is positive for s > 0 by assumption. We first show that the pressures we are dealing with are finite. As in [S99, Lemma 2], finiteness of the transfer operator acting on constant functions is sufficient to give finiteness of the pressure, i.e., since we are dealing with positive u 0 , we just need to show sup x∈Y |(R(u, s)1)(x)| < ∞ for u ≥ 0. We estimate To show that P(φ + sψ − u 0 ) ≤ 0, suppose by contradiction that P(φ + sψ − u 0 ) > 0. We use the ideas of [S99, e.g. Theorem 2] to truncate the whole system, i.e., restrict to the system to the first n elements of the partition A and the corresponding flow. We write the pressure of this system as P n (·), so [S99, Theorem 2] says that P n (φ + sψ − u 0 ) > 0 for all large n. By the definition of pressure, there exists a measureμ so that Note that τ dμ < ∞ since τ is bounded on this subsystem. Abramov's formula (3.3) implies that the projected measure µ (which relates toμ via (3.4)) has contradicting the choice u 0 = P(φ + sψ).
To show that P(φ + sψ − u 0 ) ≥ 0, we will use the continuity of the pressure function u → P(φ + sψ − u) wherever this is finite (recall from above that we have finiteness for any u ∈ [0, u 0 )). By the definition of pressure, for any ǫ ∈ (0, u 0 ), there exists µ ′ with By (3.4), any invariant probability measure for the flow corresponds to an invariant probability measureμ ′ for the map F . By Abramov's formula, regardless of whether τ dμ ′ is finite or not. By continuity in u, P(φ + sψ − u 0 ) ≥ 0 as required.
The following result is a consequence of (7.5) and of Lemma 7.4.
The proof of a(ii) goes similar with the versions of the proofs of Proposition 5.1 (i), Proposition 6.1 (i) with β < 2 replaced by the argument used in the proofs of Proposition 5.1 (i), Proposition 6.1 (i) with β = 2.
The proof of (b) goes again similarly with the proofs of Proposition 5.1 (i), Proposition 6.1 (i) replaced by the argument used in the proofs of Proposition 5.1 (ii), Proposition 6.1 (ii) .

A Derivatives ofR(u) and v(u)
RecallR(u) can be written in the form (4.1) in Section 4.3 and that v(u) is its normalised eigenfunction.
Proof Given (4.1), we write As in Remark 4.1, as u → 0, d du r 0 (u) is bounded and we take d du r 0 (u) = γ 1 . Under (H2), the first term is bounded in · Bw . Also, by (7.3) with κ = 1, The bound log(log(1/u)) is far from optimal (given that d duR (u) Bw ≪ 1), but this suffices for the present purpose. The same estimate holds for | d du µφ(u)(1)|. These together with the formula v(u) = P (u)1 µφ(u)(1) give the second part of item (i). For the estimates on the second derivatives ofR(u), we just need to differentiate once more, recall that as in Remark 4.1, d 2 du 2 r 0 (u) = γ 2 , as u → 0 and repeat the argument above with κ = 2. The estimate d 2 du 2 v(u) Bw ≪ log(1/u) follows from item (i) together with (H1), (H3) and, again, arguments similar to [KL99] and [LT16, Proof of Corollary 3.11].

B Checking (H1)-(H8) for the almost Anosov flow
We start with a technical result that will be essential in verifying (H2) and (H6).
Lemma B.1 Assume that w(x, y) is a homogeneous function of degree ρ as in Proposition 2.2. Then Since also T 0 w(q(t)) dt = o(T ), we can find N ∈ N independent of n such that τ = T 0 1 + w(q(t)) dt + O(1) varies by at least 1 (but at most by O(N)) over {n ≤τ ≤ n + N}, and therefore r varies by at least 1 on this region. It follows that for each k, {r = k} ⊂ {n ≤τ ≤ n + N} for some n = n(k) and sup {r=k} τ − inf {r=k} τ is bounded, uniformly in k.
The proof for ρ ≥ 2 goes likewise.
B.1 Verifying (H1): recalling previously used Banach spaces Charts and distances: Throughout, W s denotes the set of admissible leaves, which consists of maximal stable leaves in elements of the partition It is convenient to arrange the enumeration of Y such that Y j = {r = j} for j ≥ 2, and use indices j ≤ 1 for the remaining (parts of) P i . To define distances between leaves, as in [BT17, Section 3.1], we use charts is the length of the (largest) unstable leaf in Y j . For any leaf W ⊂ Y j , we have the parametrisation so g Y j is a parametrisation of W in the chart. Suppressing Y j and using g andg for close-by stable leaves W ⊂ Y k andW ∈ W s ∈ Y ℓ we define their by where g andg are parametrisations of W andW in the appropriate chart. Here an empty sum k j=k+1 is 0 by convention. From here on, in the notation W · dµ s , the measure µ s refers to the SRB measure µφ conditioned on the stable leaf W : similarly for m s and Lebesgue measure. Hence the norms defined below are w.r.t. the invariant measure, not the conformal measure as in [BT17]. It is possible to do this due to Lemma B.2, specifically (B.7).
Definition of the norms: Given h ∈ C 1 (Y, C), define the weak norm by Given q ∈ [0, 1) we define the strong stable norm by The strong norm is defined by h B = h s + h u .
Definition of the Banach spaces: We define B to be the completion of C 1 in the strong norm and B w to be the completion in the weak norm. As clarified in [BT17, Lemma 3.2, Lemma 3.3, Proposition 3.2], (H1) holds with α = α 1 = 1. The spaces B and B w defined above are simplified versions of functional spaces defined in [DL08], adapted to the setting of (F, Y ). The main difference in the present setting is the simpler definition of admissible leaves and the absence of a control on short leaves. This is possible due to the Markov structure of the diffeomorphism.
As this is the same Banach space as in [BT17], most parts of (H1) can be taken from there. For H1(v), which is concerned with the transfer operator w.r.t. µφ, we first need a lemma.
Lemma B.2 Let γ = γ(q) be the angle between the stable and unstable leaf at q and h u 0 = dµ u dm u be the density of µφ conditioned on unstable leaves. Then where the sum is over all preimage leaves W j = F −1 (W s ) ∩ {r = j} and is piecewise C 1 on unstable leaves.
Proof We have the pointwise formulas for R 0 : L 1 (mφ) → L 1 (mφ) and R : L 1 (µφ) → L 1 (µφ): where |DF | = J mφ = | det(DF )|. This also shows that J µφ F = J mφ h 0 h 0 •F −1 and analogous formulas hold for J µ s F and J µ u F . Integration over a stable leaf W ∈ W s with preimage leaves W j gives Since dµ u = h u 0 dm u for a C 1 density h u 0 , and dmφ = dm s dm u sin γ, whence J mφ F = J m s F · J m u F · sin γ•F sin γ , the other formula K u W = 1 follows as well. Then [BT17,Equation (18)], leading to the parametrisation of the unstable foliation of the induced map overf = Φ hor 1 , shows that this foliation is C 1 . The analogous statement holds for the stable foliation. To obtain the unstable/stable foliations of F , we need to flow a bounded and piecewise C 1 amount of time (see Lemma B.1). Therefore q → γ(q) is piecewise C 1 . Hence the latter expression of K u W shows that it is piecewise C 1 on unstable leaves. Now (H1)(v) follows as in [BT17,Lemma 3.3], with J µφ and K u W j instead of |DF | −1 and |DF | −1 J W j F ; this can be done because K u W j is piecewise C 1 on unstable leaves by Lemma B.2.
Proof By Lemma B.1, there is N (independent of t) and k(t) such that contains supp(ω(t − τ )). We split v = v + − v − into its positive and negative parts and treat them separately. Also we assume without loss of generality that 0 ≤ ω ≤ 1 (otherwise we split ω in a positive and negative part as well). Then Let W s denote the stable foliation of F and decompose the measure µφ as v + dµφ = W s W s v + dµ W s dν u . Note that E is the union of leaves inW s , so 1 E is constant on each W s ∈ W s . Take ϕ : E → R + arbitrary such that ϕ| W s ∈ C 1 (W s ) with ϕ C 1 (W s ) ≤ 1 for each W s ∈ W s ∩ E. Similar to [BT17, Proposition 3.1], we get The same holds for v − , and this ends the proof.
Proof We need to estimate the B w -norm of M(t)v = R(ω(t − τ )v). This means that we need to take some leaf W s ∈ W s , the collection of stable leaves in Y stretching across an element of the As in the proof of Proposition B.3, we can split For v + and v − together, this gives Assumption (H8) is the same as (H2), with τ replaced by ψ 0 and e −uτ by e −sψ 0 . Its verification is entirely analogous to the above.
B.3 Verifying that Rζ ∈ B for a large class of ζ (includingψ in Theorem 2.5 (b)) and completing the verification of (H4) Assumption (H4)(i) and the tail estimate part of (H4)(ii) (so that τ ∈ L 2 (µφ)) is verified in Proposition 2.4. To check the remaining part of (H4)(ii) and assumption onψ in Theorem 2.5 (b) we give a more general result in Lemma B.5 below. This result ensures that givenψ as in Theorem 2.5 (b) we have Rψ ∈ B. This is needed to verify the abstract assumptions of Theorem 8.2(b) forψ as in Theorem 2.5 (b).
Lemma B.5 For 0 < κ < β and ζ : Y → R piecewise C 1 , define Proof Recall that, by Proposition 2.2 and Lemma 2.3, r = O(τ ) and vice versa. Abbreviate Y j = {r = j}; for large j these are strips close to the stable manifold W s p of the neutral fixed point, and bounded by stable and unstable curves and both F −1 and |DF | −1 are C 1 on each F (Y j ).
For the stable norm s , choose an arbitrary stable leaf W and q-Hölder function ϕ ∈ C q (W ) with |ϕ| C q (W ) ≤ 1. Let W j = F −1 (W ) ∩ Y j . By Lemma B.2, and the fact that K u ≪ j −(1+β) and The stable part s of B is treated in the same way. Now for the unstable part u , let ϕ such that |ϕ| C 1 (W ) ≤ 1. Using [BT17, Equation (43)], which expresses W j hφ dm in terms of the parametrisation of W j , for any nearby leaves W,W ∈ Y , we define a diffeomorphism v :W → W as in [BT17, Proof of Proposition 3.2 (unstable norm part)]. Let v j = F −1 • v • F :W j → W j be the corresponding bijection between the preimage leaves W j ,W j ⊂ Y j . Then we compute, because as in the first part of this proof, the sum in the above expression is bounded. For the sum S 2 , using (B.7) we split By distortion estimate [BT17,Equation (40)], this is bounded by Cd(W,W ) 1 J u mφ for some uniform distortion constant C > 0. Therefore Now for S 3 , the weighted u part of the norm 0 gives |ζ • v j − ζ| ≪ j κ+1+β d(W j ,W j ) ≪ j κ+1+β d(W,W )L u (Y j ). As in the first half of the proof, |K u W j | ∞ ≪ j −(1+β) . We have L u (Y j ) ≪ τ 1+β due to the small tail estimates (2.3), so Therefore Rζ B < ∞ and the proof is complete.
Since C ′ −ψ ∼ Cτ κ , this corollary can also be used to show that R(ψh) ∈ B.
As in the statement of Proposition 5.1 (ii) and Proposition 5.7 we need Corollary B.7τ := τ − τ * cannot be written as h • F − h for any h ∈ B.
Proposition B.8 Assume thatψ satisfies (H5) and letR(u, s)v = R(e −uτ e sψ v). Then there exists σ 1 ∈ (0, 1) and δ, C 0 , C 1 > 0 such that for all h ∈ B, n ∈ N, 0 ≤ s < δ, and u ≥ 0, Before turning to the proof, we need another lemma (which we will apply with g ≡ 1, but the general g is needed for the induction in the proof).
Lemma B.9 Assume thatψ satisfies (H5) (in particular,ψ < C ′ − C −1 τ κ < ∞ for some C ′ , C > 0 and κ > 1/β). There exists N ∈ N depending only of F : Y → Y such that for all positive integrable functions g that are bounded and bounded away from zero, the following holds. There exist δ > 0 such that for all s ∈ [0, δ], all admissible stable leaves W ⊂ Y and all n ∈ N, where K u W ′ F n equals the analogue of K u W ′ from (B.7) for the iterate F n on the preimage leaf W ′ .
Proof Let P 1 be the partition element containing the images F ({r = k}) of the strips {r = k} k≥2 , see Figure 3. Let N ∈ N be such that F N −1 (P 1 ) intersects each P i , i ≥ 1. Let W be an arbitrary stable leaf in P i for some i ≥ 1, and let W be the collection of preimage leaves under F −N . By a change of coordinates, for any integrable function g.
Given a ≥ 1 to be chosen later, set W + = {W ′ ∈ W : inf W ′ τ κ ≥ 2NaC supψ} and W − = W \ W + . Then there is ǫ > 0 (depending only on the geometry of the Markov map F and sup g inf g ) such that W ′ ∈W + W ′ g • F N K u W ′ F N dµ s ≥ ǫ W g dµ s . Since W ′ dµ s is proportional to the measure of the element in N −1 i=0 F −i (P) that W ′ belongs to, Proposition 2.4 gives ǫ ≫ µφ({τ > (2NaC supψ) 1 κ }) ≫ Ba − 1 β ′ for some B > 0 and β ′ := κβ > 1. We haveψ N ≤ (N − 1) supψ − C −1 τ κ ≤ (N − 1 − 2Na) supψ ≤ −Na supψ on W + , andψ N ≤ N supψ on W − . Therefore β ′ −1 }, so that a 0 > 1. This means that b(x) ≤ 1 for all 1 ≤ x ≤ a 0 , i.e., for all 0 ≤ s ≤ log a 0 N supψ . Now for general n = pN + q with 0 ≤ q < N, we use induction on p. Let W p (W ) be the collection of preimage leaves of W under F −pN . Assume by induction that W ′ ∈W p−1 (W 1 ) W ′ e sψ (p−1)N g • F (p−1)N K u W ′ F (p−1)N dµ s ≤ W g dµ s for all stable leaves W 1 and g as above. Then This way we proved the lemma for all multiples of N. For 0 < q < N, we get an extra factor e sq supψ . This proves the lemma.