The Bowen–Series coding and zeros of zeta functions

We recall the following standard definition of a Fuchsian group.

For the moment, we will focus on the case when $V=\mathbb{D}/\Gamma$ is a compact surface and the associated Fuchsian group $\Gamma$ is called co-compact. In this case the limit set is the entire circle. The elements of the Fuchsian group $\Gamma$ have a few useful and important properties.

Given a Fuchsian group we want to define a piecewise continuous expanding Markov map called the Bowen–Series map defined on the boundary $\mbox{\hskip 5.30904pt{1 0 -.25 1}\hbox to0.0pt{\hss\leavevmode\hbox{\set@color$\partial$}}}\mathbb{D}$.

In order to define the map $T\colon\mbox{\hskip 5.30904pt{1 0 -.25 1}\hbox to0.0pt{\hss\leavevmode\hbox{\set@color$\partial$}}}\mathbb{D}\to\mbox{\hskip 5.30904pt{1 0 -.25 1}\hbox to0.0pt{\hss\leavevmode\hbox{\set@color$\partial$}}}\mathbb{D}$ we need to define a partition of the boundary. The construction below relies on Ford fundamental domain.

To every $\mapg\in\Gamma$ we relate an isometric circle:

$$C(\mapg)=\{z\in\mathbb{D}\mid|{}^{\prime}(z)|=1\}$$

The name isometric here refers to the fact that locally doesn’t change Euclidean length. A nice treatment of isometric circles can be found in Ford’s monograph [16].

The isometric circle of an isometry $C(\mapg)$ is a geodesic in the Poincaré metric. Indeed, assuming $\mapg(z)=\frac{\alpha_{1}z+\alpha_{2}}{\overline{\alpha_{2}}z+\overline{\alpha_{1}}}$ we get ${}^{\prime}(z)=(\overline{\alpha_{2}}z+\overline{\alpha_{1}})^{-2}$. Therefore $|{}^{\prime}(z)|=1$ defines a circle of radius $|\overline{\alpha_{2}}|^{-1}$ centred at $-\overline{\alpha_{1}}/\overline{\alpha_{2}}$ provided $\alpha_{2}\neq 0$. To see that it is a geodesic, observe that the triangle with vertices at the origin, the centre of $C(\mapg)$, and a point of the intersection of $C(\mapg)$ and $\mbox{\hskip 5.30904pt{1 0 -.25 1}\hbox to0.0pt{\hss\leavevmode\hbox{\set@color$\partial$}}}\mathbb{D}$ is right-angled. A useful property is that a map $\mapg\in\Gamma$ transfers its isometric circle to the isometric circle of its inverse: $\mapg C(\mapg)=C({}^{-1})$.

The region of the unit disk “exterior” to the isometric circles of all elements of $\Gamma$ is a fundamental domain for the Fuchsian group $\Gamma$. It is called Ford domain. It is known that every side $s$ of the fundamental domain is identified with another one by an element of $\Gamma$ that we shall denote $\mapg(s)$. Moreover,

$$\Gamma=\langle\mapg(s)\mid s\mbox{ is a side of the Ford fundamental domain of }\Gamma\rangle,$$

in other words, the elements identifying the sides of the Ford domain constitute a set of generators of $\Gamma$.

In order to guarantee that the boundary map we are about to define is Markov, we need to impose additional assumptions on the way the surface group acts on the hyperbolic plane:

• •

  Every side $s$ of a fundamental domain lies on the isometric circle of $\mapg(s)$, i.e. $s\subset C(\mapg(s))$;

• •

  All hyperbolic lines, containing sides of the fundamental domain, are entirely covered by images of the sides of the fundamental domain under action of $\Gamma$.

We summarise the main properties of the map $T$ in the following lemma.

Let us denote by $\Gamma x$ the orbit of $x$ under the action of the group, i.e., $\Gamma x=\{\mapg x\mid\mapg\in\Gamma\}$.

We begin with a topological result. Let $V$ be a compact surface of constant negative curvature. It is easy to see that there are infinitely many closed geodesics since each free homotopy class (i.e., conjugacy class) in $\pi_{1}(V)$ contains a closed geodesic. Furthermore, there are countably many closed geodesics since each free homotopy class for a negatively curved surface contains precisely one closed geodesic. By a closed geodesic we mean the directed closed geodesics, i.e., each curve considered as a set actually corresponds to two directed geodesics, which differ by orientation.

Let us denote by $\ell(\gamma)$ the length a (primitive) closed geodesic $\gamma$. Closed geodesics are in bijection with the (prime) closed orbits of the geodesics flow of the same length.

The natural proof of Huber’s theorem uses the Selberg zeta function $Z_{V}(s)$ and the location of its zeros. We recall the definition.

The convergence of the infinite product from (3) on $\Re(s)>1$ follows from the estimate in Lemma 9. The asymptotic estimates for the counting function $N(t)$ follow completely by analogy with the Prime Number Theorem for primes and the zeta function $Z_{V}(s)$ is used in place of the Riemann $\zeta$-function.

The zeta function $Z_{V}(s)$ formally defined on $\Re(s)>1$ can be extended to the entire complex plane using the famous Selberg trace formula [24]. This is explained in the book of Hejhal [19] in considerable detail. We recall the basic properties:

We want to consider error terms in the asymptotic asymptotic relation from Theorem 1. To put this into perspective we need to recall what might happen in the corresponding setting in number theory.

Let us begin by recalling some classical ideas from number theory. These are useful in understanding the role of the dynamical analogues.

Of course there is additionally another property which has not yet been established.

There is an improvement to Huber’s basic Theorem 1 at least as stated. In fact, the original result of Huber included error terms, so now we are actually stating more of his original theorem.

Note that $\varepsilon$ here is responsible for the rate of convergence, rather then “just” for the error term itself.

As is well known,

$$\mathord{\rm li}(x)\colon=\int_{2}^{x}\frac{{\mathrm{d}}u}{\log u}\sim\frac{x}{\log x}\mbox{ as }x\to+\infty.$$

Therefore this statement is consistent with the original statement of Theorem 1, but now has an additional error term.

The proof uses properties of $Z_{V}$ and the rate $\varepsilon>0$ depends on location of zeros. This is completely analogous to the situation in number theory where one studies a counting problem for prime numbers by using the Riemann zeta function, except in the present context we have stronger results on the zeta function.

For compact surfaces there is a natural self-adjoint operator. The extension and zeros of Selberg zeta function are related to the Laplacian $\Delta\colon\mathcal{L}^{2}(V)\to\mathcal{L}^{2}(V)$. This can be defined as an operator on real analytic functions $\Delta\colon C^{\omega}(V)\to C^{\omega}(V)$ and then extended to square integrable functions. Let

$$\Delta\psi_{n}+\lambda_{n}\psi_{n}=0$$

be the eigenvalue equation. There are infinitely many eigenvalues

$$0=\lambda_{0}<\lambda_{1}\leq\lambda_{2}\leq\ldots\nearrow+\infty$$

for the operator $-\Delta$. Moreover, we have the following results:

1. 1.

   $\#\{\lambda_{n}\leq t\}\sim\frac{\mathrm{Area}(V)}{4\pi}\cdot t$ as $t\to\infty$ (Weyl, 1911);

2. 2.

   The zeros of $Z_{V}(s)$ in $0<\Re(s)<1$ satisfy the equality $s_{n}(1-s_{n})=\lambda_{n}$, where $\lambda_{n}$ are eigenvalues of the Laplacian [37].

The parameter $\varepsilon>0$ for the error term for counting closed orbits in Theorem 2 can be expressed in terms of the eigenvalues of the Laplacian, namely, it corresponds to the distance of the zeros of $Z_{V}$ from the line $\Re(s)=1$ (although one may need to take $\varepsilon$ slightly smaller).

The approach and the statement fail if $V$ has infinite area. In general case there is no reason for zeros to lie on such lines. In the context of non-compact surfaces we need an alternative approach using the Bowen–Series map, that allows us to establish a correspondence between closed geodesics and closed orbits.

Recall that a geodesic is called prime if it traces its path exactly once. We say that a periodic orbit $\{x,Tx,\ldots,T^{n-1}x\mid T^{n}x=x\}$ is prime if $T^{k}x\neq x$ for all $1\leq k<n$.

An advantage of the Bowen–Series construction is that the map $T$ defined by (2) can be used to recover the lengths of closed geodesics. More precisely,

To take advantage of this correspondence to study zeta function $Z_{V}$, we introduce the family of complex Ruelle–Perron–Frobenius transfer operators (indexed by $s\in\mathbb{C}$). This is often referred to as the “Thermodynamic Viewpoint” [30]. We begin by recalling the definition.

However, in order to proceed we need to replace the big space $C^{0}(X)$ by a smaller Banach space upon which the transfer operator has good spectral estimates.

Let $\{I_{j}\}$ denote the arcs in the Bowen–Series coding (not to be confused with intervals from the definition of $T$).

We next consider a closely related problem in the general area of smooth ergodic theory. We begin with a definition.

In the case of constant negative curvature the Liouville measure corresponds to the normalised Haar measure.

Mixing is a property in the hierarchy of ergodic properties. It implies ergodicity and is in turn implied by the Bernoulli property. In the case of square integrable functions $F,G\in L^{2}(\nu)$ there is no reason to expect to be able to say anything about the speed of mixing (i.e., the rate at which $\rho(t)$ tends to zero). But since we are assuming $F$ and $G$ are smooth more can be said.

Of course the definition can also be reformulated for different invariant measures, but for our present purposes we only need to consider the measure $\nu$.

The following result dates back over sixty years and was the forerunner of method that has proved to be very successful over subsequent years.

The proof of Theorem 3 uses unitary representation of $SL(2,\mathbb{R})$ on the space $L^{2}(Y)$ given by $[Uf](x)=f({}^{-1}x)$. The representation $U$ is reducible and therefore the problem of establishing strong mixing is reduced to the understanding of the action of each of the component representations. This brings us to the next question:

The following results on mixing don’t directly rely on properties of the zeta function, but the underlying mechanisms are similar as will hopefully soon become apparent.

The argument is based on representation theory approach. In fact the correlation functions correspond to “decay of matrix coefficients” in representation theory [26].

We have used the same constant $\varepsilon>0$ for both results, in the counting geodesics problem Theorem 2, and for the speed of mixing of geodesic flow in Theorem 4. Assuming this wasn’t carelessness, one might ask the following question:

The simple answer is: “They are the same!”. To see this, we can consider the Laplace transform of the correlation function $\rho$:

$$\hat{\rho}(z)=\int_{0}^{\infty}e^{-zt}\rho(t)\mathrm{d}t,\quad z\in\mathbb{C}.$$

Thus we are introducing another complex function to describe the mixing rate. The function $\hat{\rho}$ is well-defined for $\Re(z)>0$ since the integral converges on this half-plane. However, there is much more that one can say. We recall the following properties of $\hat{\rho}$.

The existence of meromorphic extension of $\hat{\rho}$ can be obtained by establishing a connection with the resolvent of the transfer operators and using properties of the spectrum of the transfer operator. The second part follows from the observation that $Z_{V}$ has a similar connection to the spectra of a suitable transfer operator. The last part is classical harmonic analysis using the Paley–Wiener theorem.

As in the case of compact surfaces, one has in the non-compact case that there are infinitely many closed geodesics (with one in each free homotopy class). The obvious question is the following.

Assume that $V=\mathbb{D}/\Gamma$ is a surface of infinite area, where $\Gamma$ denotes the associated Fuchsian group. We first recall that $\delta$ denotes the Hausdorff dimension of the limit set and let us again denote by $\#\{\gamma\colon\ell(\gamma)\leq t\}=\colon N(t)$ the number of prime closed geodesics of length at most $t$. Now we can state the first result, which gives that $\delta$ is also the asymptotic growth rate of $N(t)$.

However, whereas the proof in the case of compact surfaces used the Selberg trace formulae this is not available to us in the case of general infinite area surfaces. Instead in the present setting the transfer operators can be used to give an extension to the zeta function $Z_{V}$ and this leads to a proof of Theorem 5. Similarly to the compact case there is a stronger form of the asymptotic formula with an error term.

Both results Theorem 6 and Theorem 5 require the dynamical approach of transfer operators. This method also applicable in the case of compact surfaces, but in that case we had the luxury of using the trace formulae, which served us even better in as much as they gave both an analytic extension and additional information on the location of the zeros. Despite its apparent greater generality, or perhaps because of it, the transfer operator approach has the downside that there is less control over the location of zeros. The next lemma is the analogue of Lemma 12 for compact surfaces.

As in the case of compact surfaces we have the following correspondence for the lengths (compare with Lemma 13).

We again define the Selberg zeta function

$$Z_{V}(s)=\prod_{n=0}^{\infty}\prod_{\gamma}\left(1-e^{-(s+n)\ell(\gamma)}\right)$$

(5)

where the product is taken over all prime closed geodesics $\gamma$ and $\ell(\gamma)$ is the length; as before in (3). Note that in the case of infinite area surfaces the closed geodesics are contained in the recurrent part of the flow. The infinite product on the right hand side of (5) converges for $\Re(s)>\delta$ as is easily seen using Theorem 5 and defines an analytic function.

To proceed we again replace $C^{0}(X)$ by a smaller Banach space upon which the transfer operator has good spectral estimates (e.g. Hölder, $C^{1}$, $C^{k}$, $C^{\omega}$, the smaller space the better the results).

The main case we want to concentrate on is the three funnelled surface (or “pair of pants”) from Example 5. This will be a paradigm for a general case.

In order to study the zeta function $Z_{V}$ using the transfer operator we need to be more restrictive in the type of Banach space upon which the transfer operator acts. We will consider a Banach space of analytic functions.

1. 1.

   Choose four disjoint open neighbourhoods $U_{j}\supset\Lambda_{j}$ of elements of partition of the Cantor set, as pictured in Figure 6;

2. 2.

   Let $B\subset C^{\omega}(\sqcup_{j}U_{j})$ be the Banach space of bounded analytic functions on the disjoint union of the disks $\sqcup_{j}U_{j}$ with the norm

   $$\|f\|=\sup_{z\in\sqcup_{j}U_{j}}|f(z)|.$$

   Of course, this is equivalent to the direct sum of spaces of analytic functions on each of the four neighbourhoods.

3. 3.

   By construction we see that the Banach space is preserved by transfer operators given by (4), i.e., ${\mathcal{L}}_{s}\colon B\to B$ for every $s\in\mathbb{C}$.

4. 4.

   On the Banach space $B$ we have that $\mathcal{L}_{s}$ is trace class, in particular, there exists only a countable set of non-zero eigenvalues which sum is finite⁴⁴4This again follows directly from the works of Grothendieck [17], [18] and Ruelle [35].. Thus we can write

   $$\det(I-\mathcal{L}_{s})\colon=\exp\left(-\sum\limits_{n=1}^{\infty}\frac{1}{n}\mathop{\rm trace}(\mathcal{L}_{s}^{n})\right)$$

   and the right hand side converges to a function analytic on $\mathbb{C}$.

5. 5.

   Furthermore using the connection between the derivatives of the Bowen–Series map and closed geodesics one can show that $Z_{V}(s)=\det(I-\mathcal{L}_{s})$.

6. 6.

   Finally, we have the classical observation: $Z_{V}(s)=0$ if and only if there exists a function $f\neq 0\in B$ such that $\mathcal{L}_{s}f=f$. Thus in order to show that there is a zero free strip it suffices to show that there exists $\delta^{\prime}<\delta$ such that for any $\delta^{\prime}<\Re(s)<\delta$ the operator $\mathcal{L}_{s}$ doesn’t have $1$ as an eigenvalue.

As we have observed already, closed geodesics in the case of non-compact surfaces lie in the recurrent part of the geodesic flow. We also adopt the convection that they are oriented geodesics (i.e., a pair of geodesics which are identical as sets but have different orientations counted as two different geodesics).

We recall basic properties of the zeta function (cf. [31]) for infinite area surfaces.

To extend the theory to the case of surfaces of infinite area we first need a new measure replacing the Haar measure. In this case we are concerned with measures supported on the recurrent part of the geodesic flow. The easiest way to construct such measures is by using the classical construction of measures on the limit set and then to convert these into measures invariant by the geodesic flow and supported on the recurrent part of the flow.

This is the standard construction of the Patterson–Sullivan measure on $\Lambda$ (cf. a book of Nicholls [29, Ch. 3], for example). Although for each parameter $t$ we obtain a measure on the countable set of points in the orbit of the point $x_{0}$, and these measure live on the open disk and give zero measure to the boundary circle (and the limit set). As $t$ decreases more weight is given to points closer to the boundary (in the Euclidean metric). In the limit these measures converge in the weak-$*$ topology of measures (on the closed unit disk) to a measure actually supported on the boundary cicle.

We use the Patterson–Sullivan measure $\nu_{\delta}$ and the Lebesgue measure along the flow direction $dt$ to define a flow invariant measure $\widehat{\nu}_{\delta}$ on the unit tangent bundle to the universal cover, which can be identified with $\bigl{(}\mbox{\hskip 5.30904pt{1 0 -.25 1}\hbox to0.0pt{\hss\leavevmode\hbox{\set@color$\partial$}}}\mathbb{D}\times\mbox{\hskip 5.30904pt{1 0 -.25 1}\hbox to0.0pt{\hss\leavevmode\hbox{\set@color$\partial$}}}\mathbb{D}\setminus\mbox{diagonal}\bigr{)}\times\mathbb{R}$ by:

$$d\widehat{\nu}_{\delta}(x,y,t)=\frac{d\nu_{\delta}\times d\nu_{\delta}\times dt}{|x-y|^{2\delta}}$$

This corresponds to an invariant measure on $\mbox{\hskip 5.30904pt{1 0 -.25 1}\hbox to0.0pt{\hss\leavevmode\hbox{\set@color$\partial$}}}\mathbb{D}\times\mbox{\hskip 5.30904pt{1 0 -.25 1}\hbox to0.0pt{\hss\leavevmode\hbox{\set@color$\partial$}}}\mathbb{D}$ under the diagonal action of $\Gamma$, i.e., $\mapg(x,y)=(\mapg x,\mapg y)$. Finally, by considering the quotient by $\Gamma$ and normalizing we arrive at a $\varphi_{t}$-invariant measure $\nu$ on the unit tangent bundle $Y$.