数理解析研究所講究録
第 1777 巻 2012 年 1-20

1Generalizations of the continued fraction
transformation and the Selberg zeta functions
Takehiko Morita
Graduate School of Science, Osaka University,
Toyonaka, Osaka 560-0043 Japan

1. INTRODUCTIONLet

be the upper half-plane in endowed with the Poncar\’e metric $ds^{2}=(dx^{2}+$and letbe the modular group$\mathbb{H}$$dy^{2})/y^{2}$$\mathbb{C}$$\Gamma_{1}=PSL(2,\mathbb{Z})$$PSL(2, \mathbb{Z})=\{(\begin{array}{ll}a bc d\end{array})$:$a,$ $b,$ $c,$ $d\in \mathbb{Z}$, $ad$-$bc$$=1\}/\{I, -I\}$

identified with the set of linear fractional transformations $z\mapsto(az+b)/(cz+d)$ as usual.
We consider the so-called continued fraction transformation (Gauss transformation)
defined by: $(0,1)arrow(0,1)$ , : $x\mapsto 1/x-[1/x]$ . It is well-known that the ergodic theoryof is closely related to the dynamical theory of the geodesic flow on the modular surface
as well as the metric theory of continued fractions ([14], [15], [16], [25], [32]).
Now we pay attention to the fact that the modular surface
has two different aspects.
The first aspect is rather classical. We consider
as an example of Riemann surfaceof finite area (possibly with cusps and elliptic singularities). The second one is slightly
fancy. We consider
as the moduli space of complex structures of surface of genus 1.
So there are at least two directions to generalize
according to which aspect we look
$T_{G}$$T_{G}$$T_{G}$$M_{1}=\mathbb{H}/\Gamma_{1}$$M_{1}$$M_{1}$$M_{1}$$M_{1}$

at. The aim of the article is to explain about the generalizations focusing the role of thecontinued fraction transformation.
In the case of the modular surface the crucial point is the existence of a natural oneto-one correspondence among the following three sets.
$CG(M_{1})$ : the totality of oriented prime closed geodesics
in.
$\bullet$$M_{1}$$\gamma$: the totality of primitive hyperbolic conjugacy classes in , i.e. can be
written as $c=(h\rangle=\{g^{-1}hg : g\in\Gamma_{1}\}$ , where is a primitive hyperbolic element in .
$\bullet$$HC(\Gamma_{1})$

$c$

$h$

$\bullet$$PO(T_{G}^{2})$: the totality of prime periodic orbits$\Gamma_{1}$$c$$\Gamma_{1}$$\tau$

of$T_{G}^{2}$, i.e.

$\tau$

can be regarded as theset of the distinct points $\tau=\{x, T_{G}^{2}x, \ldots , T_{G}^{2(p-1)}x\}$ , where is a periodic point ofand is the least period of .
To be more precise, assume that $\gamma\in CG(M_{1}),$ $c\in HC(\Gamma_{1})$ , and $\tau\in PO(T_{G}^{2})$ arecorresponding one another. Let
be the least period of, the maximal eigenvalue
$h\in
c$
of
as a matrix and $N(\tau)=J(T_{G}^{2p})(x)=|DT_{G}^{2p}(x)|$ . Note that for each periodic
$x$

$p$

$x$

$l(\gamma)$$\gamma,$$\lambda(c)$$T_{G}^{2}$

2
suchof and an hyperbolic element
belong to the. In addition we can easily show thatpoint $x\in\tau$ there exists a neighborhood
that$T_{G}^{2p}=h_{x}$in$U_{x}$$U_{x}$$h_{x}\in\Gamma_{1}$$x$$h_{x},$$\ldots,$ $h_{T_{G}^{2(p-1)}x}$

same conjugacy class. Under the notation above, the corresponding elements satisfy(1.1)$\exp(l(\gamma))=\lambda(c)=N(\tau)$.

The one-to-one correspondence enables us to express the Selberg zeta function in termsof periodic orbits of . In order to see this, we recall the definition of the Selberg zetafunction. Let $U(N)$ be the unitary group of degree $N$ and : $\Gamma_{1}arrow U(N)$ a unitary. The Selberg zeta function for with representation is formally defined$T_{G}^{2}$$\rho$$\Gamma_{1}$$\rho$$representationas$$Z(s, \rho)=\prod_{k=0}^{\infty}\prod_{c\in HC(\Gamma_{1})}\det(I_{N}-\rho(c)e^{-(s+k)l(c)})$,is a complex variable and $\det(I_{N}-\rho(c)e^{-(\epsilon+k)l(c)})$ is regarded as $\det(I_{N}-$$\rho(h)e^{-(s+k)l(c)})$ for some $h\in c$ since the determinant is a conjugacy invariant. The Selberg
with is also formally defined asfunction for

where

$s$

$L$

$\Gamma_{1}$$\rho$$L(s, \rho)=\frac{Z(s+1,\rho)}{Z(s,\rho)}=\prod_{c\in HC(\Gamma_{1})}\det(I_{N}-\rho(c)e^{-sl(c)})^{-1}$.

By virtue of (1.1), a formal calculation leads us to the identity:(1.2)$Z(s, \rho)=\exp(-\sum_{k=0}^{\infty}\sum_{n=1}^{\infty}\frac{1}{n}\sum_{x:T_{G}^{2n}x=x}$

trace $(\rho(T_{G}^{2n}|_{x})J(T_{G}^{2n})(x)^{-(s+k)})$ ,in a neighborwhich coincides with
where
denotes the hyperbolic element inhood of . Note that the formal products and formal series in the above are absolutely
convergent in the half-plane ${\rm Re} s>1$ and determine analytic functions. Classically the
Selberg zeta function is studied via the Selberg trace formula (see [7], [8]). The equation(1.2) suggests us an alternative approach to the Selberg zeta function via thermodynamic(see [31]). In fact, Pollicott [25] obtained some results on the distribution
formalism forof closed geodesics on the modular surface and quadratic irrationals and Mayer [16] gavea determinant representation of the Selberg zeta function in terms of transfer operator. Speaking of ergodic theoretic approach to thewith complex parameter associated to
distribution of closed geodesics, there is a monumental work by Margulis in 1970 concerned with asymptotics of number of closed trajectories of Anosov systems on a compact
manifold (see [12]). Note that the methods in [12] is different from ours but it is also remarked in [12] that those methods are applicable to the geodesic flows over non-compact$\Gamma_{1}$$T_{G}^{2n}|_{x}$$x$$T_{G}^{2}$$T_{G}$$T_{G}^{2n}$3Riemannian manifolds of finite volume with negative sectional curvature
is of finite
area but not compact and has an elliptic singularity). On the other hand, Parry [22]and Parry and Pollicott [23] (see also [24]) employ another approach via zeta functionsin analytic number theory to obtain the prime number type theorem for closed orbits of
Axiom A flows on compact manifolds. They represents Axiom A flows as suspension flows
over appropriately chosen subshifts of finite type and studied the analytic properties ofsome zeta functions by thermodynamic formalism.
Next let us consider the two way of generalization ofin view of the equation (1.1).
In the first generalization, we want to extend the results to the case when is replaced byany co-finite Fuchsian group F. If we have a map playing the same role as
for a generalco-finite Fuchsian group, we can follow the arguments in [15]. Fortunately we know that
Bowen and Series construct in [3] a one-dimensional Markov map whose action on an
appropriately chosen subset in is orbit equivalent to that of on RU. Inspired by[15], [16], and [23], Morita [19] considered some renormalized Boewn-Series map instead ofsubshift of finite type and studied a unified approach to the determinant representationof the Selberg zeta function and the prime number type theorem for general co-finite
Fuchsian groups via thermodynamic formalism. We have to note that there is a workof Pollicott [26] in the case when is co-compact and a similar problem for co-compact
Keinian groups are treated in [30].
In the second generalization, we consider the moduli spaceof genus $g\geq 2$ insteadof. It is known thathas a similar structure to. For example, the Teichm\"uller
space of genus , the Teichm\"uller modular group, and the Teichm\"uller geodesic flow play
the roles of the upper-half plane , the modular group, and the geodesic flow
, respectively. Note that Masur [13] and Veech [34] solved the Keane Conjectureof(see [9], [33]) independently by showing the ergodicity of the Teichm\"uller geodesic flow
with respect to a canonical invariant measure. To the question “what plays the role ofin the case of?” we do not have a satisfactory answer at present while the Rauzy
induction and its renormalizations are expected to play the role offor Teichm\"uller
modular group (see [29], [34], [35], [36], [37], [38], and [40]). Selberg
Keeping the above mentioned facts in mind, we explain about the first generalizationin Section 2. We mainly take up the determinant representation of Selberg zeta functions
for co-finite Fuchsian groups with representation . Since we already obtained such a
result in [19] in the case when is trivial, we just give some ideas to handle nontrivial. The section include some unpublished results but they might be folklore in nowadays.
Section 3 is devoted to the second generalization. We discuss about some results in [20]and [21] on distribution of periodic orbits of a class of renormalized Ruazy-Veech-Zorich$(M_{1}$$M_{1}$$\Gamma_{1}$$T_{G}^{2}$$\mathbb{R}$$\Gamma$$\{\infty\}$$\Gamma$$M_{9}$$M_{1}$$M_{9}$$M_{1}$$g$$PSL(2, \mathbb{Z})$$\mathbb{H}$$M_{1}$$T_{G}^{2}$$M_{g}$$T_{G}^{2}$$\rho$$\rho$$\rho$

4
inductions. Comparing with the first generalization, these results are not satisfactory andit seems there are many problems left unsolved.
2. GENERALIZATION I : MARKOV SYSTEMS FOR $CO$-FINITE FUCHSIAN GROUPS
Following [3] we work on the Poincar\’e disc $D=\{z\in C : |z|<1\}$ endowed withbe a co-finite subgroup of
the metric $ds^{2}=4dzd\overline{z}/(1-|z|^{2})^{2}$ in this section. Let$SU(1,1)$ . Starting with a Markov map assocated toin [3], we can construct a triplet
called a Markov system associated to . The Markov system plays the$\Gamma$$\Gamma$$\Gamma$$\mathcal{T}_{\Gamma}=(\mathcal{R}, \mathcal{P}, T)$

although it is not a single transformation. We summarize the fundamentalrole offacts on our Markov system below (for the detailed construction see [19]).
(M.1) (finiteness) $\mathcal{R}=\{A(1), \ldots, A(q)\}$ is a set of a finite number of closed arcs on thewith mutually disjoint nonempty interior.
unit circle(M.2)
is a set of closed arcs with mutually disjoint nonempty interior such that,such that int $J\subset intA$ andfor each there exists
denotes the Haar measure on the unit disc. In particular, there exists a
where
int $A(\iota(J))$ .
such that int
map :$T_{G}^{2}$$S^{1}$$\mathcal{P}=\{J\}$$J$$A\in \mathcal{R}$$| \bigcup_{A\in \mathcal{R}}A\backslash \bigcup_{J\in \mathcal{P}}J|=0$$|\cdot|$$J\subset$$\mathcal{P}arrow\{1, \ldots, q\}$$\iota$From the construction, we see that is co-compact if and only if $\#’P<$ oo.
satisfying $T_{J}J=A$
and an element, an arc(M.3) To eachsuch that $T_{J}J=$
are assigned. Namely, there exists a map :$\Gamma$$T_{J}\in\Gamma$$A\in \mathcal{R}$$J\in \mathcal{P}$$\tau$$A(\tau(J))$$\mathcal{P}arrow\{1, \ldots, q\}$.

is an a.e. defined transformation on $X=u_{A\in R}$ such that $T|_{intJ}=T_{J}|_{1ntJ}$ for.
each
We need some notation. For closed arc , we denote by $[A)$ (resp. $(A])$ the half-open) is a(resp.
arc obtained by taking the right (resp. left) endpoint away from A.
right continuous modification (resp. a left continuous modification) of .
(M.4)

$A$

$T$

$J\in \mathcal{P}$$A$$T_{L}$$T_{R}$$T$(M.5) (orbit equivalence) For for any(resp.

$x$

there exists $g\in\Gamma$ (resp.

$g’\in\Gamma$) such that$y\in$(resp.
and for any, we can fined $m,$ $n\geq 0$ such thatsatisfying $y=gx$ for some.
resp.
. For $n\geq 2$ , letof . Let
Next we introduce the notion of n-fold iteration
be the family of closed arcs with nonempty interior having the form$($

$T_{L}^{m}x=T_{L}^{n}y)$$\mathcal{T}_{\Gamma}^{n}$$\mathcal{P}_{1}=P$$\mathcal{T}_{\Gamma}$$J$$\mathcal{P}_{n}$

(2.1)

for

$x,$

$g\in\Gamma$$\bigcup_{A\in \mathcal{R}}(A])$$T_{R}^{m}x=T_{R}^{n}y$$x,$ $y \in\bigcup_{A\in R}[A)$$g’x \in\bigcup_{A\in \mathcal{R}}(A])$$gx \in\bigcup_{A\in \mathcal{R}}[A)$$J=J(i_{0})\cap T_{J(i_{0})}^{-1}J(i_{1})\cap T_{J(i_{0})}^{-1}\circ T_{J(i_{1})}^{-1}J(i_{2})\cap\cdots\cap T_{J(i_{0})}^{-1}oT_{J(i_{1})}^{-1}o\cdots oT_{J(i_{n-2})}^{-1}J(i_{n-1})$$J(i_{0})$

,

...,

. Such a is often denoted bydefined by
we assign an element$J(i_{n-1})\in \mathcal{P}$$J(i_{0}, \cdots i_{n-1})\in \mathcal{P}_{n}$$J(i_{0}, \cdots i_{n-1})$$J$$T_{J}^{n}\in\Gamma$. To each $J=$.

$T_{J}^{n}=T_{J(i_{n-1})}\circ\cdots oT_{J(i_{0})}$5Then n-fold iteration: $Xarrow X$ of is defined a.e. so thatfor each. Then we can easily see that the triplet
satisfies (M.1), (M.2), (M.3),
and (M.4). We call it the n-fold iteration ofand denote it by. Note that and:are defined in the same way as in (M.2) and (M.3), respectively.
By virtue of Proposition 3.2 in [19], we may assume without loss of generality that theMarkov system
satisfies the following mixing condition.
(M.6) (mixing) There exists a positive integer
such that for each
we can findsuch that
andhold for each $j=1$ .
.
$T^{n}$$T$$J\in \mathcal{P}_{n}$$T^{n}|_{intJ}=T_{J}^{n}|_{intJ}$$(\mathcal{R}, \mathcal{P}_{n}, T^{n})$$\mathcal{T}_{\Gamma}$$\mathcal{T}_{\Gamma}^{n}$$\tau$$\iota$$\mathcal{P}_{n}arrow\{1, \ldots, q\}$$\mathcal{T}_{\Gamma}$$J\in \mathcal{P}$$n_{0}$$J_{1},$$\ldots,$$J_{q}\in \mathcal{P}_{n_{0}}$$J_{j}\subset J$$T_{J_{j}^{0}}^{n}J_{j}=A(j)$$\ldots,$$q$

The following are needed to investigate the analytic properties of the Selberg zetafunction for .
(M.7) (expanding) There exists a positive integer such that$\Gamma$$k$$\inf_{J\in \mathcal{P}_{k}}\inf_{x\in}|DT_{J}^{k}(x)|>1$(M.8) (R\’enyi condition) For any positive integer$n$.

we have$R( \mathcal{T}_{\Gamma}^{n})=\sup_{J\in \mathcal{P}_{n}}\sup_{x\in J}\frac{|D^{2}T_{J}^{n}(x)|}{|DT_{J}^{n}(x)|^{2}}<\infty$

Moreover,(M.9) For each $j=1,2$ , . . . ,$\sup_{n}R(\mathcal{T}_{\Gamma}^{n})/n<$

$q$

.

oo,

, there exist simply connected domains $B(j)$ and $D(j)$in complex plane such that$(M.9,i)A(j)\subset B(j)\subset\subset D(j)$;(M.9.ii) $T_{J}^{-1}D(\tau(J))\subset B(\iota(J))$ for any $J\in P$ ;( $M.9$ .iii) for any $\delta>1/2$ , we have$\sum_{J\in’\mathcal{P}}\sup_{z\in D(\tau(J))}|DT_{J}(T_{J}^{-1}z)|^{-\delta}<\infty$

Now we introduce some notions concerned with$\mathcal{T}_{\Gamma}$.

-periodic orbit. We write as$\mathcal{P}=$for the sake of convenience. A sequenceis called admissible
iffor any . To a point $x\in X=u_{I=1}^{q}A(i)$ , one can assign anadmissible sequencecalled -itinerary of such that $x\in J(\xi_{0})$ and. . . $oT_{J(\xi_{0})}x\in J(\xi_{k+1})$ for each $k\geq 0$ . It is denoted as $\xi(x)=(\xi_{0}(x)\xi_{1}(x)\cdots)$ . A point
having a periodic -itinerary is called a -periodic point. A set of distinct points
is a -periodic orbit of period if there exists a permutation of theset $\{0,1, \cdots , p-1\}$ such that -itineraries
satisfyof$mod p$ , whereis the shift transformation on the sequence space such that$\{J(j)\}_{j=1}^{\#\mathcal{P}}$$\xi=(\xi_{0}\xi_{1}\cdots)$$T_{J(\xi)}:J(\xi_{i})\supset J(\xi_{i+1})$$i$$\xi=(\xi_{0}\xi_{1}\cdots)$$x$$\mathcal{T}_{\Gamma}$$T_{J(\xi_{k})^{\circ}}$$x$$\mathcal{T}_{\Gamma}$$\{x_{0}, x_{1}, \ldots, x_{p-1}\}$$\mathcal{T}_{\Gamma}$$\mathcal{T}_{\Gamma}$$\mathcal{T}_{\Gamma}$$\sigma$$\tau=$$p$$p$$\xi(x_{\pi(i)})$$\pi$$x_{\pi(i)}$$\sigma\xi(x_{\pi(i)})=\xi(x_{\pi(i)+1})$$(\sigma\xi)_{k}=\xi_{k+1}$

6
for any $k\geq 0$ . Note for any admissible sequence with least period , there exists a
unique -periodic point $x\in X$ such that $\xi(x)=\xi$ by virtue of the expanding condition(M.7). In particular, the set $\tau(x)=\{x, T_{J(\xi_{0}(x))}x, \cdots, T_{J(\xi_{p-1}(x))}o\cdots oT_{J(\xi_{0}(x))}x\}$ forms
the totality of -periodic orbits.
a -periodic point of period . We denote by
Then we have the following.
$p$

$\xi$

$\mathcal{T}_{\Gamma}$$PO(\mathcal{T}_{\Gamma})$$p$$\mathcal{T}_{\Gamma}$

LEMMA 2.1. (1) For$\tau=\{x, \ldots, x_{p-1}\}\in PO(\mathcal{T}_{\Gamma})$

is a primitive hyperbolic element in .
(2) If we define a map:

with$\mathcal{T}_{\Gamma}$$\xi(x)=(\dot{j}_{0}, \ldots,\dot{j}_{p-1}),$$T_{J(i_{0},\ldots,i_{p-1})}^{p}$$\Gamma$$PO(\mathcal{T}_{\Gamma})arrow HC(\Gamma)$$\Phi$

by$\Phi(\tau)=(T_{j()}^{p}i_{0},\ldots,i_{p-1})$,

then we have,$\lambda(\Phi(\tau))=|DT_{J(i_{0},\ldots,i_{p-1})}^{p}(x_{0})|$where$\tau$

is wriuen as in (1) above and

The setlowing hold.
(3)(S. i)$PO(\mathcal{T}_{\Gamma})$$(\gamma)$

denotes the conjugacy class ofis divided into four subsets as$\gamma\in\Gamma$.

$PO( \mathcal{T}_{\Gamma})=\bigcup_{i=0}^{3}PO(\mathcal{T}_{\Gamma})_{i}$the fol-(disjpoint union).
are injective.

$\Phi(PO(\mathcal{T}_{\Gamma})_{0})\cup\Phi(PO(\mathcal{T}_{\Gamma})_{1})=HC(\Gamma)$(S. ii) The restri ctions$\Phi|_{PO(\mathcal{T}_{\Gamma})_{:}}$

are finite sets having the common image under . In
has genus .
particular, these are empty if the corresponding surface$(3.iii)PO(\mathcal{T}_{\Gamma})_{i}(i=1,2,3)$$\Phi$$0$$to\Gamma$

For the proof see Lemma 4.2 in [19].
We need some function spaces. For a bounded domain $D$ in and a positive integer$N,$valued holomorphic functions
denote the totality of
andvalued bounded holomorphic functions on $D$ , respectively.
on $D$ and its subspace of, where $K$
is a nuclear Fr\’echet space with semi-norms$\mathbb{C}$$A(D;\mathbb{C}^{N})$$\mathbb{C}^{N}$$\mathcal{A}_{b}(D;\mathbb{C}^{N})$$\mathbb{C}^{N}$$p_{K}(f)= \sup_{x\in K}|f(z)|_{\mathbb{C}^{N}}$$\mathcal{A}(D;\mathbb{C}^{N})$is a Banach.
is the usual norm on
is any compact subset of anddenotes. For a compact set $K,$
space with the norm.
the space of allvalued continuous functions with the norm
valuedin and a positive integer $N$ , a
For a finite number of domains
resp.
is said to be an element infunction on$D$$\mathbb{C}^{N}$$\mathcal{A}_{b}(D;\mathbb{C}^{N})$$|\cdot|_{\mathbb{C}^{N}}$$C(K;\mathbb{C}^{N})$$\Vert f\Vert_{\infty}=\sup_{z\in D}|f(z)|_{\mathbb{C}^{N}}$$\Vert f\Vert_{\infty}=\sup_{x\in K}|f(x)|_{\mathbb{C}^{N}}$$\mathbb{C}^{N}$$D_{1},$

$f$

if
$($

$f|_{D_{j}}$

resp.

is in

$D_{k}$$A(D_{j};\mathbb{C}^{N})$$\mathcal{A}_{b}(U_{j=1}^{k}D_{j};\mathbb{C}^{N}))$by$\mathbb{C}^{N}$$\mathbb{C}$$A(u_{j=1}^{k}D_{j};\mathbb{C}^{N})$$u_{j=1}^{k}D_{j}$$\oplus_{j=1}^{k}\mathcal{A}_{t}(D_{j};\mathbb{C}^{N}))$$f|_{D_{j}}$$\ldots,$(resp.

$\mathcal{A}_{b}(D_{j};\mathbb{C}^{N})$) for each$j$

. The space

is naturally identified with the space$\mathcal{A}(u_{j=1}^{k}D_{j};\mathbb{C}^{N})$$\oplus_{j=1}^{k}\mathcal{A}(D_{j};\mathbb{C}^{N})$$\mathcal{A}(u_{j=1}^{k}D_{j};\mathbb{C}^{N})\ni f\mapsto\oplus_{j=1}^{k}f_{j}\in\oplus_{j=1}^{k}\mathcal{A}(D_{j};\mathbb{C}^{N})$for each . Finally a finite number of compact sets$j$$\mathcal{A}_{b}(u_{j=1}^{k}D_{j};\mathbb{C}^{N}))$$($$K_{1},$$\ldots$,$K_{k},$(resp.

, where $f_{j}=$$C(u_{j=1}^{k}K_{j};\mathbb{C}^{N})$7denotes the totality of functions

and$C(\lfloor\rfloor_{j=1}^{k}K_{j};\mathbb{C}^{N})$$f$defined on

is identified with$u_{j=1}^{k}K_{j}$with$\oplus_{j=1}^{k}C(K_{j};\mathbb{C}^{N})$$f|_{K_{j}}\in C(K_{j};\mathbb{C}^{N})$for each$j$.

Put $B(X)=u_{j=1}^{q}B(j)$ and $D(X)=u_{j=1}^{q}D(j)$ , where $B(j)$ ’s and $D(j)$ ’s are thedomains appearing in (M.9). For$(w, s)arrow G_{J}(s)(w)\in \mathbb{C}$ defined by$J\in \mathcal{P}$, consider a holomorphic function$B(\iota(J))\cross \mathbb{C}\ni$$G_{J}(s)(w)=(DT_{J}(w) \frac{w}{T_{J}w})^{-8}$Note that if $w\in S^{1}$ , we have $G_{J}(-1)(w)=(DT_{J}(w))(w/T_{J}(w))=|DT_{J}(w)|$ .
Now we are in a position to define a family of twisted transfer operators with complex
parameter. It is shown that the family determines a meromorphic function which takes
values in the space of nuclear operators acting on an appropriate chosen Banach space.
The main purpose of this section is to represent the Selberg zeta function by using theFredholm determinants of these operators. Let : $\Gammaarrow U(N)$ be a unitary representation.
For an elementor, we define an operator
formally by$\rho$$C(X;\mathbb{C}^{N})$$f=\oplus_{j=1}^{q}f_{j}\in \mathcal{A}(D(X);\mathbb{C}^{N})$(2.2)$\int,$$\rho$.

$( \mathcal{L}(s, p)f)_{i}(z)=\sum_{J\in \mathcal{P},\tau(J)=i}G_{J}(s)(T_{J}^{-1}z)\rho(T_{J})f_{i}(T_{J}^{-1}z)$

By virtue of the condition (M.9), for ${\rm Re} s>1/2$ the formally defined operator$\mathcal{L}(s, \rho)$

can be realized as an element in,and, wheredenotes the space of bounded linear operators fromtofor topological linear spaces
andand. We see that for
as above, the n-fold iterationof the operator
can be written$\mathcal{L}(\mathcal{A}_{b}(B(X);\mathbb{C}^{N});\mathcal{A}_{b}(D(X);\mathbb{C}^{N})),$$\mathcal{L}(C(X;\mathbb{C}^{N}))$$\mathcal{X}$$\mathcal{Y}$$\mathcal{L}(\mathcal{X};\mathcal{Y})$$\mathcal{X}$$f=\oplus_{j=1}^{q}f_{j}$$\mathcal{L}(\mathcal{A}_{b}(D(X);\mathbb{C}^{N}))$$\mathcal{Y}$$\mathcal{L}(\mathcal{X})=\mathcal{L}(\mathcal{X};\mathcal{X})$$\mathcal{L}(s, \rho)^{n}$$\mathcal{L}(s, \rho)$as$( \mathcal{L}(s,\rho)^{n}f)_{i}(z)=\sum_{J\in p_{\backslash ,\tau(J)=i}}G_{n,J}(s)(T_{J}^{-n}z)\rho(T_{J}^{n})f_{i}(T_{J}^{-n}z)$where$T_{J}^{-n}=(T_{J}^{n})^{-1}$,and $G_{n,J}(s, w)$ is given by$G_{n,J}(s)(w)=G(s)(w)G(s)(T_{J(i_{0})}w)G(s)(T_{J(i_{0}i_{1})}^{2}w)\cdot\cdot\cdot\cdot$$\cdot$$G(s)(T_{J(i_{0}i_{1},\ldots,i_{n-2})}^{n-1}w)$

if
has the form as in (2.1).
The following theorem is easily proved in the similar way to Theorem 5.1 and its corollaryin [19],$J\in \mathcal{P}_{n}$THEOREM 2.2. An analytic function
has a meromorphic extension to the entire s-plane which takes values in the space of$\{s\in \mathbb{C} :{\rm Re} s>1/2\}\ni s\mapsto \mathcal{L}(s,\rho)\in \mathcal{A}_{b}(D(X);\mathbb{C}^{N})$8nuclear opemtors of order . The candidates of poles are the points $s=-k/2,$ $k=$$-1,0,1,2,$ . In particular, the Fredholm determinant $Det(I-\mathcal{L}(s, \rho))$ extends to ameromorphic function to the entire s-plane and the candidates of its poles are the same
, possibly with different order.
as those of$0$$\cdots$$\mathcal{L}(s,\rho)$

Now we can state the following.
THEOREM 2.3. Let
is the Markov system associated to co-finite Rachsian group .
. For with ${\rm Re} s>1$ , the Fredholm determinant
Consider a representation :$Det(I-\mathcal{L}(s,\rho))$is represented
with respect toof the twisted tmnsfer operator
by an absolutely convergent series as$\Gamma$$\mathcal{T}_{\Gamma}$$\rho$$\Gammaarrow U_{N}$$s$$\mathcal{L}(s, \rho)$$Det(I-\mathcal{L}(s, \rho))=\exp(-\sum_{k=0}^{\infty}\sum_{n=0}^{\infty}\sum_{J\in \mathcal{P}_{n}T_{J}^{n}J\supset J}\frac{1}{n}$$=\exp$where$x_{J}$$(- \sum_{\tau\in PO(\mathcal{T}_{\Gamma})}\sum_{k=0}^{\infty}\sum_{n=0}^{\infty}\frac{1}{n}$is the unique fixed poin

$t$

of

$T_{J}^{n}$for

trace $(\rho(T_{J}^{n}))|DT_{J}^{n}(x_{J})|^{-(s+k)})$trace$J\in P_{n}$$\mathcal{T}_{\Gamma}$$(\rho(\Phi(\tau))^{n})\lambda(\Phi(\tau))^{-(e+k)n})$with

Proof. By Theorem 2.2 and the estimate of the norm of.

with trivial representation
with operator norm less than$\mathcal{L}(s, 1)$

in [19], we see thatis a nuclear operator of order
${\rm
Re}
s>1$
1 if
. Therefore we have$\mathcal{L}(s,\rho)$$T_{J}^{n}J\supset J$,$0$$Det(I-\mathcal{L}(s, \rho))=\exp(-\sum_{n=0}^{\infty}\frac{1}{n}trace\mathcal{L}(s, \rho)^{n})$.

On the other hand we have$trace\mathcal{L}(s,\rho)^{n}=\sum_{J\mathcal{P}_{n},T_{J}^{n}\supset J}trace\mathcal{L}(n, s, J, \rho)$where$\mathcal{L}(n, s, J, \rho)$

is the nuclear operator on$\mathcal{A}_{b}(D(\iota(J));\mathbb{C}^{N})$$\mathcal{L}(n, s, J, \rho)f(z)=G_{n,J}(s)(T_{J}^{-n}z)\rho(T_{J}^{n})f(T_{J}^{-n}z)$,of order$0$defined by$(z\in D(\iota(J)))$. Therefore we can obtain the theorem in the same way as
for
Theorem 7.2 in [19] if we verify the validity of the following lemma.
$f\in \mathcal{A}_{b}(D(\iota(J));\mathbb{C}^{N})$$\square$into itself with $\sup_{z\in D}|D\varphi(z)|<$be a univalent function on a domain
LEMMA 2.4.
. Let $a\in D$ be a unique fixed point of . Given an analytic function on $D$ with $F(a)\neq$and a unitary matrix of order $N$ , consider the opemtor :$D\subset \mathbb{C}$$Let\varphi$

$1$

$0$

$F$

$\varphi$

$U$

$L$

$\mathcal{A}_{b}(D;\mathbb{C}^{N})arrow \mathcal{A}_{b}(D;\mathbb{C}^{N})$9defined by$Lf(z)=F(\varphi(z))Uf(\varphi(z))$.

Then we have $Spec(L)\backslash \{0\}=\{F(a)(D\varphi)(a)^{k}\lambda_{j}$ : $j=1,2,$
where
are the eigenvalues of .
$\lambda_{1},$, $N$ and $k=0,1,$ . . ,
$\}$

$,$

$U$

$\ldots\lambda_{N}$Pmof. We write$\ldots$as, Sinceis a unitary matrix, there
such that, whereis a orthonormal basis of
consisting of eigenvectors of . Consider$V^{-1}LV$
the operator
which is spectrally equivalent to . Since $V^{-1}LV^{t}(f_{1}, \ldots, f_{N})=$, we reduce our problem to the cases when
$N=1$ . Thus we obtain the desired result from Lemma 7.1 in [19].
$f\in \mathcal{A}_{b}(D;\mathbb{C}^{N})$exists a unitary matrix$f={}^{t}(f_{1},$$V$$\ldots,$$f_{N})$$U$$VUV^{-1t}(e_{1}, \ldots, e_{N})={}^{t}(\lambda_{1}e_{1},$$\{e_{1}, \ldots, e_{N}\}$$\ldots,$$\lambda_{N}e_{N})$$U$$\mathbb{C}^{N}$$L$${}^{t}(\lambda_{1}F(\varphi(z))f_{1}(\varphi(z)),$$\ldots,$$\lambda_{N}F(\varphi(z))f_{N}(\varphi(z)))$$\square$

In order to rewrite the Selberg zeta function $Z(s, p)$ in terms of the $\mathbb{R}edholm$ determinant of, we introduce the following.
$\mathcal{L}(s, \rho)$(2.3)$—(s, \rho)=\prod_{k=0}^{\infty}\prod_{c\in\Phi(P\circ 1(\Gamma))}\det(I_{N}-\rho(c)e^{-(\epsilon+k)l(c)})$

.

Since
is a finite set, the analytic properties of:are easily investigated.
Combining Theorem 2.2 with Theorem 2.3 we obtain our main result.
$\Phi(PO_{1}(\Gamma))$$(s, \rho)$

THEOREM 2.5. For ${\rm Re} s>1$ , we have$Z(s, \rho)$ 三 $(s, \rho)^{2}=Det(I-\mathcal{L}(s,\rho))$.

In particular, $Z(s, p)$ is an analytic function in the half-plane ${\rm Re} s>1$ without zerohaving a meromorphic extension to the entire s-plane. Moreover, the candidates of poles
are located on the ${\rm Re} s=-k/2,$ $k=-1,0,1,$.
$\ldots$

Furthermore we can show the following theorem for the L-functions without consulting
the Selberg trace formula.

THEOREM 2.6 (cf.[l]). The Selberg L-function has the following properties.
(L.1) In the half-plane ${\rm Re} s>1,$ $L(s, \rho)$ is absolutety convergent and analytic.
(L.2) $L(s, \rho)$ has a meromorphic extension to the entire s-plane.
(L.3) In the closed half-plane$L(s, \rho)$ has no zeros.
(L.4) Letbe a group homomorphism such that the image of those elements
whose conjugacy classes contains elements of $HC(\Gamma)$ genemtes G. Let : $Garrow U(N)$be any nontrivial irreducible representation. Consider a representation given by.
Then $L(s, \rho)$ is analytic in the half-plane ${\rm Re} s>1/2$ .
(L.5) $L(s, 1)$ has a simple zero at $s=1$ and is analytic in the half-plane ${\rm Re} s>1/2$ .
${\rm Re} s\geq 1,$$\varphi:\Gammaarrow G$$\psi$$\rho=\psi\varphi$

10
As a corollary to Theorem 2.6, we can show the following Chebotarev type density
theorem (cf, [1], [28]).
THEOREM 2.7. Let be a normal subgroup ofclass $[g]\in[\Gamma/G]$ , Then we have$G$$\Gamma$with finite index. For any conjugacy$\#\{c\in HC(\Gamma):\pi_{G^{C\in}}[g], l(c)\leq t\}\sim\frac{\#[g]}{[\Gamma,G]}\frac{\exp(t)}{t}$where$\pi_{G}$:$\Gammaarrow\Gamma/G$$\lim_{tarrow\infty}A(t)/B(t)=1$$(tarrow\infty)$,

is the natural projection and $A(t)\sim B(t)(tarrow\infty)$ means.

Sketch of Proof of Theorem 2.6. If ${\rm Re} s>1$ we see that
$L(s, \rho)=\frac{Z(s+1,\rho)}{Z(s,\rho)}$holds. Thus the assertions (L.1), (L.2), and (L.3) follow from Theorem 2.5. The validityof the assertion (L.5) is verified in Theorem 7.4 in [19]. It remains to show the assertion
given by (2.3) is an analytic(L.4). By virtue of Theorem 2.5 and the fact that
${\rm
Re}
s>0$
, we see that each pole of $L(s, \rho)$ in thefunction without zeros in the half-plane
half-plane ${\rm Re} s>1/2$ is located on the axis ${\rm Re} s=1$ and for with ${\rm Re} s=1$ is a pole if. Therefore we have only to show that the following
and only if 1 is an eigenvalue oflemma. If $N=2$$\Xi(s,\rho)$$s$$\mathcal{L}(s, \rho)$

LEMMA 2.8. Let :${\rm Re} s=1$ , we denote bydefined by (2.2). Then$\rho$$\Gammaarrow U_{N}$be a unitary representation appearing in (L.4). For with:the twisted tmnsfer opemtor
has no eigenvalues of modulus 1.

$\mathcal{L}_{X}(s, \rho)$$s$$C(X;\mathbb{C}^{N})arrow C(X;\mathbb{C}^{N})$$\mathcal{L}_{X}(s, \rho)$

Idea of Proof We do not have enough space to give the proof. So we just explain about
how to prove it. First of all, we note that Lemma 6. 2 in [19] is strong enough that we canshow the lemma in the case when $N=1$ i.e. is a character. In the case when $N=2$ ,has an eigenvalue of modulus 1 and is the corresponding
we can show that if
eigenvector, then there exists a function : $X\cross Xarrow S^{1}$ such that $f(y)=\alpha(x,y)f(x)$for any $(x, y)\in X\cross X$ . This contradicts the irreducibility of . Note that the proof ofthis step is carried out following the method proving Proposition 4.4 in[l].
$\rho$$f$$\mathcal{L}_{X}(s,\rho)$$\alpha$$\rho$

3. GENERALIZATION II: RENORMALIZED RAUZY-VEECH-ZORICH INDUCTIONSIn this section, we consider renormalized Rauzy-Veech-Zorich inductions as generalizations of the continued fraction transformation. The definition of these transformations isbased on the ergodic property of Rauzy induction established by Veech [34] and Masur [13]

11in their way to solve the Keane conjecture on interval exchange transformations. Therefore
we start with the definition of interval exchange transformations ([9], [33]). Let $d\geq 2$ be
an integer. Consider the cone:for eachand thesymmetric groupof degree . For, we definethatfor $0\leq j\leq d$ . Consider a partitionof the interval
into subintervals. Let.
Then the interval exchange transformation:
is defined by$\Lambda_{d}=\{\lambda=(\lambda_{1},$$d$$\mathfrak{S}_{d}$$\lambda_{d})^{t}\in \mathbb{R}^{d}$$\ldots,$$(\lambda, \pi)\in\Lambda_{d}\cross \mathfrak{S}_{d}$$\beta_{j}(\lambda)=\sum_{i=1}^{j}\lambda_{j}$$\beta(\lambda)\in\{0\}\cross\Lambda_{d}.so$$X(\lambda)=[0, |\lambda|_{1})$$\alpha(\lambda)$$X_{j}(\lambda)=[\beta_{j-1}(\lambda),\beta_{j}(\lambda))(1\leq j\leq d)$$T_{(\lambda.\pi)}$$j\}$$\lambda_{j}>0$$\lambda^{\pi}=(\lambda_{\pi^{-1}1}, \ldots, \lambda_{\pi^{-1}d})^{t}$$X(\lambda)arrow X(\lambda)$.

$T_{(\lambda,\pi)}x=x+ \sum_{j=1}^{d}(\beta_{\pi j-1}(\lambda^{\pi})-\beta_{j-1}(\lambda))I_{X_{j}(\lambda)}(x)$By definition
maps the j-th intervalinonto -th intervalinisometrically preserving the orientation. Thus the Lebesgue measure $m$ restrictedtois an invariant measure for. Keane conjectured that for fixed irreducible
is uniquely ergodic Lebesgue almost every.
Next we recall the definition of Rauzy induction:
for ourconvenience. Consider the following
matrices $L(\pi)$ and $R(\pi)$$X_{j}(\lambda)$$\tau_{(\lambda.\pi)}$$\pi j$$\alpha(\lambda)$$X_{\pi j}(\lambda^{\pi})$$\alpha(\lambda^{\pi})$$X(\lambda)$$\pi\in \mathfrak{S}_{d},$$T_{(\lambda.\pi)}$$T_{(\lambda.\pi)}$$\lambda\in\Lambda_{d}$$\mathcal{T}_{0}$$\Lambda_{d}\cross \mathfrak{S}_{d}arrow\Lambda_{d}\cross \mathfrak{S}_{d}$$d\cross d$$L(\pi)=(e_{d-1}(\pi^{-1}j)^{t}I_{d-1}$

where

,

$0_{d-1,1})$$R(\pi)$$=(\begin{array}{ll}I_{\pi^{-1}d} K_{\pi^{-1}d,d-\pi^{-1}d}O_{d-\pi^{-1}d,\pi^{-1}d} J_{d-\pi^{-1}d}\end{array})$,is the $k\cross k$ identity matrix,
is $d-1$ -dimensional zero column vector,$e_{d-1}(\pi^{-1}j)$ is the $d-1$-dimensional unit vector whose-th component is 1,
is$I_{k}$$0_{d-1}$$\pi^{-1}j$the$k\cross l$zero matrix and$(d-\pi^{-1}d)\cross(d-\pi^{-1}d)$$K_{\pi^{-1}d,d-\pi^{-1}d}$and$J_{d-\pi^{-1}d}$$O_{k,l}$are $\pi^{-1}d\cross(d-\pi^{-1}d)$ matrix andmatrix given by$K_{\pi^{-1}d,d-\pi^{-1}d}=(\begin{array}{llll}0 0 \cdots 0\vdots \vdots \ddots \vdots 0 0 \cdots 0l 0 \cdots 0\end{array}),$$J_{d-\pi^{-1}d}=(\begin{array}{llll}0 1 \cdots 0\vdots \vdots \ddots \vdots 0 0 \cdots 11 0 \cdots 0\end{array})$

respectively. In addition we consider two transformations $L,$ $R$ :,$\mathfrak{S}_{d}arrow \mathfrak{S}_{d}$defined by$\sigma j$$(\sigma j\leq\sigma d),$$(L\sigma)j=\{$ $\sigma d+1$$\sigma j+1$

For$(\lambda, \pi)\in\Lambda_{d}\cross \mathfrak{S}_{d}$with$(\sigma j=d)$$(R\sigma)j=\{\begin{array}{ll}\sigma j (j\leq\sigma^{-1}d)\sigma d (j=\sigma^{-1}d+1)\sigma(j-1) otherwise\end{array}$otherwise$\lambda_{\pi^{-1}d}\neq\lambda_{d}$, we put$L(\pi)$$A(\lambda, \pi)=\{$$(if \lambda_{d}>\lambda_{\pi^{-1}d})$$R(\pi)$$(if \lambda_{d}<\lambda_{\pi^{-1}d})$’$D(\lambda)$$=\{\begin{array}{l}L (if \lambda_{d}>\lambda_{\pi^{-1}d})R (if \lambda_{d}<\lambda_{\pi^{-1}d}).\end{array}$

12
Then the Rauzy inductions
are defined forwith$(\lambda, \pi)$(3.1)$\mathcal{T}_{0}$:$\Lambda_{d}\cross \mathfrak{S}_{d}arrow\Lambda_{d}\cross \mathfrak{S}_{d}$$\lambda_{\pi^{-1}d}\neq\lambda_{d}$and$\mathcal{T}$:$\Delta_{d-1}\cross \mathfrak{S}_{d}arrow\Delta_{d-1}\cross \mathfrak{S}_{d}$by$\mathcal{T}_{0}(\lambda, \pi)=(A(\lambda, \pi)^{-1}\lambda, D(\lambda)\pi)$,$\mathcal{T}(\lambda, \pi)=$$( \frac{A(\lambda,\pi)^{-1}\lambda}{|A(\lambda,\pi)^{-1}\lambda|_{1}}, D(\lambda)\pi)$.

A permutation
is called irreducible if $\pi\{1, \ldots, k\}=\{1, \ldots, k\}$ yields $k=d$ . Fix. Consider the Rauzy classintroduced in [29].
a irreducible element
and the counting measure on ,andbelow denote the volume measure on.
:
respectively. We need the following result on$\pi\in \mathfrak{S}_{d}$$\mathcal{R}=\mathcal{R}(\pi_{0})$$\pi_{0}\in \mathfrak{S}_{d}$$\mathcal{R}$$\triangle_{d-1}$$\#\Re$$\omega_{d-1}$$\mathcal{T}$$\triangle_{d-1}\cross \mathcal{R}arrow\Delta_{d-1}\cross \mathcal{R}$

THEOREM 3.1 (Veech [34]). There exists a invariant measure equivalent to, the densityonwhich makes both conservative and ergodic. For eachis given by the restriction of a function on
onwith respect towhich is mtional, positive, and homogeneous of degree-d.
$\omega_{d-1}\cross\#_{R}$$\mathcal{T}$

$\mu$

$\mu$

$\pi\in \mathcal{R}$$\mathcal{T}$$\Delta_{d-1}\cross \mathcal{R}$$\Delta_{d-1}(=\Delta_{d-1}\cross\{\pi\})$$\omega_{d-1}$$\Lambda_{d}$Theorem 3.1 implies that satisfies the Poincar\’e recurrence. Thus we can define jumptransformations and induced transformations for . Recall these notions briefly. Let$\mathcal{T}$$\mathcal{T}$be ameasure space and : $Xarrow X$ a -nonsingular transformation
satisfying the Poincar\’e recurrence i.e. almost every $x\in X$ has the property that for any
with $\mu(E)>0,$ $T^{n}x\in E$ holds for infinitely many $n\geq 0$ . Then for any $E,$
with $\mu(E)>0$ and $\mu(F)>0$ , we put for $x\in E$$(X, \mathcal{B},\mu)$$\sigma-finite$

$T$

$\mu$

$\mu$

$F\in \mathcal{B}$$E\in \mathcal{B}$$n(E, F;x)= \inf\{n\geq 1 :T^{n}x\in F\}$ .

In the case when $E=F$ we just write as $n(E;x)=n(E, E : x)$ . From our assumption$n(E, F;x)<\infty\mu-$ a.e. Thus we obtain almost everywhere defined transformation:$Earrow F$ called the jump transformation of
from to by$T_{E,F}$

$T$

$F$

$E$

$T_{E,F}x=T^{n(E,F;x)_{X}}$.

and called the induced transformation of
In the case $E=F,$
is denoted by$E$$E$or the first return map of to . Roughly speaking, ‘renormalization of thetotransformation ’ means the procedure of constructing a new transformation by producing
jump transformations and their compositions.
:
We now consider the renormalization of the Rauzy induction
given by (3.1). Set$T_{E}$$T_{E,F}$

$T$

$T$

$T$

$\triangle_{d-1}\cross \mathcal{R}arrow\Delta_{d-1}\cross \mathcal{R}$$\mathcal{T}$$\triangle(R, \pi)=\{\lambda\in\triangle_{d-1}:\lambda_{d}<\lambda_{\pi^{-1}d}\}\cross\{\pi\}$$\triangle(L, \pi)=\{\lambda\in\Delta_{d-1}:\lambda_{d}>\lambda_{\pi^{-1}d}\}\cross\{\pi\}$$\Delta(L)=\bigcup_{\pi\in \mathcal{R}}\Delta(L, \pi)$,$\Delta(R)=\bigcup_{\pi\in’\mathcal{R}}\triangle(R, \pi)$.

13Note that the sets$\triangle(L, \pi)$andare expressed by$\triangle(R, \pi)$$\Delta(L, \pi)=(L(\pi)\Lambda_{d-1}\cap\triangle_{d-1})\cross\{\pi\}$

We consider the jump transformations,$\Delta(R, \pi)=$:$\mathcal{T}_{\Delta(L),\Delta(R)}$$(R(\pi)\Lambda_{d-1}\cap\triangle_{d-1})\cross\{\pi\}$$\Delta(L)arrow\triangle(R)$

and

.
:

$\mathcal{T}_{\Delta(R),\Delta(L)}$. The Rauzy-Veech-Zorich induction : $\Delta(L)\cup\triangle(R)arrow\Delta(L)\cup\Delta(R)$ is
the transformation such thatand
. The transformation:
is a typical example of the renormalized
Rauzy-Veech-Zorich induction. Note that if $d=2$ and $\pi=(21)$ , then $S$ is conjugate with$\Delta(R)arrow\Delta(L)$$\mathcal{G}$$\mathcal{G}|_{\Delta(L)}=\mathcal{T}_{\Delta(L),\Delta(R)}$$S=\mathcal{T}_{\Delta(L),\Delta(R)}\circ \mathcal{T}_{\Delta(R),\Delta(L)}$$T_{G}^{2}$$\mathcal{G}|_{\Delta(R)}=\mathcal{T}_{\Delta(R),\Delta(L)}$$\triangle(L)arrow\triangle(L)$. Therefore we would like to consider the renormalizations of the Rauzy induction as

generalizations of the continued fraction transformation.
The rest of the section is devoted to the study of a special class of renormalized RauzyVeech-Zorich inductions whose members play the same role asin our argument. Let$T_{G}^{2}$

be such that is irrational, i.e. the entries of are linearly independent. Then the corresponding interval exchange transformation
is minimal by$(\hat{\lambda}, \pi_{0})\in\triangle(L, \pi_{0})$over$\mathbb{Q}$$\hat{\lambda}$$\lambda$$T_{(\hat{\lambda},\pi 0)}$

the result in [9]. Therefore we can find $N\geq 2$ such that

$A_{N}(\hat{\lambda}, \pi_{0})>0$

by virtue of theremark after Proposition 3.30 in [33]. We denoteby for the sake of simplicity.
Consider the set
and. We are interested in the
induced transformationof to the set. We regard
as a transformationonin a natural way. Note that since
is an renormalization of , it is also a
renormalization of . In particular,
andcoincides in this case.
For a nonnegative invertible matrix , letand define the map:
by $\overline{A}x=Ax/|Ax|_{1}$ for. Then we have the following.
$B$$A_{N}(\hat{\lambda}, \pi_{0})$$\triangle_{B}=B\Lambda_{d}\cap\triangle_{d-1}$$\Delta(B, \pi)=\triangle_{B}\cross\{\pi\}$$S$$S_{B}$$\triangle(B, \pi)$$\Delta_{B}$$S_{B}$$S$$S_{B}$$S_{B}$$\mathcal{G}$$\mathcal{G}_{B}$$A$$\triangle_{A}=A\Lambda_{d}\cap\Delta_{d-1}$

へ$A$$\triangle_{d-1}arrow\triangle_{d-1}$$x\in\triangle_{d-1}$

LEMMA 3.2 (Lemma 3.1 in [21]). Let
nonnegative integml matrices$S_{B}$$\mathcal{A}=\{A^{(k)}\}$

be as above. There exist sequences of distinct
and $C=\{C^{(k)}\}$ satisfying the following:(1)$A^{(k)}B=BC^{(k)}$and $\det A^{(k)}=\det C^{(k)}=\pm 1$ .

(2)

$S_{B}|_{\Delta_{AB}}=\overline{A^{-1}},$

$i.e$

.

$S_{B}x= \frac{A^{-1_{X}}}{|A^{-1}x|_{1}}$each.
(3) The family of the set

for $A\in A$ . In particular,$S_{B}\Delta_{AB}=\Delta_{B}$for$A\in \mathcal{A}$

$i.e$

.

$\mathcal{P}=\{\Delta_{AB} :$\omega_{B}(\triangle_{AB}\cap\Delta_{A’B})=0$

where

for

$A,$

A\in \mathcal{A}\}$$A’\in A$

withforms a measumble partition of$A\neq A’$and$\Delta_{B}$$\omega_{B}(\triangle_{B}\backslash \bigcup_{A\in A}\Delta_{AB})=0$,,$\omega_{B}=\omega_{d-1}(\triangle_{B})^{-1}\omega_{d-1}|_{\Delta_{B}}$

Next we introduce the Hilbert projective metric on. Note that the results on theHilbert projective metrics that we need as well as their application to the study of ergodic
behavior of dynamical systems are summarized in [11]. For, we write $x\leq y$ if$\Delta_{d-1}$$x,$$y\in\Lambda_{d}$14each entry of $y-x$ is nonnegative. Put$\alpha(x, y)=\sup\{a\geq 0:ax\leq y\}$$\Theta(x, y)=\log\frac{\beta(x,y)}{\alpha(x,y)}$$\beta(x,y)=\inf\{b\geq 0 :y\leq bx\}$ ,,.

such thatis a pseudo-metric on
is called the Hilbert projective metric on .
$\Theta(x, y)=0$ if and only if $x=cy$ holds for some $c>0$ . Thus
is a metric on the projective
space. We summarize the basic properties of the renormalized Rauzy-Veech-Zorich
induction
as the following lemma.
$\Theta$$\Theta$$\Lambda_{d}$$\Lambda_{d}$$\Theta$$\Delta_{d-1}$$S_{B}$LEMMA 3.3 (Lemma 3.4 in [21]). Let$S_{B}$

be as above. Then we have the following.

(1) (Markov property) For any $n\geq 1$ we have$P_{n}=n-1k=0\vee S_{B}^{-k}\mathcal{P}$for any$\triangle_{AB}\in \mathcal{P}_{n}$. In particular(2) (expanding) There eanst$C_{1}>0$

and$S_{B}^{n}$and:$S_{B}^{n}\triangle_{AB}=\triangle_{B}$is a homeomorphism.
such that for any $n\geq 1$$\Delta_{AB}arrow\Delta_{B}$$\theta\in(0,1)$$\Theta(S_{B}^{n}x,S_{B}^{n}y)\geq C_{1}^{-1}\theta^{-n}\Theta(x, y)$.
holds for any(3) (finite distortion) There exists $C_{2}>0$ such that for any $n\geq 1$$x,$$y\in\triangle_{AB}\in \mathcal{P}_{n}$$| \log\frac{J(S_{B}^{n})(x)}{J(S_{B}^{n})(y)}|\leq C_{2}\Theta(S_{B}^{n}x,S_{B}^{n}y)$holds for any $y\in\triangle_{AB}\in P_{n}$ , where.
respect to$\delta\in(0,1)$(4) There exist
and $C_{3}>0$ such that

$J(S_{B}^{n})$$x,$denotes the Jacobian of$S_{B}^{n}$vnth$\omega_{B}$$\sum_{A\in A}\sup_{x\in\Delta_{B}}\frac{1}{|Ax|_{1}^{d(1-\delta)}}<C_{3}$.

Note that for the proof of the assertion (4) in Lemma 3.3, we need some results in
Bufetov [4].
be the renormalized RauzyNow we introduce a family of transfer operators. Letwith ${\rm Re} s>1-\delta$
Veech-Zorich induction defined just before Lemma 3.2 above. For, we put
and a complex-valued function on$S_{E}$$s\in \mathbb{C}$$\Delta_{B}$$\mathcal{L}(s)f(x)=\sum_{A\in A}\frac{1}{|Ax|_{1}^{ds}}f(\overline{A}x)$,15where $\delta\in(0,1)$ is as in Lemma 3.3. Let
be the Banach space of complex-valued
continuous functions on
endowed with the supremum norm
and letbethe Banach space of complex-valued Lipschitz continuous functions onwith respect
to the projective metric endowed with the norm
$C(\triangle_{B})$$\Delta_{B}$$\Vert\cdot\Vert_{\infty}$$F_{\Theta}(\triangle_{B})$$\Delta_{B}$$\Theta$$\Vert g\Vert_{\Theta}=[g]_{\Theta}+\Vert g\Vert_{\infty}$

where

,

and $[g]_{\Theta}= \sup_{x,y\in\Delta_{B}:x\neq y}|g(x)-g(y)|/\Theta(x,y)$ i.e. theLipschitz constant of with respect to .
anddenote thesubspaces of real-valued elements ofand, respectively. In [20] we proveda weak version of local central limit theorem for the partial sum
with $f\in$using the methods in [18]. If the perturbed $Perron-\mathbb{R}obenius$ operators given by$\Vert g\Vert_{\infty}=\sup_{x\in\Delta_{B}}|g(x)|$$\Theta$$g$$C(\triangle_{B})$$C(\triangle_{B}arrow \mathbb{R})$$F_{\Theta}(\triangle_{B}arrow \mathbb{R})$$F_{\Theta}(\triangle_{B})$$\sum_{k=0}^{n-1}foS_{B}^{k}$$F_{\Theta}(\Delta_{B})$$\mathcal{L}(1)(e^{\sqrt{-1}tf}g)(x)=\sum_{A\in A}\frac{\exp(\sqrt{-1}tf(\overline{A}x))}{|Ax|_{1}^{d}}g(\overline{A}x)$

form an analytic family of bounded linear operators on, it seems that the same
technique does work.. The assertion (4) in Lemma 3.3 guarantees that if we choose$F_{\Theta}(\triangle_{B})$$f(x)=\log|A^{-1}x|_{1}$operators onfollowing;, the family$F_{\Theta}(\Delta_{B})$$\mathcal{L}(1)(e^{\sqrt{-1}tf}\cdot)$

although$f$becomes an analytic family of bounded linear. Anyway we can show theis not an element in$F_{\Theta}(\triangle_{B})$PROPOSITION 3.4 (Proposition 4.4 in [21]). There exist a neighborhood of the half-plane
and the open discwith mdius
centered at 1 such that theanalytic familyof bounded linear opemtors on
satisfies thefollowing,.
$U$${\rm Re} s\geq 1$$r_{0}\in(0, \delta)\subset U$$\{\mathcal{L}(s) :(1) For $s\in D(1, r_{0}),$$r_{0}<\delta$s\in D(1, r_{0})\}$$\mathcal{L}(s)$$F_{\Theta}(\triangle_{B})$has the spectml decomposition$\mathcal{L}(s)^{n}=\lambda(s)^{n}E(s)+R(s)^{n}$for eachwhereis a simple eigenvalue ofwith maximal modulus,$E(s)$ is the projection onto the one-dimensional eigenspace corresponding to,and $R(s)$ is a bounded linear opemtor with spectml mdius less thanfor some$r_{1}\in(0,1)$ independentof $s\in D(1, r_{0})$ .
(2) For $s\in U\backslash D(1, r_{0})$ , the spectml mdius ofis less than 1.
(3)
in the assertion (1) is analytic function on $D(1, r_{0})$ such that $\lambda(1)=1$
and is decreasing on $[1-r_{0},1]$ .
$n\in \mathbb{N}_{f}$$\lambda(s)$$\mathcal{L}(s)$$\lambda(s)$$r_{1}$$\mathcal{L}(s)$$\lambda(s)$

$a$

16
(4)

and $R(s)$ in the assertion (1) are analytic functions on $D(1,r_{0})$ with values
given by the Dunford integrals
in bounded linear opemtors on$E(s)$$F_{e}(\triangle_{B})$,$E(s)= \frac{1}{2\pi\sqrt{-1}}\int_{|z-1|=r_{2}}R(\mathcal{L}(s), z)dz$$R(s)^{n}= \frac{1}{2\pi\sqrt{-1}}\int_{|z|=r1}z^{n}R(\mathcal{L}(s), z)dz$

for each $n\in N$ , where $0<r_{1},$ $r_{2}<1$ are independent of $s\in D(1,r_{0})$ satisfying.
$r_{1}+r_{2}<1$ anddenotes the resolvent opemtor of$R(\mathcal{L}(s), z)=(zI-\mathcal{L}(s))^{-1}$Moreover, we can show that there exists$\delta_{1}>0$with ${\rm Re} s>1-\delta_{1}$ as follows. For each $A\in A_{m},$byin. Defined an operator on$\Delta_{B}$$\mathcal{K}_{n}$$\mathcal{L}(s)$is quasicompact forsuch that
denotes the unique fixed point of$s$$\mathcal{L}(s)$$\overline{A}$$x_{A}$$F_{\Theta}(\Delta_{B})$$\mathcal{K}_{n}g(x)=\sum_{A\in A_{\hslash}}\frac{1}{|Ax|_{1}^{d\epsilon}}g(x_{A})$for$f\in F_{\Theta}(\Delta_{B})$. Let$r_{0}$

and$\lambda(s)$be as in Proposition 3.4, we show the following.

PROPOSITION 3.5 (Proposition 4.5 in [21]). There ext. positive constants
we havesuch that for any with $Res>1-r_{0}$ and$ts$$C_{4}$

and$C_{5}$$g\in F_{e}(\Delta_{B})$$s$,$\Vert(\mathcal{L}(s)^{n}-\mathcal{K}_{n})g\Vert_{\infty}\leq C_{4}\lambda({\rm Re} s)^{n}\theta^{n}[g]_{\Theta}$$[(\mathcal{L}(s)^{n}-\mathcal{K}_{n})g]e\leq C_{5}(|s|+1)\lambda({\rm Re} s)^{n}\theta^{n}[g]_{\Theta}$

In particular$\mathcal{L}(s)$is quasicompact as far as$\lambda({\rm Re} s)\theta<1$holds.

Let $PO(S_{B})$ denote the totality of prime periodic orbits...,$S_{B}^{p-1}\lambda\}\in PO(S_{B})$.

$\tau$

of$S_{B}$. For$\tau=\{\lambda,$, put$N(\tau)=J(S_{B}^{p})(\lambda)^{\frac{1}{d}}$Let us consider the following zeta function given by the formal Euler product(3.2)$\zeta(s)=\prod_{\tau\in PO(S_{B})}(1-N(\tau)^{-ds})^{-1}$.

A formal calculation leads us to the equation$\zeta(s)=\exp(\sum_{n=1}^{\infty}\frac{1}{n}\sum_{x:S_{B}^{n}x=x}J(S_{B}^{n})(x)^{-e})=\exp(\sum_{n=1}^{\infty}\frac{1}{n}\sum_{A\in A_{n}}|Ax_{A}|_{1}^{-d\epsilon})$(3.3)$= \exp(\sum_{n=1}^{\infty}\frac{1}{n}\sum_{A\in \mathcal{A}_{m}}\lambda_{A}^{-ds})$.

$S_{B}\lambda$,

17
The main theorem in this section is the following.

THEOREM 3.6 (cf. Theorem 5.1 in [21]). The infinite product in the reght hand side of(3.2) is absolutely convergent for with ${\rm Re} s>1$ and defines an analytic function urthout$zem$ . In addition, the series in (3.3) are absolutely convergent
and the equations are all
justified. Moreover there exists
such thathas the meromorphic extension to$s$$\delta_{1}>0$$\zeta(s)$the half-planesatisfying the following:(1) $s=1$ is the unique pole on the axis ${\rm Re} s=1$ and it is simple.
(2) In the half-plane ${\rm Re} s>1-\delta_{1},$does not have zeros.
(3) There exists
such that in the half-plane ${\rm Re} s>1-\delta_{2z}s=1$ remains tobe the unique pole of. $i.e$ .
tums out to be a pole free region${\rm Re} s>1-\delta_{1}$$\zeta(s)$$\delta_{2}\in(0, \delta_{1})$$\zeta(s)$for$\zeta(s)$$\{s:{\rm Re} s>1-\delta_{2}\}\backslash \{1\}$.

Sketch of Proof The reader familiar with the transfer operator approach to dynamicalzeta function will notice that Proposition 3.4 and Proposition 3.5 imply the validity of
the assertions (1) and (2). In order to prove the assertion (3) we consult the results in [2]concerned with exponential decay of correlations of(see also [27]).
$S_{B}$

It is well known that Theorem 3.6 provides us with enough information in order toprove the following.

THEOREM 3.7 (cf. [27]). Thee exists$\alpha\in(0, d)$such that$\#\{\tau\in PO(S_{B}):\log N(\tau)\leq t\}=\frac{e^{dt}}{dt}+O(e^{\alpha t})$$(tarrow+\infty)$.

Finally we explain about an geometric interpretation of Theorem 3.7 following Veech[34] and Mosher [17]. Let $g\geq 2$ be an integer.
and $Mod_{g}$ denote the Teichm\"uller space
and the mapping class group of genus , respectively. Consider the following sets.
$T_{9}$$g$: the totality of oriented prime closed geodesics
ichm\"uller metric in .
$\bullet$$CG(T_{g})$$\gamma$with respect to the Te-$T_{g}$: the totality of primitive hyperbolic conjugacy classes

in $Mod_{g}$ , i.e.
can be written as $c=\langle h\rangle=\{g^{-1}hg : g\in Mod_{g}\}$ , where is a primitive hyperbolic
element in in $Mod_{9}$ whose representative is a pseudo-Anosov diffeomorphism.
$\bullet$$HC(Mod_{g})$

$c$

$c$

$h$

For $\gamma\in CG(T_{g})$ and$\bullet$$l(\gamma)$$c=\langle h\rangle$, we put: the least period of .
$\gamma$$\bullet$$\lambda(c)$: the dilatation of .
$h$Then there exists a natural one-to-one correspondence between these sets such that$\exp(l(\gamma))=\lambda(c)$ holds if $\gamma\in CG(T_{9})$and $c\in HC(Mod_{g})$ are corresponding each other. If

18
we consider an analogue of the prime number theorem for length spectrum of Teichm\"uller
space of genus greater than 1, we arrive at a difficulty that there is no results for the zetafunction which plays the role of the Selberg zeta function for the modular surface. On
the other hand, if we look at the renormalized Rauzy-Veech-Zorich induction which is asort of generalization of the continued fraction transformation, we notice the following, such that the eigenvector
there exists
facts. For any periodic point ofcoincides with . By way of zippered
corresponding to the Perron-Frobenius root
rectangles in [34] (see also [39], and [10]), there exist an positive integer depending only
on the irreducible permutation , a closed Riemann surface of genus , a holomorphic
is the dilatation ofl-form , and a pseudo-Anosov diffeomorphism on such thatchoosing
an appropriatebyis obtained
and the interval exchange transformationtransversal to the measured foliation determined by . Therefore we see that for each$\tau\in P(S_{B})$ , we can find a Teichm\"uller closed geodesic
and a hyperbolic conjugacy classsuch that $\exp(l(\gamma))=\lambda(c)=N(\tau)$ . Note that the number of intervals turnsofout to be the dimension of the corresponding moduli space of Abelian differentials. Although is a primitive periodic orbit, and are not necessarily so. We have to note thatrecently Eskin and Mirzakhani [6] establish remarkable result. They prove an analogue ofthe prime number theorem for closed Teichm\"uller geodesics on the principal stratum of
the moduli space of quadratic differentials using the method in Margulis [12]. As a trial
Theorem 3.7 may be still interesting, but it seems that we need more ideas to establish an
analogue of the prime number theorem for closed Teichm\"uller geodesics on any stratumof the moduli space of quadratic differentials.
$A \in\bigcup_{n=1}^{\infty}A$$S_{B}$$x$$x$$\lambda_{A}$$x_{A}$$g$$R$$\pi_{0}$$\omega$$\varphi$

$R$

$g$

$\lambda_{A}$$\varphi$$T_{(x_{A},\pi_{0})}$$\omega$$\gamma$$c$$d$$Mod_{9}$$\tau$$\gamma$$c$

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