q$. \cr} $$ Then $\a=q$ and $c_n=q^n$ so that $$ \zeta (f,z) ={1\over 1-z} $$ which is analytic in the entire $z$-plane except for a simple pole at $z=1$. In this example the constants $\varrho_n$ of Corollary 3.1 vanish identically for $n>0$ and the decay of correlations is faster than any exponential. \vskip 0.5cm {\bf Example 5.2.} Set $$ f(x)= \cases{x/(1-x) &if $x\leq 1/2$, \cr 2x-1 &if $x>1/2$. \cr} $$ Then $\a=1$ and $c_n = 1/(n+1)$ so that the invariant measure $\nu$ is infinite. For the piecewise affine approximation we find the zeta function $$ \zeta (\hf,z) = {z \over (1-z)^2 \log{1/(1-z)} } $$ which has a logarithmic branch point at $z=1$. Moreover, it is analytic in the entire $z$-plane with a branch cut along the ray $(1, +\infty)$. In a neighbourhood of $z= 1$ we have $(1-z)\zeta (\hf,z) \to \infty$ but $(1-z)^{2}\zeta (\hf,z) \to 0$. We point out that in this case one can obtain the analytic continuation in the cut plane also for $\zeta (f,z)$. Indeed, one can easily check that $$ e(x)={d\nu \over dm}(x) = {1\over \log 2}\cdot {1\over x} $$ so that $d_n = {\log\left(n+2/ n+1\right)/ \log 2}$ and the analytic continuation is given by Theorem 3.1 and formula (3.1) with $$ d(x)={\log \left(x+2/x+1\right)\over \log 2}. $$ Finally, using the above zeta functions and Karamata's theorem, one shows that both $f$ and $\hf$ have wandering rate $w_n = \CT \, \log n$ and return sequence $a_n \sim \CT \, n/\log n$. \vskip 0.5cm {\bf 6. The coding.} \vskip 0.2cm We now construct a coding. Let $\O$ be the set of one-sided sequences $\o = (\o_0\o_1\dots )$, $\o_i\in\{1,2,\dots\}$ satisfying the compatibility condition: given $\o_i$ then either $\o_{i-1}=\o_i +1$ or $\o_{i-1}=1$. Then, the map $\xi:\O \rightarrow \ui$ defined by $$ \xi(\o) =x\quad\hbox{according to}\quad f^j(x)\in A_{\o_j},\;\; j\geq 0 $$ is a bijection between $\O$ and the points of $\ui$ which are not preimages of the origin. In other words, to any sequence $\o \in \O$ corresponds, via the map $\xi$, a point $x\in \ui \setminus \{c_k\}_{k\geq 0}$, and viceversa. Moreover, $\xi$ conjugates the map $f$ with the shift $T$ on $\O$. \vskip 0.2cm Let us consider the infinite sequence $\{ t_j\}_{j\geq 1}$ of successive entrance times in the state $1$: $t_1(\o)=\inf\{i\geq 0\;:\; \o_i=1\}$ and, for $j\geq 2$, $t_j(\o)=\inf\{i>t_{i-1}\;:\; \o_i=1\}$. Furthermore, we define a sequence of integer valued random variables by $$ \s_j(\o)=t_{j+1}-t_j,\;\; j\geq 0 $$ with the convention that $t_0=-1$. It is then easy to realize that for any $\o\in \O$, we have $g^j(x)\in A_{\s_j}$, $j\geq 0$, where $x=\xi(\o)$ and the integers $\s_j=\s_j(\o)$ are defined above. Let $\S$ be the set of {\it all} one-sided sequences $\s$ of the form $\s =(\s_0\s_1\dots )$, $\s_j\in\{1,2,\dots\}$. Then, the map $\pi :\S \to \ui$ defined by $$ \pi(\s) =x\quad\hbox{according to}\quad g^j(x)\in A_{\s_j},\;\; j\geq 0 $$ is a bijection between $\S$ and the points of $\ui\setminus \{c_k\}_{k\geq 0}$. Moreover, $\pi$ conjugates the map $g$ with the shift $T$ on $\S$. Observe that the map $\eta := \pi^{-1}\circ \xi$ determines a bijection between $\O$ and $\S$. \vskip 0.2cm {\bf Remark 1.} We can represent the dependence of the $\s_j$'s on $\o$ recursively as follows: $$ \s_0=\o_0\quad\hbox{and}\quad\s_j = \o_{s_j}\quad\hbox{for}\quad j>0, \quad\hbox{where}\quad s_j=\sum_{i=0}^{j-1}\s_i. $$ Notice however that this rule may associate to a periodic sequence an eventually periodic one. More precisely, let $\o \in \O$ be a periodic sequence of period $n$ for the shift $T$. We write it in the form $\o=({ \overline {\o_0 \o_1 \dots \o_{n-1} }})$. If $\o_0 =1$ then we may write, for some $k\geq 0$, $$ \o_0 \o_1 \dots \o_{n-1}\, = \,{\underbrace{1\dots 1}_{r_0\geq 1}}\,\, {\underbrace{l_1l_1-1\dots 1}_{l_1\geq 1}}\,\, {\underbrace{1\dots 1}_{r_1\geq 0}} \,\, \ldots \,\,{\underbrace{l_kl_k-1\dots 1}_{l_k\geq 1}}\,\, {\underbrace{1\dots 1}_{r_k\geq 0}}\, $$ where $r_0+l_1+\dots +r_k=n$. Hence, the above rule gives $$ \s_0 \s_1 \dots \s_{m-1} \, = \,{\underbrace{1\dots 1}_{r_0\geq 1}}\,\,l_1\,\, {\underbrace{1\dots 1}_{r_1\geq 0}} \,\, \ldots \,\,l_k\,\, {\underbrace{1\dots 1}_{r_k\geq 0}} $$ so that $\s(\o)=({ \overline {\s_0 \s_1 \dots \s_{m-1} }})$ is periodic of period $m= k+ r_0+\dots +r_k$ and satisfies $n=\s_0+\dots +\s_{m-1}$. On the other hand, if $\o_0 >1$, it may happen that, for some $k\geq 1$, $$ \o_0 \o_1 \dots \o_{n-1}\, = \,{\underbrace{l_0l_0-1\dots 1}_{l_0\geq 1}}\,\, {\underbrace{1\dots 1}_{r_1\geq 0}} \,\, {\underbrace{l_1l_1-1\dots 1}_{l_1\geq 1}}\,\,\ldots \,\, {\underbrace{1\dots 1}_{r_k\geq 0}}\,\, {\underbrace{l_k+l_0\dots l_0+1}_{l_k\geq 0}} $$ and, according to the above rule, one would find the eventually periodic sequence $\s(\o)=({\s_0 \overline {\s_1 \s_2 \dots \s_{m} }})$ where $$ \s_0 = l_0\quad\hbox{and}\quad \s_1 \s_2 \dots \s_{m}\, = \, {\underbrace{1\dots 1}_{r_1\geq 0}} \,\, l_1\,\,\ldots \,\, {\underbrace{1\dots 1}_{r_k\geq 0}}\,\, l_k+ l_0 $$ whose ultimate period is $m= k+ r_1+\dots +r_k$ and which satisfies $n=\s_1+\dots +\s_m$. Thus, in the latter case, in order to obtain a correspondence between periodic sequences it is necessary to apply the aformentioned rule to some iterate of the original sequence. By the way, this operation does not modify the weight associated to the periodic sequence (see below) and thus it has no influence in what follows. \vskip 0.2cm {\bf Remark 2.} For every integer $j\geq 0$, denote by $x_j$ the projection on the $j^{th}$ symbol, i.e. $x_j(\o)=\o_j$. Then the stochastic process on $\O$ given by $x_j(\o)=\o_j$, $j\geq 0$, is a Markov chain with transition probabilities $p_{ij}= |\psi_0(A_j)\bigcap A_i|/|A_i|$. An easy consequence is that the random variables $\s_j$ are independent and identically distributed. Their common law is given by: ${\rm Prob}(\s_j=k)=\b_k$ for any $j\geq 0$ and $k\geq 1$. Notice that, for $\a=1$, the state $1$ of the Markov chain is positive-recurrent when $s<1$ and null-recurrent when $s\geq 1$. \vskip 0.5cm {\bf 7. Interactions, zeta functions and transfer operators.} \vskip 0.2cm We now use the maps $\xi$ and $\pi$ to lift the functions $\log \df$ and $\log \dg$, respectively, up to some `interactions' on the symbol spaces. More precisely, given $\o \in \O$ and $\s\in \S$, we set $$ V(\o) = \cases{V_0(\o)=\log [(\psi_0)^{\prime}(\xi(\o_1 \o_2\dots ))], &if $\o_0 >1$ \cr V_1(\o)=\log [(\psi_1)^{\prime}(\xi(\o_1 \o_2\dots ))], &if $\o_0=1$ \cr} \eqno(7.1) $$ and $$ W(\s) = \log [(\phi_{\s_0})^{\prime}(\pi (\s_1 \s_2\dots ))].\eqno(7.2) $$ Given $0<\t <1$ we define a metric on $\S$ by setting $d_{\t}(\s,\sp)=\t^n$ where $n$ is such that: $\s_j=\sp_j$ for $0\leq j\leq n$. Moreover, for any continuous function $\Phi: \S \to \C$ and integer $n\geq 0$, set: $$ {\rm var}_n\Phi= \sup \{\, |\Phi(\s)-\Phi(\sp)|\,: \, \s_j=\sp_j,\, 0\leq j\leq n\, \} $$ and $$ |\Phi |_{\t} = \sup \{ {{\rm var}_n\Phi \over \t^n}, n\geq 0 \}\eqno(7.3) $$ Finally, we denote by ${\cal{F}}_{\t}$ the space of all Lipschitz functions on $\S$ with respect to the metric $d_{\t}$, that is all continuous function $\Phi$ on $\S$ satisfying ${\rm var}_n\Phi\leq C\t^n$ for some constant $C>0$ (so $|\Phi |_{\t}$ is the least Lipschitz constant). With the norm $\Vert \Phi \Vert_{\t} =|\Phi |_{\t} + |\Phi |_{\infty}$, one makes $\Ft$ a Banach space. A direct consequence of the assumptions made above on the map $f$ is the following result: \vskip 0.2cm {\bf Lemma 7.1.} {\it $W(\s)\in \Ft$ for any $\t \geq \beta$.} \vskip 0.2cm {\it Proof.} The proof is a trivial adaptation to the present context of the argument given in [CI1], Lemma 2.1. $\qed$ \vskip 0.2cm We now write the dynamical zeta functions for the map $f$ and $g$ in the following way: $$ \zeta (f,z) = \exp \sum_{n=1}^{\infty} {z^n\over n} Z_n(f), \qquad \zeta (g,z) = \exp \sum_{n=1}^{\infty} {z^n\over n} Z_n(g) \eqno(7.4) $$ where $$ Z_n(f) =\sum_{\scriptstyle \o\in \O \atop \scriptstyle T^n\o=\o} \exp \sum_{j=0}^{n-1}V(T^j\o) + \a^n \eqno(7.5) $$ and $$ Z_n(g) =\sum_{\scriptstyle \s\in \S \atop \scriptstyle T^n\s=\s} \exp \sum_{j=0}^{n-1}W(T^j\s) \eqno(7.6) $$ The term $\a^n$ in (7.5) accounts for the contribution of the fixed point in $0$. Using the functions $\hV$ and $\hW$ one obtains the corresponding quantities for the affine model. \vskip 0.2cm Let us now examine how $\zeta (f,z)$ and $\zeta (g,z)$ are related to one another. First, let $\o\in \O$ be a periodic sequence of period $n$ for the shift $T$. Moreover, let $\s (\o )$ be a periodic sequence of period $m$ in the space $\S$ corresponding to $\o$ or to some iterate of it (see Section 6, Remark 1). Thus $s_m(\s)=\sum_{j=0}^{m-1}\s_j = n$. >From (7.1) and (7.2) we then have $$ \sum_{j=0}^{n-1}V(T^j\o) = \sum_{j=0}^{m-1}W(T^j\s) $$ Using this fact we write $Z_n(f)$ as follows: $$ Z_n(f) = \a^n +\sum_{m=1}^n {n\over m} \sum_{\scriptstyle \s\in \S ,\, \scriptstyle T^m\s =\s \atop \scriptstyle s_m(\s)=n } \exp \sum_{j=0}^{m-1}W(T^j\s) \eqno(7.7) $$ The second sum ranges over the $n-1 \choose m-1$ ways to write the integer $n$ as a sum of $m$ positive integers, counting all permutations. Therefore, $$\eqalign{ \sum_{n=1}^{\infty}{z^n\over n} Z_n(f) &= \log ({1\over 1-\a z}) + \sum_{n=1}^{\infty}\sum_{m=1}^{n} {1\over m} \sum_{ \scriptstyle \s\in \S,\, \scriptstyle T^m\s =\s \atop \scriptstyle s_m(\s)=n } z^n \exp \sum_{j=0}^{m-1}W(T^j\s) \cr &= \log ({1\over 1-\a z}) + \sum_{m=1}^{\infty} {1\over m} \sum_{\scriptstyle \s\in \S \atop \scriptstyle T^m\s=\s } z^{s_m(\s)}\exp \sum_{j=0}^{m-1} W(T^j\s) \cr} $$ Putting together these observations we have the following \vskip 0.2cm {\bf Proposition 7.1.} {\it Consider the zeta function of two variables given by $$ Z (f,w,z) = \exp \sum_{n=1}^{\infty} {w^n\over n} \sum_{\scriptstyle \s\in \S \atop \scriptstyle T^n\s=\s } z^{s_n(\s)}\exp \sum_{j=0}^{n-1}W(T^j\s) $$ where $s_n(\s)=\sum_{j=0}^{n-1}\s_j$. Then $$ Z (f,1,z) = \z (f,z)(1-\a z) \quad\hbox{and}\quad Z (f,w,1) = \z (g,w) \eqno(7.8) $$ wherever the series expansions converge absolutely.} \vskip 0.2cm We shall now study the relationships between the zeta functions we have introduced above, in particular the two-variables zeta function defined in Proposition 7.1, and some transfer operators acting on the space $\Ft(\S)$. For $W\in \Ft$ and $z \in \C$, let ${\L}_{W,z} : \Ft (\S) \to \Ft (\S)$ be the operator-valued power series defined by $$ {\L}_W(z) = \sum_{k=1}^{\infty} z^k\, {\L}_{W,k}\eqno(7.9) $$ where $$ ({\L}_{W,k}\, v)(\s) = e^{W(k\s)}\, v(k\s)\eqno(7.10) $$ and $k\s$ denotes the sequence $(k\s_0\s_1\dots )$. Notice that for $z=1$ one recovers the (symbolic version of the) Ruelle transfer operator ${\L}_W$ corresponding to the first passage map $g$. \vskip 0.2cm {\bf Lemma 7.2.} {\it The radius of convergence of ${\L}_W(z)$ is bounded below by $1/\a$.} \vskip 0.2cm {\it Proof.} According to (7.9), the radius of convergence of ${\L}_W(z)$ is given by $\lim_{k\to \infty}\Vert {\L}_{W,k} \Vert_{\t}^{-1/k}$. We have $$ |{\L}_{W,k} v(\s)| \leq e^{W(k\s)} |v|_{\infty} $$ and also $$ |{\L}_{W,k} v(\s) - {\L}_{W,k} v(\sp) | \leq \, e^{W(k\s)}\, \left( \, \t \, |v|_{\t} + \CT \, \t \, |v|_{\infty}\, \right). $$ Hence, $$ \Vert {\L}_{W,k} \Vert_{\t} \leq \CT \, \sup_{\s}e^{W(k\s)} $$ On the other hand, from Lemmata 2.1 and 7.1 it follows that for $\a <1$ one has $\sup_{\s}e^{W(k\s)} \leq \CT \, \a^k$; whereas, for $\a =1$, one finds $\sup_{\s}e^{W(k\s)} \leq \CT \, k^{-(1+1/s)}$. $\qed$ \vskip 0.2cm The (simbolic version of the) transfer operator associated to the map $f$ is given by $${\L}_V u(\o ) = ({\L}_0 + {\L}_1 ) u(\o )$$ with $$\cases{ {\L}_0 u(\o )=e^{V_0((\o_0 +1) \o )} u((\o_0 +1)\o )\cr {\L}_1 u(\o )=e^{V_1(1 \,\o )} u(1\, \o )\cr } $$ We now compare the action of ${\L}_V$ and ${\L}_W(z)$ by means of the bijection $\eta = \pi^{-1}\circ \xi : \O \to \S$. Let $\I_{\eta}:C(\S)\to C(\O)$ be defined by $\I_{\eta}\psi = \psi \circ \eta$. If $\psi \in \Ft (\S)$ and $|z|\leq 1/\a$, it is then easy to check that ${\L}_W(z) \psi \in \Ft (\S)$ and also $\I_{\eta}^{-1}{\L}_V \I_{\eta} \psi \in \Ft (\S)$. Set moreover $$ \Bt = \{\, \phi \in C(\O)\,:\, \I_{\eta}^{-1}\phi \in \Ft(\S)\,\} $$ and $$ {\M}_W(z) = \I_{\eta}\, {\L}_W(z)\, \I_{\eta}^{-1}.\eqno(7.11) $$ Now observe that if $\o = \eta^{-1} (\s)$ then we can write $$ W(\s)=\sum_{j=0}^{\s_0-2}V_0(T^j\o) + V_1(T^{\s_0-1}\o). $$ This yields the representation $$ {\M}_W(z) = \sum_{k=1}^{\infty} z^{k}\, {\L}_1 {\L}_0^{k-1} =z {\L}_1 (1-z {\L}_0)^{-1} \eqno(7.12) $$ from which one deduces an algebraic relation between the operators ${\M}_W(z)$ and ${\L}_V$, which can be viewed as the counterpart of Proposition 7.1 (see [Pr] for related results): \vskip 0.2cm {\bf Proposition 7.2.} {\it For any $0< |z| \leq 1/\a$ we have} $$ (\, 1-{\M}_W(z)\, )\, (\, 1-z{\L}_0 \, ) = 1-z\, {\L}_V. $$ \vskip 0.2cm {\bf Remark.} In [CI1] we have constructed the absolutely continuous invariant probability measure $\rho (dx) = h(x)\, m(dx)$ for the dynamical system $(\ui , g)$ as a Gibbs state on $\S$ for the function $W(\s)$, whose density $h(\s)$ is in $\Ft (\S)$ and satisfies $d^{-1} < h < d$ for some $d>0$. Taking $z=1$ in Proposition 7.2 one deduces that if ${\M}_W h = h$ and ${\L}_V e = e$ then $h$ and $e$ are related by $h=(1-{\L}_0)\, e$ or else $e = \sum_{k=0}^{\infty} {\L}_0^kh$ (with a slight abuse of notation we have denoted by the same symbols $h$ and $e$ the function $h\circ \xi^{-1}$ and $e\circ \xi^{-1}$). This correspondence has been used in [CI2] and [CI3] to study some ergodic properties of the $\s$-finite absolutely continuous measure $\nu (dx) = e(x)\, m(dx)$ invariant for the dynamical system $(\ui , f)$ with $\a =1$ and $s=1$. We can actually say more. \vskip 0.2cm {\bf Proposition 7.3.} {\it Let $0<|z|\leq 1/\a$. Then $1\in {\rm sp}\, ({\M}_W(z))$ if and only if $1/z \in {\rm sp}\, ({\L}_V)$, and they have the same geometric multiplicity. Furthermore, the corresponding eigenfunctions $e_z$ of ${\L}_V$ and $h_z$ of ${\M}_W(z)$ are related by $h_z = (1-z{\L}_0) e_z$ or else $e_z = \sum_{k=0}^{\infty} z^k{\L}_0^k h_z$.} \vskip 0.2cm {\it Proof.} Assume that ${\M}_W(z)h_z = h_z$. Then, from Proposition 7.2 it follows that $(1-z{\L}_V)\sum_{k=0}^{\infty} z^k{\L}_0^k h_z = 0$. Conversely, assume that $z{\L}_V e_z = e_z$, then $(1-{\M}_W(z))(1-z{\L}_0)e_z=0$. $\qed$ \vskip 0.2cm Now set $$ R_n(W,z) = \sup_{\s \in \S} \sum_{k_1=1}^{\infty}\dots \sum_{k_n = 1}^{\infty} z^{k_1+\dots + k_n} \exp \sum_{i=1}^nW(k_i\dots k_n\s) $$ and $$ P(W,z) = \lim_{n\to \infty}{1\over n}\log R_n(W,z).\eqno(7.13) $$ For any fixed $0< |z| \leq 1/\a$ the quantity $P(W,z)$ is the {\sl pressure} associated to the interaction $W(\s)+\s_0\log z$. In particular $P(W,1)=0$. \vskip 0.2cm {\bf Lemma 7.3.} {\it The following formal identity holds:} $$ 1-\exp P(W,z) = (1-z)\sum_{n=0}^{\infty}d_nz^n $$ \vskip 0.2cm {\it Proof.} We first notice that the derivative $$ DP(W,1) = {d\over dz} P(W,z)\biggr|_{z=1} $$ gives the mean return time in the state $1$ with respect to the Gibbs measure on $\S$ for the function $W$ ([R5], Chapter 5). In other words we have $$ DP(W,1) =\sum_{n=1}^{\infty}n \rho (A_n) = \taun \eqno(7.14) $$ where (2.5) and the subsequent comment have been used. Therefore, by (2.3), one finds $$ 1-\exp P(W,z) = 1-\sum_{n=1}^{\infty}\rho (A_n)z^n =(1-z)\sum_{n=0}^{\infty}d_nz^n.\qquad \qed $$ \vskip 0.2cm {\bf Remark.} We know from Section 2 that $\taun$ is finite in the two cases: $\a <1$ and $\a =1$, $s<1$. If $\a =1$ and $s\geq 1$ then $\taun = \infty$. According to (7.14), this fact can be interpreted as a phase transition characterized by the coexistence, for $\a=1$ and $s \geq 1$, of two equilibrium states for $(f,\ui)$: the $\s$-finite measure $\nu$ and the Dirac delta measure concentrated at the neutral fixed point (see [Me]). \vskip 0.2cm We now get some more information about the spectrum of ${\L}_W(z)$. Let us recall that the spectrum ${\rm sp}\, (P)$ of a bounded linear operator $P$ can be decomposed into a discrete part, made up of isolated eigenvalues of finite multiplicity, and its complement, the essential spectrum, denoted by $\ess (P)$ (see, e.g., [DS]). The essential spectral radius is then defined as $\ress (P) = \sup \, \{ \, |\lambda| \, : \, \lambda \in \ess (P) \, \}$. Moreover, we shall denote by $r(P)$ the spectral radius of $P$. \vskip 0.5cm {\bf Theorem 7.1.} {\it For any $0< |z| \leq 1/\a$, the spectrum of ${\L}_W(z):\Ft \to \Ft$, as well as that of ${\M}_W(z): \Bt \to \Bt$, consists of two disjoint parts: \item{1)} every point in the disk $\{\, \lambda \, : \, |\lambda | \leq \t \, \exp P(W,|z|) \, \}$; \item{2)} isolated eigenvalues in the annulus $\{\, \lambda \, : \, \t \, \exp P(W,|z|) < |\lambda | \leq \exp P(W,|z|) \, \}$. If $z$ is real and positive the spectral radius coincides with $\exp P(W,z)$ which is also a maximal simple eigenvalue.} \vskip 0.2cm {\bf Remark.} According to Lemma 4.1, the smallest essential spectral radius is attained by taking $\t = \beta$. \vskip 0.2cm {\it Proof.} It is easy to check that for all $v\in \Ft$ $$ |{\L}_W(z)^n v|_{\infty}\leq R_n(|z|)\, |v|_{\infty} \leq R_n(|z|)\, \Vert v \Vert_{\t} $$ and $$ |{\L}_W(z)^n v|_{\t}\leq R_n(|z|)\, (\CT\, |v|_{\infty}+\t^n |v|_{\t}) \leq R_n(|z|)\, (\CT +1)\Vert v \Vert_{\t}. $$ Therefore $\Vert {\L}_W(z)^n \Vert_{\t} \leq (\CT+2)R_n(|z|)$, where we have also denoted by $\Vert \,\,\, \Vert_{\t}$ the operator norm. Thus, the spectral radius formula implies that $$ r({\L}_W(z))\leq \exp P(W,|z|) $$ Moreover, repeating the argument used in ([Po1], p.151; see also [K]), one finds $$ \ress ({\L}_W(z)) \leq \t \, \exp P(W,|z|) $$ and also shows that the disk of radius $\t \, \exp P(W,|z|)$ is included in ${\rm sp}\, ({\L}_W(z))$. Now, if $z$ is real and positive we have $$ r({\L}_W(z)) =\lim_{n\to \infty} \left( \, \Vert {\L}^n_{W,z} \Vert_{\t}\, \right)^{1/n} \geq \lim_{n\to \infty} \left(\, |{\L}^n_{W,z}1 |_{\infty}\, \right)^{1/n} = \exp P(W,z) $$ and therefore $r({\L}_W(z)) = \exp P(W,z)$. It remains to show that $\exp P(W,z)$ is a maximal simple eigenvalue of ${\L}_W(z)$. Let ${\Psi}_z$ be such that ${\L}_W(z){\Psi}_z = \l_z {\Psi}_z$ where $\l_z$ is the (simple) eigenvalue with largest modulus of ${\L}_W(z)$. Reasoning as in ([CI1], Theorem 2.1) it is not difficult to show that ${\Psi}_z \in \Ft (\S)$ and $d_z^{-1} \leq {\Psi}_z \leq d_z$ for some positive constant $d_z$. Then, $$ \eqalign{ \log \l_z &= \limsup_{n\to \infty} {1\over n}\log {\L}_W(z)^n{\Psi}_z \cr &=\limsup_{n\to \infty} {1\over n}\log {\L}_W(z)^n 1 =P(W,z). \qquad\qquad \qed \cr } $$ Now choose $\Theta > \t$ and write $\P_z$ for the projection corresponding to the part of the spectrum of ${\L}_W(z)$ in the disk of radius $\Theta \, \exp P(W,z)$. Let moreover $\l_1(z), \dots ,\l_M(z)$ be the eigenvalues outside this disk, ordered with decreasing modulus, and $m_1(z), \dots ,m_M(z)$ their multiplicites. Then, for any $n>0$, we have the following decomposition: $$ {\L}^n_W(z)v = \sum_{i=1}^M\l_i^n(z) \cdot \Psi_{z,i} L_{z,i}^n \Psi_{z,i}^* v + \P_z {\L}^n_W(z) \, v \eqno(7.15) $$ where the row vector $\Psi_{z,i}$ and the column vector $\Psi_{z,i}^*$ span the generalized eigenspaces of ${\L}_W(z)$ and ${\L}^*_W(z)$ corresponding to the eigenvalue $\l_i(z)$, and the matrices $L_{z,i}$ can be assumed in Jordan normal form, so that $m_i(z)= {\rm tr}\, L_{z,i}$. In the same manner, for the operator ${\M}_W(z):\Bt \to \Bt$ we have $$ {\M}^n_W(z)\, v = \sum_{i=1}^M\l_i^n(z) \cdot h_{z,i} L_{z,i}^n m_{z,i} v + \P_z {\M}^n_W(z) \, v \eqno(7.16) $$ where $h_{z,i}=\I_{\eta}\circ \Psi_{z,i}$ and $m_{z,i}=\Psi_{z,i}^*\circ \I_{\eta}^{-1}$. >From the above spectral properties of ${\L}_W(z)$ we get the following result: \vskip 0.2cm {\bf Theorem 7.2.} {\it The two-variables zeta function defined in Proposition 7.1 has the following analytic properties: \item{(i)} for any $0< |z|\leq 1/r({\L}_0)$, $1/Z(f,w,z)$, considered as a function of the variable $w$, is holomorphic in the disk of radius $1/\t \exp P(W,|z|)$. Its zeroes in this disk, counted with multiplicity, are the inverses of the eigenvalues of ${\L}_W(z):\Ft (\S) \to \Ft(\S)$ in the corresponding annulus. Moreover, the zero of smallest modulus is simple and located at $1/\exp P(W,z)$; \item{(ii)} for any $0<|w| \leq 1$, $1/Z(f,w,z)$, considered as a function of the variable $z$, is holomorphic in the disk of radius $1/|w|r({\L}_0)$. Its zeroes in this disk are located at those values of $z$ such that ${\L}_W(z):\Ft (\S) \to \Ft(\S)$ has $1/w$ as an eigenvalue.} \vskip 0.2cm {\it Proof.} We shall follow Haydn ([H1], Theorem 4; see also [BK] and [R2]). Fixing $n>0$ we let $\sum_{\nu}$ be the sum over words $\nu$ of length $n$, i.e. words of the form $\nu = (\s_0, \dots ,\s_{n-1})$, and denote by $\s^{(\nu)}$ the periodic concatenation $({ \overline {\s_0 \s_1 \dots \s_{n-1} }})$. Let moreover $\chi_{\nu}\in \Ft (\S)$ be such that $\chi_{\nu}(\s) = 1$ if $\s$ begins with the word $\nu$, $\chi_{\nu}(\s) = 0$ otherwise. Then we have the following key relation: $$ {\Lambda}_n(f,z):=\sum_{\scriptstyle \s\in \S \atop \scriptstyle T^n\s=\s } z^{s_n(\s)}\exp \sum_{j=0}^{n-1}W(T^j\s) = \sum_{\nu} ({\L}^n_W(z) \chi_{\nu})(\s^{(\nu)})\eqno(7.17) $$ Inserting (7.15) into (7.17) we get $$\eqalign{ {\Lambda}_n(f,z) &= \sum_{\nu} \sum_{i=1}^M\l_i^n(z) \cdot \Psi_{z,i}(\s^{(\nu)}) L_{z,i}^n \Psi_{z,i}^* \chi_{\nu} + \sum_{\nu}(\P_z {\L}^n_W(z) \chi_{\nu})(\s^{(\nu)}) \cr &= \sum_{i=1}^M \l_i^n(z) \cdot (L_{z,i}^n\Psi_{z,i}^*)^{\rm tr} (\sum_{\nu} \Psi_{z,i}(\s^{(\nu)})\cdot \chi_{\nu})^{\rm tr} +\sum_{\nu}(\P_z {\L}^n_W(z) \chi_{\nu})(\s^{(\nu)}) \cr &={\Lambda}_n^{(0)}(f,z) + {\Lambda}_n^{(1)}(f,z) + {\Lambda}_n^{(2)}(f,z) \cr } $$ where $$\eqalign{ &{\Lambda}_n^{(0)}(f,z) = \sum_{i=1}^M m_i(z)\, \l_i^n(z) \cr &{\Lambda}_n^{(1)}(f,z) = \sum_{i=1}^M \l_i^n(z) \cdot (L_{z,i}^n\Psi_{z,i}^*)^{\rm tr} (\sum_{\nu} \Psi_{z,i}(\s^{(\nu)})\cdot \chi_{\nu} - \Psi_{z,i} )^{\rm tr} \cr &{\Lambda}_n^{(2)}(f,z) = \sum_{\nu} (\P_z{\L}^n_W(z) \chi_{\nu})(\s^{(\nu)}) \cr } $$ and $\Psi^{\rm tr}$ denotes transposition. The first term gives the contribution: $$ \exp -\sum_{n=1}^{\infty} {w^n \over n} {\Lambda}_n^{(0)}(f,z) =\prod_{i=1}^M(1-w\l_i(z))^{m_i}. $$ Now, according to Theorem 7.1, if $z$ is real and positive $\l_1(z)$ is simple and equals $\exp P(W,z)$. On the other hand, for any $w\in \C$, the function $$ Q(W,z,w) : = 1- w\exp P(W,z)\eqno(7.18) $$ extends uniquely to a function holomorphic in the disk $\{z: |z|\leq 1/\alpha \}$. Therefore, in the domain $\{z: |z|\leq 1/\alpha \} \times \{w: |w|\leq 1 \}$ we have $$ \exp -\sum_{n=1}^{\infty} {w^n \over n} {\Lambda}_n^{(0)}(f,z) =Q(W,z,w)\cdot \prod_{i=2}^M(1-w\l_i(z))^{m_i}.\eqno(7.19) $$ Finally, proceeding as in [H1], one can show the existence of two constants $R_1,R_2 >0$ such that $$ |{\Lambda}_n^{(1)}(f,z)| \leq R_1 \Theta^n e^{nP(W,|z|)} \quad\hbox{and}\quad |{\Lambda}_n^{(2)}(f,z)| \leq R_2 \Theta^n e^{nP(W,|z| )} $$ and the Theorem is proved. $\qed$ \vskip 0.2cm Putting together Proposition 7.1, Proposition 7.3 and Theorem 7.2 (with the subsequent Remark) we obtain the following, \vskip 0.2cm {\bf Corollary 7.1.} {\it \item{(i)} $1/\zeta (g,z)$ is holomorphic in the disk of radius $1/\beta$. Its zeroes in this disk, counted with multiplicity, are the inverses of the eigenvalues of ${\L}_{W}$ in the annulus $\{\, \lambda \, : \, \beta < |\lambda | \leq 1\, \}$. \item{(ii)} $1/\zeta(f,z)$ is holomorphic in the disk of radius $1/\a$. In this disk, $1/\zeta(f,z)=0$ if and only if $1/z \in {\rm sp}({\L}_{V})$.} \vskip 0.2cm {\bf Remark.} When $\a =1$ the above result yields no zeroes of $1/\zeta (f,z)$ but the point $z=1$. In this case one expects that the eigenvalue $1$ of ${\L}_{V}$ is not isolated (i.e. there is no `gap'). Nevertheless, one may consider a generalised transfer operator corresponding to the interaction $\beta\, V$, for some $\beta \in \C$, and then study the spectral gap and the analytic properties of the pressure $P(\beta \, V)$ as functions of $\beta$ (see [Pr],[L]). \vskip 0.5cm {\bf 8. Proofs of the main results.} \vskip 0.2cm {\it Proof of Theorem 3.1.} Set $$ Q(W,z) \equiv Q(W,z,1) = 1- \exp P(W,z) $$ According to Lemma 7.3 we have $$ Q(W,z) = (1-z)\sum_{n=0}^{\infty}d_nz^n $$ and the assertion follows putting together (7.19), with $w=1$, and Proposition 7.1. $\qed$ \vskip 0.2cm {\it Proof of Theorem 3.2.} We first notice that having fixed $0<\gamma \leq 1$ one can find $0<\t <1$ such that $$ u\circ \pi^{-1} \in \Ft\quad\hbox{and}\quad u\circ \xi^{-1} \in \Bt \quad\hbox{if}\quad u \in \F $$ and the linear maps $\F \to \Ft$ and $\F \to \Bt$ thus defined are continuous. This enables us to study the problem in the spaces $\Ft$ or $\Bt$. The property of having compact support on $]0,1]$ of a given $u\in \F(\ui)$ translates, for instance, into that of having compact support on $\O\setminus \O_{\infty}$, where $\O_{\infty}:=\cup_{K>0} \{\, \o \in \O\,:\, \o_i > K, \, \forall i\geq 0\, \}$, for $u\circ \xi^{-1} \in \Bt$. We now use the results of Section 7. In particular, we shall consider the decomposition (7.16) where, for the simplicity of formulae, we assume that the eigenvalues $\l_i$ of ${\M}_W(z):\Bt \to \Bt$ with modulus $>\Theta \, \exp P(W,z)$ are simple. For $z$ real and positive, let $h_{z}\equiv h_{z,1}$ and $m_{z}\equiv m_{z,1}$ be normalized so that $m_z(h_z)=1$ and set $\rho_z=h_z\, m_z$. Moreover, for $z=1$, set $m\equiv m_{1}$ and $h\equiv h_{1}$. Then $\rho =h\, m$ is the symbolic version of the unique absolutely continuous probability measure invariant for the dynamical system $(\ui ,g)$ (denoted with the same symbol with slight abuse of language). With the same abuse of language we denote by the symbol $e$ the eigenvector of ${\L}_V$ to the eigenvalue $1$ and $\nu = e\, m$. According to Proposition 7.3, we also define $e_{z,i}= (1-z{\L}_0)^{-1}\, h_{z,i}$, for $i>1$. Let now $u,v \in \F (\ui)$. For simplicity of notation (and without fear of confusion) we shall also denote with the symbols $u,v$ the functions $u\circ \xi^{-1},v\circ \xi^{-1}\in \Bt$. >From Lemma 2.6 and the above observations it follows that if $u \in \Bt$ then $u\cdot e \in \Bt$ as well. Then, using Proposition 7.2, Proposition 7.3 and the decomposition (7.16), we have the following calculation: $$\eqalign{ s_{uv}(z) &=\sum_{n=0}^{\infty} z^n \, \nu (u\cdot v\circ T^n) \cr &=\sum_{n=0}^{\infty} z^n \, m (v \cdot {\L}_V^n (u\cdot e)\, ) \cr &= m (v\cdot (1-z{\L}_V)^{-1} (u\cdot e)\, ) \cr &= m (v \cdot (1-z{\L}_0)^{-1}(1-{\M}_W(z))^{-1}(u\cdot e)\, ) \cr &= {m_z(u\cdot e)\, m(e_z\cdot v)\over 1-\lambda_1(z) } +\sum_{i=2}^M{m_{z,i}(u\cdot e)\, m(e_{z,i}\cdot v)\over 1-\lambda_i(z)}\cr &\qquad \qquad + m(v \, (1-z{\L}_0)^{-1}(1-{\M}_W(z))^{-1}\P_z(u\cdot e)\, ) \cr } $$ For $0n}d_l$. 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