Quadratic irrationals and linking numbers of modular knots

A closed geodesic on the modular surface gives rise to a knot on the 3-sphere with a trefoil knot removed, and one can compute the linking number of such a knot with the trefoil knot. We show that, when ordered by their length, the set of closed geodesics having a prescribed linking number become equidistributed on average with respect to the Liouville measure. We show this by using the thermodynamic formalism to prove an equidistribution result for a corresponding set of quadratic irrationals on the unit interval.


Introduction
Let M = PSL 2 (Z)\H denote the modular surface; here H denotes the upper half plane endowed with the hyperbolic metric, and PSL 2 (Z) acts by isometries on H via linear fractional transformation. The closed geodesics are the periodic orbits for the geodesic flow on the unit tangent bundle T 1 M. Following the results of Ghys [Gh07] and Sarnak [Sa10] on the linking numbers of modular knots, we continue the study of the set of primitive closed geodesics having a prescribed linking number, and in particular their distribution on T 1 M.
In [Gh07], Ghys showed that T 1 M is homeomorphic to the threesphere without a trefoil knot, and hence, a closed geodesic can be thought of as a knot in this space. He then computed the linking number of a such a knot with the trefoil knot in terms of a certain arithmetic invariant. If γ denotes an oriented primitive closed geodesic in T 1 M, we will denote this linking number by link(γ) and we refer to it as the linking number of this primitive geodesic.
Motivated by this result, Sarnak [Sa10] indicates how a careful analysis of the Selberg trace formula for modular forms of real weights can be used to study the number of primitive closed geodesics with a prescribed linking number and bounded length (see also [Mo10]). In particular, if C n (T ) denotes the set of primitive closed geodesics, γ, Date: May 1, 2014. This work was partially supported by NSF grant DMS-1237412. 1 with linking number link(γ) = n and length ℓ(γ) ≤ T , then his analysis implies that |C n (T )| ∼ e T 3T 2 (that is |C n (T )|/ e T 3T 2 → 1 as T → ∞). For comparison we recall that the prime geodesic theorem states that |C(T )| ∼ e T T , where C(T ) is the set of primitive closed geodesics with ℓ(γ) ≤ T .
In order to study the average distribution of these closed geodesics we adapt the approach of Pollicott [Po86], using the correspondence between closed geodesics and quadratic irrationals together with the thermodynamic formalism for the Gauss map.
There is a well known correspondence, going back to Artin [Ar24] and beautifully explained by Series [Se85], between the periodic orbits of the Gauss map and the primitive closed geodesics on the modular surface. This correspondence gives a natural ordering of the quadratic irrationals, that is, for x ∈ Q we let ℓ(x) denote the length of the corresponding primitive closed geodesic and we order them according to this length.
Using Mayer's [Ma76] thermodynamic formalism for the Gauss map, in [Po86] Pollicott showed that, with this ordering, the quadratic irrationals become equidistributed on (0, 1) with respect to the Gauss measure given by dν = 1 log(2) dx 1+x . Then, using the correspondence between the Gauss map and the geodesic flow, he deduced the average equidistribution of the full set of closed geodesics when ordered by length.
Remark 1.1. There is another natural, and well known, correspondence between primitive closed geodesics on the modular surface and equivalence classes of binary quadratic forms (see [Sa82]). Instead of ordering the geodesics by length one can order them according to their discriminant (that is, the discriminant of the corresponding quadratic form). In this setting, Duke's theorem [Du88] shows that the set of closed geodesics of a given discriminant also become equidistributed in T 1 M as the discriminant goes to infinity. Since closed geodesics with the same discriminant also have the same length, Duke's theorem also implies that the set of closed geodesics with a fixed length become equidistributed when the length goes to infinity, which also implies the average equidistribution.
The linking number of a closed geodesic can be computed (up to a sign) from the corresponding continued fraction expansion. Specifically, we will show below that if x ∈ Q has an even minimal periodic expansion x = [a 1 , . . . , a 2n ], then its T-orbit corresponds to two primitive geodesics γ,γ (related by orientation reversal symmetry) having the same length and opposite linking numbers given by On the other hand, if x ∈ Q has an odd minimal expansion x = [a 1 , . . . , a n ] (with n odd), then it corresponds to a single primitive geodesic (invariant under orientation reversal symmetry) having length and linking number zero.
To any quadratic irrational x ∈ Q with a minimal even expansion x = [a 1 , . . . , a 2n ] we attach the alternating sum Combining the dynamical approach of [Po86] with the results from [Sa10] we show that the quadratic irrationals with a given alternating sum (up to a sign) become equidistributed on average with respect to the Gauss measure. Specifically, for any n ∈ Z let where Q + n (T ) = Q n ∪ Q −n . The set of quadratic irrationals x ∈ Q with Alt(x) = 0 contains the set Q odd of quadratic irrationals with an odd periodic expansion. We show that this (much smaller) set is also equidistributed.
Theorem 2. Let Q odd (T ) denote the set of quadratic irrationals with an odd minimal expansion and ℓ(x) ≤ T . Then |Q odd (T )| ∼ 3 log(2) π 2 e T /2 T and for any f ∈ C([0, 1]) we have Remark 1.2. We note that the reason that Q odd (T ) is much smaller than Q even (T ) is an artifact of our ordering. In a slightly different ordering given byl(x) = −2 n j=1 log(T j x) (which is more natural when considering only the minimal expansion x = [a 1 , . . . , a n ]) we get that roughly half the points are odd, half are even, and each is equidistributed.
These equidistribution results of quadratic irrationals imply the equidistribution of the corresponding sets of closed geodesics. Given a test function F ∈ C c (T 1 M) and a closed geodesic γ we denote by where ϕ t : T 1 M → T 1 M is the geodesic flow and x ∈ T 1 M is any point on the geodesic. As a consequence of Theorem 1 we get Theorem 3. |C n (T )| ∼ e T 3T 2 and for any F ∈ C c (T 1 M) where C + n (T ) = C n (T ) ∪ C −n (T ). Remark 1.3. It is likely that the same equidistribution result holds for the smaller set C n (T ) instead of C + n (T ). Note that the time reversal of a primitive geodesic has the same length and an opposite linking number. Since a closed geodesic and its time reversal have the same projection to the base manifold M, our result implies that the closed geodesics in C n (T ) at least become equidistributed on the base manifold.
Remark 1.4. In addition to the thermodynamic formalism, another crucial ingredient in the proof of Theorem 1 (and hence also Theorem 3) is the estimate (2.1) proved in [Mo10,Sa10] using the trace formula for modular forms of real weight. Consequently, our result does not give an independent proof for the asymptotics |C n (T )| ∼ e T 3T 2 . It would be interesting if one could obtain such a result using an entirely dynamical approach.
Remark 1.5. One should compare Theorem 3 to analogous results on the counting and equidistribution of closed geodesics on a compact hyperbolic surface lying in a prescribed homology class. In this case, Katsuda and Sunada [KS90], Lalley [La89], and Pollicott [Po91] obtained similar results using an entirely dynamical approach (which works also in variable negative curvature). The asymptotics for the number of closed geodesics in a fixed homology class were previously obtained by Phillips and Sarnak [PS87] using the Selberg trace formula (for compact hyperbolic surfaces). In [Ze89] Zelditch generalized the trace formula to give another proof of the equidistribution of closed geodesics in a fixed homology class (with explicit bounds on the rate of equidistribution). It should be possible to further generalize Zelditch's approach to give another proof for the equidistribution of closed geodesics with a prescribed linking number (such an approach would also give bounds for the rate of equidistribution).
Let C i (T ) ⊂ C 0 (T ) denote the subset of inert geodesics, that is, the primitive geodesics that are left invariant under orientation reversal symmetry. From Theorem 2 we get Remark 1.6. The result in Theorem 4 is not new. In [Sa07], using the Selberg trace formula for PGL 2 (Z), Sarnak showed that |C i (T )| ∼ e T /2 T . He also showed there, that if a primitive geodesic is inert, then all other primitive geodesics with the same discriminant are also inert. Consequently, the average equidistribution of the inert geodesics already follows from Duke's Theorem. We note however that the proof we give here is entirely dynamical and does not rely on the Selberg trace formula nor on Duke's theorem.

Background and notation
We write A B (or A = O(B)) to indicate that A ≤ cB (or |A| ≤ c|B|) for some constant c. We use the notation A(T ) ∼ B(T ) to indicate that A(T )/B(T ) → 1 as T → ∞.
2.1. The modular surface. We denote by H = {z = x + iy : y > 0} the upper half plane. With the identification T z H ∼ = C, the hyperbolic metric on T z H is given by ξ, η z = ℜ(ξη) y 2 , and the unit tangent bundle is T 1 H = {(z, ξ) ∈ H × C : |ξ| = Im(z)}. In this model the geodesics are either semi-circles orthogonal to the real line or vertical lines. The group PSL 2 (R) acts on the hyperbolic plane by isometries via linear fractional transformations, that is, We can thus identify T 1 H ∼ = PSL 2 (R) and the unit tangent bundle of the modular surface M = PSL 2 (Z)\H as T 1 M ∼ = PSL 2 (Z)\PSL 2 (R). With these identification the Liouville measure µ on T 1 M is the projection of the Haar measure of PSL 2 (R). This measure projects down to the hyperbolic area on M normalized so that Area(M) = 1. Specifically, in the coordinates (z, ξ) = (z, ye iθ ) we have dµ(x, y, θ) = 3 π 2 dxdy y 2 dθ. There is a one-one correspondence between oriented (primitive) closed geodesics on T 1 M and (primitive) hyperbolic conjugacy classes in PSL 2 (Z), and we denote by {A γ } the conjugacy corresponding to a closed geodesic γ. Here a closed geodesic is called primitive if it wraps once around and a hyperbolic element is primitive if it is not the power of some other hyperbolic element.
Recall that a hyperbolic element A ∈ PSL 2 (Z) has two fixed points on the real line (which are conjugate quadratic irrationals). We note that if A ∈ {A γ } then γ has a liftγ to the upper half plane hitting the real line at the two fixed points of A (different lifts will correspond to different conjugates of A γ ).
If (z, ξ) ∈ T 1 M is a point on a closed geodesic, γ, then (z, −ξ) and (−z, −ξ) are points on two different closed geodesic we call the time reversal, γ −1 , and the orientation reversal,γ, of γ respectively. If {A γ } is the conjugacy class in PSL 2 (Z) corresponding to γ, then . Following [Sa07], we call a primitive geodesic inert ifγ = γ and reciprocal if γ −1 = γ. We note for future reference that γ is inert if and Page 227]).
2.4. Linking numbers. We recall that a closed geodesic, γ, on T 1 M gives rise to a knot in the 3-sphere with a trefoil knot removed, and that the linking number, link(γ), is the linking number of this knot with the trefoil knot. In [Gh07], Ghys showed that link(γ) = Ψ(A γ ), where Ψ : PSL 2 (Z) → Z denotes the Rademacher function.
The Rademacher function Ψ(A) depends only on the conjugacy class of A, and can be computed by expressing A as a word in the generators The reader can consider the following as the definition of Ψ(A) (see [RG72, Page 54] for another equivalent definition and some properties of the Rademacher function). Any element A ∈ PSL 2 (Z) is conjugated to either S, U, U −1 or From this, together with the relation wSU = SU −1 w, we get that the linking number changes sign under time reversal or orientation reversal, that is link(γ) = link(γ −1 ) = −link(γ). In particular, if γ is inert or reciprocal then link(γ) = 0.
The Rademacher function also comes up in the multiplier system of the Dedekind eta function

Specifically, for
with ν 1/2 (A) = e iπΨ(A)/12 . Consequently, for any real r ∈ (−6, 6) we have that ν r (A) = e iπrΨ(A)/6 is a multiplier system for modular forms of real weight r. Using this observation, in [Sa10] Sarnak shows how a careful analysis of the Selberg trace formula for modular forms of real weight (see [He83, Chapter 9]) implies Some of the delicate estimates needed for the proof were done by Mozzochi in [Mo10]. From this, the estimate |C n (T )| ∼ e T 3T 2 (and much more) is obtained by integrating (2.1) against e −πinr/6 . Remark 2.1. We note for future reference that (2.1) still holds if we replace the sum over primitive geodesics by a sum over all closed geodesics. This is because the prime geodesic theorem implies that the contribution of the non-primitive geodesics to such a sum is bounded by O(T e T /2 ).

Closed geodesics and quadratic irrationals
In this section we recall the results of Series [Se85] on coding the geodesic flow as a suspended flow over the Gauss map, and use it to reduce the problem of equidistribution of closed geodesics to the equidistribution of quadratic irrationals.
3.1. The Gauss map and continued fraction. Any x ∈ (0, 1) has a continued fraction expansion (which terminates if and only if x is rational). When the expansion is periodic, that is, if there is n ∈ N such that a n+j = a j for all j, we write x = [a 1 , . . . , a n ]. The points with such a periodic expansion are precisely the quadratic irrationals x ∈ (0, 1) whose conjugate satisfies x < −1. Here the conjugatex of a quadratic irrational is the second root of the same quadratic polynomial.
Let T : (0, 1] → [0, 1) denote the Gauss map given by T(x) = {1/x} where {x} denotes the fractional part of x. The Gauss map acts on the continued fraction expansion by a shift to the left, hence, the periodic points for T are the points with a periodic expansion. LetT . The extended mapT acts on the continued fraction of both points together by a shift HenceT is invertible and there is a correspondence betweenT-closed orbits and T-closed orbits. Specifically, the T-orbit of x = [a 1 , . . . , a n ] corresponds to theT-orbit of (x, −1/x) = ([a 1 , . . . , a n ], [a n , . . . , a 1 ]).
We recall that the Gauss measure ν on [0, 1], given by is the unique T-invariant probability measure on [0, 1] equivalent to Lebesgue measure. We extend the Gauss measure to the measureν on [0, 1] × [0, 1] given by We note thatν isT-invariant, that its projection to each factor is ν, and that this is the only measure satisfying these properties.
3.3. Periodic orbits. The above coding gives a period preserving correspondence between primitive closed geodesics and primitive closed ψ t -orbits. Moreover, the closed ψ t -orbits are easily classified in terms of the periods of the Gauss map, that is, the purely periodic quadratic irrationals.
Specifically, to any purely periodic quadratic irrational x ∈ (0, 1) we have two closed ψ t -orbits, the orbits of (x, −1/x, 0) and of (x, −1/x, 1) (clearly these only depends on the T-orbit of x). We note that the two closed geodesics corresponding to the same T-orbit are a pair, γ,γ, of orientation reversed geodesics. For x ∈ Q with a minimal even expansion x = [a 1 , . . . , a 2n ], the length of each of the corresponding closed geodesics is given by (see [Se85, Section 3.2]).
Remark 3.1. In this formula it is important that the expansion x = [a 1 , . . . , a 2n ] is the minimal even expansion of x, in the sense that there is no n ′ < n with x = [a 1 , . . . , a 2n ′ ]. If x has an odd minimal expansion, x = [a 1 , . . . , a n ] with n odd, the length is still computed using the minimal even expansion x = [a 1 , . . . , a n , a 1 , . . . , a n ].
3.4. Linking numbers. The linking number of a closed geodesic can be computed from the minimal continued fraction expansion of the corresponding quadratic irrational (up to a sign). To do this we will use the following lemma.
Lemma 3.1. Let x = [a 1 , . . . , a n ] be a minimal expansion of a quadratic irrational. Then the cyclic group of matrices in PGL 2 (Z) fixing x is generated by B a 1 · · · B an with B a = 0 1 1 a .
Proof. Let B ∈ PGL 2 (Z) be a matrix fixing x. If we have that B = B b 1 · · · B b k for some k ∈ N and b 1 , . . . , b k ∈ N then [a 1 , . . . , a n ] = x = Bx = [b 1 , . . . , b k , a 1 , . . . , a n ], which can only happen if k = nm and b nl+j = a j for all 0 ≤ ℓ ≤ m and 1 ≤ j ≤ n, so that B = (B a 1 · · · B an ) m . Consequently, we need to show that either B or B −1 can be expressed as such a product. First, if one of the coefficients of B is zero then it must be in one of the forms ( 0 ±1 1 d ) , ( ±1 0 c 1 ) , ( ±1 b 0 1 ), or ( a ±1 1 0 ). Of these the only ones having two fixed points x ∈ (0, 1) andx ∈ (−∞, −1) are ( 0 1 1 d ) = B d with d ∈ N and ( −a 1 1 0 ) = B −1 a with a ∈ N. We may thus assume from now on that B has non-zero coefficients.
Next, we show that either ±B or ±B −1 has positive coefficients.
so we still have a matrix with positive coefficients. We can reiterate this until we get to a matrix with a ′ = 0 or b ′ = 0 (this will terminate after at most a steps). We thus see that there are b 1 , . . . , b k ∈ N ∪ {0} such that We can easily exclude the case of ( 1 0 c ′ 1 ) by considering the action on the fixed pointx ∈ (−∞, −1) (notice that ( 1 0 . This just leaves us with the two cases of B = B b 1 · · · B b k 0 ±1 1 d ′ . We note that ( 0 1 1 a ) ( 0 1 1 0 ) ( 0 1 1 b ) = ( 0 1 1 a+b ), so we may assume that all b j ∈ N except perhaps b k and b 1 which could also be zero.
In the first case, to reduce this to the first case of Bx < −1 so we must have thatd ≥ 1 which is the case we dealt with above.
Proposition 3.2. Let x = [a 1 , . . . , a 2n ] be the minimal even expansion of a quadratic irrational, let γ be one of the corresponding primitive closed geodesics, and let {A γ } denote the corresponding hyperbolic conjugacy class. Then Proof. Let A = B a 1 · · · B a 2n ∈ PSL 2 (Z). Since x = [a 1 , . . . , a 2n ] is the minimal even expansion, by Lemma 3.1 we have that A, A −1 are the unique primitive hyperbolic elements in PSL 2 (Z) fixing x,x. Since (some conjugate of) A γ also has these fixed points then either A ∈ {A γ } or A −1 ∈ {A γ }, and hence Ψ(A γ ) = ±Ψ(A).
Let V = US and note that B a = wSV a = SV −a w so that Using the relations SV a S = (SU) a and V −a = (SU −1 ) a , we can write to get that indeed Ψ(A) = −a 1 + a 2 − a 3 + . . . + a 2n .
Note that if x = [a 1 , . . . , a n ] is a minimal odd expansion then its minimal even expansion is x = [a 1 , . . . , a n , a 1 , . . . , a n ]. In this case, if A ∈ PSL 2 (Z) is the primitive element fixing x then the above argument shows that Ψ(A) = 0. Consequently, a primitive geodesic corresponding to x ∈ Q odd has linking number zero. We now show that these are precisely the inert primitive geodesics. Proof. Let γ be a closed geodesic, x = [a 1 , . . . , a n ] a quadratic irrational in the corresponding T-orbit, and A ∈ {A γ } the primitive hyperbolic element fixing x. We recall that γ is inert if and only if A = B 2 for some B ∈ PGL 2 (Z) with det(B) = −1. In this case, since any B ∈ PGL 2 (Z) with det(B) = −1 has a fixed point on the real line and any fixed point of B is also a fixed point of A, the fixed points of B are also x andx. Lemma 3.1, now implies that B = B a 1 · · · B an and hence det(B) = (−1) n so n must be odd.
3.5. Equidistribution. We can use the correspondence between closed geodesics and quadratic irrationals to relate equidistribution of closed geodesics on T 1 M to equidistribution of quadratic irrationals on [0, 1].
Let C ′ ⊆ C denote any set of oriented primitive closed geodesics which is invariant under orientation reversal symmetry (that is, γ ∈ C ′ if and only ifγ ∈ C ′ ). Let Q ′ ⊆ Q denote the corresponding set of quadratic irrationals (i.e., the set of points obtained as endpoints of lifts of geodesics from C ′ ). Let C ′ (T ) (respectively Q ′ (T )) denote the set of γ ∈ C ′ with ℓ(γ) ≤ T (respectively x ∈ Q ′ with ℓ(x) ≤ T ). Consider the counting function The following proposition reduces Theorems 3 and Theorem 4 to Theorem 1 and Theorem 2 respectively.
For k = (k 1 , . . . , k n ) consider the open interval It is thus sufficient to consider test functions of the form f = 1 1 I k ×I k ′ for any k = (k 1 , . . . , k n ), k ′ = (k 1 , . . . , k n ′ ). That is, we need to show that lim SinceT acts on the continued fraction by a shift, for f = 1 1 I k ×I k ′ we have that f •T n ′ = 1 1 I k ′′ ×[0,1] where k ′′ = (k ′ n ′ , . . . , k ′ 1 , k 1 , . . . , k n ). On the other hand, since acting byT only permutes the periodic Torbits (which all have the same length) and the set Q ′ is by definition composed of complete T-orbits, then any function f on [0, 1] × [0, 1] satisfies 1 In particular for f = 1 1 We can now approximate 1 1 I k ′′ by continuous functions get that where the last equality follows from the invariance ofν underT.
We then have and for any pair γ,γ of orientation reversed geodesics where Q γ ⊂ (0, 1) is the set of endpoints (in (0, 1)) of lifts of γ to H.
Next, we show that (3.4) implies (3.5). Let Summation by parts gives The bound |C ′ (T )| T α e cT implies that G 1 (T ) T α+1 e cT . We can thus bound |G F (t)| G 1 (t) t α+1 e ct under the integral in (3.6) to get that Using the lower bound |C ′ (T )| T β e cT we get that H F (T )

Proof of main Theorems
Theorems 3 and 4 follow directly from Theorems 1 and 2 (together with Proposition 3.4). In order to prove Theorems 1 and 2 it is enough to establish (1.3) and (1.4) for a dense set of test functions in C([0, 1]). Let B denote the Banach space of holomorphic functions on the disc D 3/2 (1) = {z ∈ C : |z −1| < 3/2} that are continuous on the boundary. The family of test functions {f | [0,1] |f ∈ B} is clearly dense in C([0, 1]) (as it contains all polynomials). For these test functions we will follow the approach of [Po86], that is, we study the analytic continuation of a certain η-unction and use a suitable Tauberian theorem.

We will show
Proposition 4.1. The series η odd f (s) absolutely converges for ℜ(s) > 1, has a meromorphic continuation to the half plane ℜ(s) > 1/2 with one simple pole located at s = 1 and residue Res s=1 η odd f = 6 log(2) π 2 1 0 f dν. Proposition 4.2. The series η f,θ (s) absolutely converges for ℜ(s) > 1, has a meromorphic continuation to the half plane ℜ(s) > 1/2 with at most one simple pole in the half plane ℜ(s) > 3/4 located at s θ = 1−3|θ| (when |θ| < 1/12). Moreover, in this case the residue Remark 4.1. With a little more work one can replace the error term of O(|θ| 1/2 ) by O(|θ|| log 1 |θ| | 2 ), but for our purpose this weaker bound will be sufficient. π 2 1 0 f 0 dν (see the proof of Proposition 4.5). The proof of the general case will follow from these two cases together with a perturbation theory argument.
4.2. The Tauberian Theorem. We postpone the proof of Propositions 4.1 and 4.2 to the following sections. We will now show how to use them together with an appropriate Tauberian theorem to prove Theorems 1 and 2. Specifically, we will need the following version of the Wienner-Ikehara Tauberian Theorem (see [Ko04,§III,Theorem 4
Proof of Theorem 1. We will apply the Wiener-Ikehara Tauberian theorem for S(t) = S f,n (t) given by By adding to f a sufficiently large multiple of f 0 (x) = −2 log(x) we can insure that S(t) is nondecreasing. The Laplace transform of S(t) is given by which absolutely converges for ℜ(s) > 1. We can obtain g(s) by integrating the derivative of η f,θ against e −iπnθ , specifically we have Using Proposition 4.2 we can write with φ(s) holomorphic in ℜ(s) > 3/4 and R f (θ) = 0 for |θ| > 1/12. Plugging this back in (4.4) we get g(s) = c with Φ(s) holomorphic in ℜ(s) > 3/4 and c = 1/2 n = 0 1/4 n = 0 .
The bound implies that the function g 1 (y) = c is in L 1 (−λ, λ) for any λ > 0 and that g x → g 1 in L 1 (−λ, λ). The Wiener-Ikehara Tauberian theorem now implies that Since the contribution of all k > 1 to this sum is bounded by O(T 2 e T /2 ) we get and summation by parts gives In particular, taking f = 1 we get that |Q + n (T )| ∼ A 1 e T T and since |Q n (T )| = |Q −n (T )| (as the Gauss map gives a bijection between them) we get that Finally, for any other f ∈ B, Proof of Theorem 2. From Proposition 4.1 we can write with φ(s) holomorphic in ℜ(s) > 1 2 . The Wiener-Ikehara Tauberian theorem now implies that 2 k odd y∈Q odd and as before we can ignore the contribution of k > 1 to get y∈Q odd ℓ(x)≤T f (y) ∼ 3 log(2) π 2 f dν e T /2 .

4.3.
The transfer operator. In order to obtain the analytic continuation of η f,θ and η odd f , we relate them to the Fredholm determinant of a suitable Ruelle-Perron-Frobenius transfer operator.
Fix a test function f ∈ B. For any complex numbers s, w ∈ C with ℜ(s) > 1 2 let χ s,w (z) = z 2s e wf (z) . For any θ ∈ [−1, 1] we define the Ruelle-Perron-Frobenius operator L θ s,w : B → B by and we denote by T θ s,w = L θ s,w L −θ s,w . For small θ (and w) we think of L θ s,w as a perturbation of L 0 s,w (and L 0 s,0 ) studied by Pollicott [Po86] and Mayer [Ma76,Ma91].
The same arguments as in [Ma76] show that this operator is a nuclear operator. Specifically we show the following. where Alt(a) = 2n j=1 (−1) j a j and ℓ(a) = −2 2n j=1 log(T j [a 1 , . . . , a 2n ]). (Note that the expansion [a 1 , . . . , a 2n ] here is not necessarily minimal.) Proof. For any a ∈ N n consider the operator L a : B → B given by L a g(z) = g([a 1 , . . . , a n ; z]) n j=1 χ s,w ([a j , . . . , a n ; z]), where we use the notation [a 1 , . . . , a n ; z] = Using the theory of Fredholm determinants for nuclear operators on Banach space we get Corollary 4.1. The function Z θ (s, w) = det(1 − T θ s,w ) is holomorphic in w, s for ℜ(s) > 1 2 and is non-zero unless 1 is an eigenvalue for T θ s,w . Moreover, for ℜ(s) > 1 it is given by [a 1 , . . . , a 2n ]) .
Next we consider the logarithmic derivatives and note that it is closely related to the function η f,θ defied in (4.1).
Proposition 4.5.η f,θ (s) has a meromorphic continuation to ℜ(s) > 1 2 with at most one simple pole in ℜ(s) > 3 4 located at s θ = 1 − 3|θ| when |θ| < 1 12 and no poles in ℜ(s) > 3 4 otherwise. Proof. Since Z θ (s, w) is analytic in ℜ(s) > 1 2 we get thatη f,θ (s) is meromorphic there, and it can have a pole only at values of s for which 1 is an eigenvalue of T θ s,0 (notice that this no longer depends on the test function f ). Next, note that 1 is a (simple) eigenvalue of Let ζ θ (s) =ζ θ (s) −ζ θ (s + 1) and note that, using the correspondence between closed geodesics and quadratic irrationals, we can also write this as ζ θ (s) = Now (2.1) implies that | e s θ t s θ − Λ θ (t)| = O(e 3t/4 ) so the integral on the right absolutely converges for ℜ(s) > 3/4, and hence defines an analytic function there.
Proposition 4.6. There is some δ > 0 such that For the proof we will use some perturbation theory of compact operators (see e.g. [Ka82] for some of the standard arguments). In particular, we will need the following estimates for the modulus of continuity of the operator T θ s,w with respect to the θ parameter, as well as the modulus of its partial derivatives. Proof. Since T θ s,w = L θ s,w L −θ s,w it is enough to prove these bounds for L θ s,w . For ℜ(s) ≥ c > 3/4 for any g ∈ B with g = 1 we can bound Since this holds for all z ∈ D 3/2 (1) and all g ∈ B with g = 1 indeed ∂ ∂s (L θ s,0 − L 0 s,0 ) |θ| 1/2 . The arguments for L θ s,0 and ∂ ∂w L θ s,w are identical.