p$. \cr} $$ Then $\a=p$ and $c_n=p^n$ so that $$ \zeta (f,z) = {1\over 1-z} $$ which is analytic in the entire $z$-plane except for a simple pole at $z=1$. \vskip 0.2cm {\bf Example 5.2.} Set $$ f(x)= \cases{x/(1-x), &if $x\leq 1/2$; \cr 2x-1, &if $x>1/2$. \cr} $$ Then $\a=1$ and $c_n = 1/(n+1)$ so that $$ \zeta (\hf,z) = {z \over (1-z)^2 \log{1/(1-z)} } $$ which is analytic in the entire $z$-plane with a cross cut along the ray $(1, +\infty)$. Furthermore, in a neighborhood of the point $z= 1$ we have $(1-z)\zeta (\hf,z) \to \infty$ but $(1-z)^{2}\zeta (\hf,z) \to 0$. \vskip 0.2cm More generally, according to Lemma 2.1 we have the following behaviour of $\zeta(\hf ,z)$: \vskip 0.2cm {\bf Proposition 5.2.} {\it Under the hypotheses (1)-(6) on the map $f$ we have: \item{a)} if $\alpha <1$ then $1/\zeta(\hf ,z)$ is holomorphic in the disk $|z| <1/\a$ with only one zero; this zero is simple and located at $z=1$; \item{b)} if $\alpha =1$ then $1/\zeta(\hf ,z)$ is holomorphic in the unit disk and can be continued analytically in the entire $z$-plane with a cross cut along the ray $(1,+\infty )$. The analytic continuation is given by the formula, valid for any $\delta \geq 0$, $$ 1/\zeta(\hf ,z) ={(1-z)^2 \over 2\pi i}\int_1^{ +\infty} \int_{\delta -i\infty}^{\delta +i\infty} c(x){t^{-x}\over t-z}dx dt \eqno(5.1) $$ where $c(x)$ is a regular function in the half-plane ${\rm Re}\, x \geq 0$ such that $c(n)=c_{n}$ for $n\geq 0$. } \vskip 0.2cm {\it Proof.} The first part is immediate. The proof of (b) follows by putting together Lemma 2.1 and standard techniques of analytic continuation of power series based on the use of the Mellin transform (see, e.g., [E], Theorem 6.1). q.e.d. \vskip 0.5cm {\bf Relating $\zeta(\hf,z)$ to $\zeta(f,z)$.} \vskip 0.2cm {\bf Theorem 5.1.} {\it Under the hypotheses (1)-(6) on the map $f$ we have: \item{a)} if $\alpha <1$ then $1/\zeta(f ,z)$ is holomorphic in the disk $|z| <1/\a$ with only one zero; this zero is simple and located at $z=1$; \item{b)} if $\alpha =1$ then $1/\zeta(f ,z)$ is holomorphic in the unit disk and can be continued analytically in the entire $z$-plane with a cross cut along the ray $(1,+\infty )$. In addition, we have the asymptotic behaviour} $$ 1/\zeta(f,z) = \CO (1) \, (1-z)^2\, \sum_{n=0}^{\infty}c_nz^n \quad \hbox{as}\quad z\to 1_- . $$ \vskip 0.2cm {\bf Remark 5.2.} If $\a =1$ and $s<1$, so that $\sum c_k < \infty$, then $\z (f,z)$ diverges as $(1-z)^{-2}$ when $z\to 1_-$. Instead, if $\a=1$ and $s\geq 1$, so that $\sum c_k = \infty$, then $(1-z)\zeta (\hf,z) \to \infty$ but $(1-z)^{2}\zeta (\hf,z) \to 0$ when $z\to 1_-$ (cf Remark 4.3). However, the asymptotic behaviour of $\z (\hf, z)$ in a neighborhood of $z=1$ and in particular the order of the branch point, may be inspected with the help of formula (5.1). We conclude by observing that from this behaviour one can obtain information on the behaviour of the partial sums $\sum_{n=1}^Nc_n$, which may be useful for studying the mixing properties of the map $f$. This can be achieved by using Karamata's theorem in a suitable way [T]. However, an upper bound can be simply obtained as follows. Set $$ L(z)=\sum_{n=0}^{\infty}c_n\, z^n = {1\over 2\pi i}\int_1^{ +\infty} \int_{\delta -i\infty}^{\delta +i\infty} c(x){t^{-x}\over t-z}dx dt. $$ Then, for any $N>1$, $$\eqalign{ \sum_{n=1}^Nc_n &\leq \left(1-{1\over N}\right)^{-N}\sum_{n=1}^N c_n\left(1-{1\over N}\right)^n \cr &\leq \CO (1) \sum_{n=0}^{\infty} c_n\left(1-{1\over N}\right)^n \cr &= \CO (1) L(1-{1\over N}).\cr } $$ For instance, for the map in Example 5.2 (for which $\a=1$ and $s=1$) one finds $L(z) = {1\over z} \log ({1\over 1-z})$ so that $\sum_{n=1}^Nc_n \leq \CO(1) \log N$. \vskip 0.2cm {\it Proof of Theorem 5.1.} The first part is the analogous of Theorem 5.29 in [R3]. We then prove (b). Set $$\eqalign{ Q(W,z) &:= 1- \exp P(W,z) \cr Q(\hW,z) &:= 1- \exp P(\hW,z) = (1-z)\sum_{k=0}^{\infty}c_kz^k \cr } $$ It is easy to check that these functions satisfy $$ Q(W,1)=Q(\hW,1) =0\quad\hbox{and}\quad Q(W,0)=Q(\hW,0) =1. $$ We then have $$ P^{\prime}(W,1)=\lim_{z\to 1_-} {Q(W,z)\over 1-z}, \qquad P^{\prime}(\hW,1)=\lim_{z\to 1_-} {Q(\hW,z)\over z-1}. $$ Now, as we have already stressed, $P^{\prime}(W,1)$ and $P^{\prime}(\hW,1)$ give the mean return time in the state $1$ with respect to the Gibbs measures on $\S$ for the functions $W$ and $\hW$, respectively. They are given formally by (cf (4.14)): $$ P^{\prime}(W,1) = \sum_{k=1}^{\infty}k \rho (A_k)\quad\hbox{and}\quad P^{\prime}(\hW,1)= \sum_{k=1}^{\infty}k \b_k $$ On the other hand, under the assumptions (1)-(6) for the map $f$ we have ([CI2], Lemma 2.4): $$ \sum_{k=1}^{\infty}k \rho (A_k) = \CO (1)\sum_{k=1}^{\infty}k \b_k $$ Therefore, the following limit exists: $$ \lim_{z\to 1_-}{Q(W,z)\over Q(\hW,z)}= {P^{\prime}(W,1) \over P^{\prime}(\hW,1) }, $$ and moreover $$ Q(W,z)= \CO(1)\, Q(\hW,z)\quad\hbox{as}\quad z\to 1_-. $$ Putting together Proposition 4.1, equation (4.16) with $w=1$ and the above we get the announced asymptotic behaviour of $\zeta(f,z)$. 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