q>1$, in fact $q' > \eta q$ for each $1\le \eta < 1+q/(n-q)$. \demo{Proof of Lemma 3} The second claim of \therosteritem{1} follows from usual properties of the convolution (see e.g. [Sch, p.151]) since $\chi \sigma_k \in L^t$ for all $1\le t < n/(n-1)$. Noting that the kernel $\sigma_k$ is of weak-type ${n\over n-1}$, we may apply the Hardy-Littlewood-Sobolev fractional integration theorem (see e.g. [St, V.Theorem~ 1 and Comment~1.4]; the comment explains why Theorem~ 1 extends from the Riesz potential to more general weak-type kernels), which yields that $\SS_k$ is a bounded operator from $\AA_{k+1,L^q}$ to $\AA_{k,L^{q'}}$ if $1 < q < n$. It remains to prove \therosteritem{2}, i.e., study $\partial_i\SS_k$. We first claim that $\partial_i \SS_k$ can be written as the sum of, on the one hand a convolution operator with kernel a form having coefficients which are linear combinations of expressions of the type $$ {x_i x_j \over \|x\|^{n+2}} \, , {\|x\|^2-n x_i^2\over \|x\|^{n+2}} \, , 1\le i,j\le n \, , \tag{2.14} $$ and on the other hand a distribution $\nu_k$ which extends to a bounded operator on $\AA_{k+1,L^q}$. Indeed, computing the functional partial derivatives $\partial_i={\partial\over \partial x_i}$ of \thetag{2.9} produces \thetag{2.14}, so that we only need to study the distributional contribution to $\partial_i \SS_k$. For this, let us take a kernel singularity $x_j/\|x\|^{n}$, and for each $t>0$ let $\chi_t:\real\to \real$ be a $C^\infty$ function, identically zero on $[-t,t]$ and which is identically equal to $1$ outside of $[-2t,2t]$. We may assume that $\sup |\chi'|\le 2/t$. Let us consider $$ \partial_i \left ( \chi_t(\|x\|) x_j \over \|x\|^{n} \right ) = {x_j \partial_i (\chi_t(\|x\|)) \over \|x\|^{n}} + \chi_t(\|x\|) \partial_i \left ({x_j\over \|x\|^{n}} \right ) \, . $$ Since the second term of the above sum converges to the already mentioned functional partial derivative as $t\to 0$, we should check that the distribution with kernel corresponding to the first term, which is just $\chi'_t(\|x\|) x_i x_j/\|x\|^{n+1}$, acts boundedly on $\AA_{k+1,L^q}$, uniformly in $t\ge 0$. By density and completeness, it is enough to consider $\phi \in \AA_{k+1,L^q}\cap C^\infty$ and take $\varphi=\phi_\ell$ for $\ell\in I(k+1)$. We may formally write for every $w\in \real^n$ $$ \eqalign { |\nu_k(\varphi)(w)| &= \left | \int_{\real^n} \chi'_t(\|x\|) {x_i x_j \over \|x\|^{n+1}}\varphi(w-x) \, dx \right |\le {2\over t} \int_{\|x\|\le t} \left | {x_i x_j\over \|x\|^{n+1}}\varphi(w-x)\, dx \right |\cr &\le {2\over t} \sup_{\|x\| \le t} |\varphi(w+x)| \int_{\|x\|\le t} \biggl | {x_i x_j\over \|x\|^{n+1}}\biggr | \, dx \, . }\tag{2.15} $$ Going to polar coordinates, it is not difficult to see that there is a constant $C>0$ so that $\int_{\| x\|\le t} |x_i x_j| \|x\|^{-n-1} \, dx \le C t$. We end our analysis of the distributional contribution by noting that $ \lim_{t\to 0} \int \sup_{\|x\|\le t} |\varphi(w+x)|^q \, dw =\int |\varphi(w)|^q \, dw $. (Since our assumptions imply that $|\varphi(w+x)-\varphi(w)| \le t \sup \|D \varphi\|$, the above is easily checked.) As pointed out to us by S.~Gou\"ezel, a more careful analysis shows that in fact $\nu_k$ is zero if $i\ne j$ and is a scalar multiple of the Dirac mass if $i=j$. We next observe that the expressions \thetag{2.14} in the functional term of the kernel of $\partial_j \SS_k$ exhibit the same kind of singularity as the Riesz transform. More precisely, this kernel is a linear combination of $\Omega_\ell(x)\over \|x\|^n$ where each $\Omega_\ell$ is homogeneous of degree zero, has vanishing integral on the unit $(n-1)$-dimensional sphere and is $C^1$ on this sphere. We may thus apply e.g. Theorem ~3 in Chapter II of [St], which immediately guarantees that $d_k \SS_k$ extends to a bounded operator on $\AA_{k+1,L^q}$, for $1< q < \infty$. \qed\enddemo \proclaim{Lemma 4 (Properties of the $\SS_k$: algebra)} Let $1< q < \infty$ . \noindent (1) $\SS_{k-2} \SS_{k-1}=0$ on $\AA_{k,L^q}$. \noindent (2) On $\AA_{k, L^q}$, and suppressing the indices for simplicity, $d \SS=(d \SS)^2$, and $( \SS d)^2= \SS d$, $d \SS \SS d=0= \SS d d\SS$. In other words, $d\SS$ and $\SS d$ are two orthogonal bounded projectors onto $\Imm d=\Ker d$, respectively $\Imm \SS=\Ker \SS$, in $\AA_{k, L^q}=\Imm d \oplus \Ker \SS =\Imm \SS \oplus \Ker d$. \endproclaim \demo{Proof of Lemma 4} The facts that $\SS d$ is onto $\Ker \SS$ and that $\Ker \SS \subset \Imm \SS$ do not depend on $ \SS^2=0$: Use $\phi=\SS d\phi$ if $\phi \in \Ker \SS$. To prove that $\SS^2=0$, we shall give an equivalent definition of $\SS$, which was indicated to us by D.~ Ruelle. Using Lemmas ~0 and ~3, it suffices to show the result on $C^\infty$ compactly supported forms (by density). Recall that there is a scalar (or hermitian) product on the space of compactly supported $C^\infty$ $k$-forms $\AA^C_{k,C^\infty}$, defined by: $$ < \phi(x)dx_{i_1}\wedge\cdots\wedge dx_{i_k} \, |\, \varphi(x)dx_{j_1}\wedge\cdots\wedge dx_{j_k}> = \left\{ \eqalign{ &\int \phi\cdot\varphi \, \text{ if $i_1,\dots,i_k=j_1,\dots,j_k$} \cr &0 \, \text{ otherwise.} } \right. \tag{2.16} $$ (The scalar product is still defined if only one of $\phi,\varphi$ is compactly supported.) Now, $d:A^C_{k,C^\infty}\to \AA_{k+1,C^\infty}$ defines a dual $d^*:\AA^C_{k+1,C^\infty}\to \AA_{k,C^\infty}$ by $=<\phi,d\varphi>\, . $ The Laplace-Beltrami operator is defined by $\Delta = dd^*+d^*d$ where $d$ is the exterior derivative of forms. In $\real^n$, we have (see, e.g., [Li]): $$ \Delta \left[ \sum_{j\in I(k)}\phi_j(x) dx_j \right] = -\sum_{j\in I(k)} (\sum_{i=1}^n \frac{\partial^2}{\partial x_i^2} \phi_j)(x) dx_j \, .\tag{2.17} $$ Define the Green kernel to be $E(x)=-\frac{\Gamma(n/2)}{(n-2)2(\pi)^{n/2}}\frac{1}{||x||^{n-2}}dx_1 \wedge\cdots\wedge dx_n$ if $n\ge 3$, and $E(x)=\frac{1}{2\pi}\log(||x||)dx_1\wedge dx_2$ if $n=2$. Then, let $E_k(x,y)$ be the $k$-form in $x$ and $n-k$-form in $y$ such that $E(x-y)=\sum_{k=0}^n (-1)^{n}E_k(x,y)$. It is well known (see [Sch]) and has been used in [Bai] that $\Delta E = \delta(x) dx_1\wedge\cdots\wedge dx_n$ (as a current acting on $C^\infty$ forms vanishing at infinity, or compactly supported $C^\infty$forms). One thus sees easily that the operator $G_k:\phi(x)\mapsto\int_y E_k(x,y)\wedge\phi(y)$ satisfies $\Delta G_k=id$ on $\AA^C_{k,C^\infty}$. Using that $\frac{\partial^2}{\partial y_i^2}E(x-y)=\frac{\partial^2} {\partial x_i^2}E(x-y)$, one checks that $G_k\Delta=id$ on the same space. The operator $\SS_k$ defined in [Bai] is in fact $d^*G_{k+1}$. Indeed, $$ d^*\left(\phi(x)dx_1\wedge\cdots\wedge dx_n\right)= \sum_{j=1}^k(-1)^{j+1}\partial_j\phi(x)dx_1\wedge\cdots{d\hat{x}_j} \cdots\wedge dx_n \, . \tag{2.18} $$ Thus, $d^*E(x)=\sigma(x)$, where $\sigma$ was defined in \thetag{2.6}. Since $\SS_k$ is defined in \thetag{2.7} as the convolution with $\sigma_k$, the equality is immediate. By Lemma 3, the composition $\SS_{k-1}\SS_{k}=d^*G_kd^*G_{k+1}$ is well defined on $L^q$ forms, and thus on $\AA^C_{k,C^\infty}$. We shall show that $d^*G_k=G_{k-1}d^*$, which implies $\SS_{k-1}\circ\SS_{k}=0$ since $d^*d^*=0$. Since $E(x)$ is an $n$-form, $d E(x)=0$. Using $d=d^x+d^y$, we obtain $d^xE_{k-1}(x,y)=-d^yE_k(x,y)$. Integrating by parts, $dG_{k-1}=G_{k}d$ (recall that $E_k(x,y)$ is an $n$-form). Notice now that $\Delta$ is auto-dual by definition, and thus $G_k$ is also auto-dual. Hence, $d^*G_k=G_{k-1}d^*$. \qed\enddemo \smallskip %%%%%%%%%%%%%%%%%%%%%%%%% \head 3. Spectral interpretation of the zeroes of the sharp determinant \endhead {\bf The Axioms} \smallskip Let $r$, $\Omega$, $\psi_\omega$, $g_\omega$, $K=\{ x\in \real^n \mid |x|\le T\}$, $\MM_k$, $\SS_k$, $\sigma_k$, $\NN_k$, and $\DD_k(z)$ be the objects from Section~ 2. Let $1_K$ be the characteristic function of $K$. We fix once and for all $$ K' =\{ x\in \real^n \mid |x|< 2 T\}\, , $$ and a radial $C^\infty$ function $\chi_K=\chi_{K,K'}$ supported in $K'$ and identically equal to $1$ on $K$. Note that $\NN_k$ and $\MM_k$ are bounded from $k$-forms to $k+1$-forms, respectively $k$-forms, with coefficients in $L^q(K')$. We use the notation $\rho(\PP)$, $\rho_{ess} (\PP)$ for the spectral and essential spectral radii of a bounded linear operator $\PP$. We are now ready to state our three assumptions: \smallskip {\bf Axiom 1:} % s< n could be reintroduced For $k=0, \ldots, n$ and $1 < t < \infty$, there are Banach spaces $\BB_{k,t}$ of distributions on $\real^n$ so that: \roster \item There are real self-adjoint invertible pseudodifferential convolution operators $\widetilde \JJ_{k}$, $\widetilde \JJ_{k}^{-1}$ of order (at most) $r$ so that $ \BB_{k, t}$ is defined by the Banach space isomorphism $$ \widetilde \JJ_{k}(L^t(\real^n)) = \BB_{k, t} \, .\tag{3.1} $$ \item If $\psi$ is a $C^r$ local diffeomorphism and and $g$ is a $C^{r}$ function supported in $K$ then $\varphi\mapsto g \cdot \psi ^* (\varphi)$ is a bounded operator on each $\BB_{k,t}$. \item Each operator $d_k :\AA_{k, \BB_{k,t}} \to \AA_{k+1, \BB_{k+1,t}}$ is bounded. \endroster {\bf Axiom 2:} There is $0\le \widetilde R < 1$ so that for $k=0, \ldots, n$: \roster \item $\MM_k$ is bounded on $\AA_{k,\BB_{k,t}}$ with $\rho (\MM_k|_{\AA_{\BB_{k,t}}}) \le 1$, $\rho_{ess} (\MM_k|_{\AA_{\BB_{k,t}}}) < \widetilde R$, and there are no eigenvalues of modulus $\widetilde R$, for each $1< t< \infty$. \item Let $\Pi_{k,t}$ be the (finite-rank) spectral projector on $\AA_{k,\BB_{k,t}}$ associated to the spectrum of $\MM_k$ outside the disc of radius $\widetilde R$. Let $\{\varphi_{k,t,s}\}$ be a basis for the corresponding generalised eigenspace. (Our assumptions imply that the dimension does not depend on $1 0$ so that $\max_{k,s} \sup_{1 < t < \infty} \| \varphi_{k,t,s} \|_{\BB_{k,t}}\le C$, and, for all $k$, $j$ , we have $$ \sup_{1 < t < \infty} \| \MM_k^j \|_{\BB_{k,t}}\le C\, , \quad \sup_{1< t < \infty} \|\MM_k ^j (\Id-\Pi_{k,t})\|_{\BB_{k,t}} \le C \widetilde R^j \, . $$ \endroster {\bf Axiom 3:} There is $\EE_0 > 0$ so that, for all $0<\EE < \EE_0$, each $j\ge 1$ and each admissible composition $\psi^{j}_{\vec \omega}$, letting $V_{\vec \omega,\EE}$ be the set of $x$ in the domain of definition of $\psi^{j}_{\vec \omega}$ so that $\|\psi^{j+\ell}_{\vec \omega}(x)-x\| < \EE$, the map $\psi_{\vec \omega}^{j+\ell}- \Id$ is injective on $V_{\vec \omega,\EE}$. Also, for each $\eta >0$ there is $C$ so that for all $k$, every sequence $\{\EE_j\}$ with $0<\EE_j <\EE_0$, and each $k$-form $\bold 1_k= dx_{i_1} \wedge \cdots \wedge dx_{i_k}$ $$ \sum_{\vec \omega \in \Omega^j} \sup_{x \in V_{\vec \omega,\EE_j}} {|g_{\vec \omega}^{(j)} (\psi^{j}_{\vec \omega} x)| \over |\det (D\psi_{\vec \omega}^{j}(x) -\Id ) | } \bigl | (\psi_{\vec \omega}^{j})^* (\bold 1_k) (x) / \bold 1_k(x) \bigr | \le C \exp(\eta j) \, , \, \forall j\, . $$ %could exploit alternating sum in Axiom 3 \smallskip \remark{Consequences of Axiom 1} $C^{\infty}$ functions with compact support are contained in $\BB_{k,t}$ and distributions in $\BB_{k,t}$ have order at most $r$. Also, $\BB_{k,t}\subset \BB_{k,t'}$ boundedly if $t >t'$. Since $\widetilde \JJ_k$ is a convolution operator it commutes (at most up to sign) with $\SS$, $d\SS$, $\SS d$ and with convolution operators having a smooth kernel $\delta_\epsilon(\|x-y\|)$. Thus, by Lemma ~3, for all $1< t < n$ the operators $\SS_k :\AA_{k+1, \BB_{k,t}}\to \AA_{k, \BB_{k,tn/(n-t)}}$ are bounded. Also, for all $1 < t <\infty$ the operators $\SS_k:\AA_{k+1, \BB_{k+1,t}}\to \AA_{k, \BB_{k,t}}$ are bounded, and both operators $d_k \SS_k$, $\SS_{k+1}d_{k+1}$ are bounded on $\AA_{k+1, \BB_{k+1,t}}$. Finally, there are $C^\infty$ mollifiers $\delta_\epsilon(\|x-y\|)$ on $K'\times K'$ with $\|\int \delta_\epsilon (\cdot,y)\varphi(y) \, dy - \varphi\|_{\BB_{k,t}}\to 0$ as $\epsilon \to 0$ for all $\varphi$ [Ad, Lemma 2.18]. By Lemma~0 \therosteritem{1}, for each fixed $y\in K$, the form $\sigma^{(r)}_k(x,y)=\widetilde \JJ_{k} (1_K(\cdot) \sigma_k(\cdot, y))$ belongs to $\AA_{k,\BB_{k,t}}$ for all $1 \le t < n/(n-1)$ and each $k$, with $$ \max_k \sup_{y\in K} \| \widetilde \JJ_{k} (1_K(\cdot) \sigma_k(\cdot, y)) \|_{\AA_{k,\BB_{k,t}}} < \infty \, , \forall \, 1 \le t < n/(n-1)\, . \tag{3.2} $$ Write $\SS^{(r)}_k$ for the convolution operator corresponding to $\sigma^{(r)}_k(x,y)$. Up to a sign, this is just $1_K \widetilde K \JJ_k \SS_k$. Hence, there is $\upsilon_k\in\{+1,-1\}$ so that for all $1 1$ and $\xi < 1/\widetilde R$ there is $L \ge 1$ so that for each $|z| < \xi$ and all $t$, $k$, we have $\|\MM_{k,L} (z)|_{Im(\Id-\Pi_{k,t})}\|_{\AA_{k,\BB_{k,t}}}\le {|z|^L \widetilde R^L\over C}$ and $$ \|\NN_{k,L} (z)|_{Ker(\Pi_{k,t})\cap Ker(\Pi_{k+1,t} \circ d_k)}\|_{\AA_{k,\BB_{k,t}}, \AA_{k+1,\BB_{k+1,t}}}\le {|z|^L \widetilde R^L\over C} \, . $$ Up to taking a slightly larger value of $\widetilde R$, we may assume that $L$ does not depend on $\xi$. \endproclaim \demo{Proof of Lemma 5} Formally, $$ \MM_{k,\ell}(z)= \Id -(\Id-z\MM_k)\exp[-\log(\Id-z \MM_k)-\sum_{j\ge \ell} {z^j\MM_k^j\over j}] =\Id-\exp\sum_{j\ge \ell}- {z^j \MM_k^j\over j} \, . $$ Then, we use that for any $0<\theta < 1$ and $C > 1$ there is $L$ so that for all $\ell\ge L$ $$ \sum_{j\ge \ell} {\theta^j\over j} \le { \theta^\ell\over \ell} \sum_{j\ge 0} \theta^j\le {\theta^\ell \over \ell (1-\theta)} \le {\theta^\ell\over 2 C} \, , \hbox{ while } 1-\exp(-{\theta^\ell\over 2C}) \le {\theta^\ell\over C} \, . $$ This gives the bound for $\MM_{k,\ell}(z)$ by Axiom~2\therosteritem{1}. To estimate $\NN_{k, \ell}(z)$ note that $$ \NN_{k,\ell}(z) =\Id-\exp\sum_{j\ge \ell}- {z^j \NN_k^{(j)}\over j} \, ,\tag{3.8} $$ with $\NN^{(j)}_k=d_k\MM_k^j - \MM^j_{k+1} d_k$, and use Axiom~1\therosteritem{3}. \qed \enddemo Whenever the value of $t$ is clear from the context, we write $\Pi_{k}$ instead of $\Pi_{k,t}$ for simplicity. Note that $\Pi_k =\Pi_k \chi_K=\chi_K \Pi_k$. It follows from Lemma~5 that the essential spectral radius of $\MM_{k,\ell}(z)$ on any $\AA_{k,\BB_{k,t}}$ is not larger than $|z|^\ell \widetilde R^\ell< 1$ for all $\ell$ and every $|z|< 1/ \widetilde R$. It is then an easy algebraic exercise to see that for any $|z|< 1/ \widetilde R$, the complex number $1/z$ is an eigenvalue of $\MM_k$ (on $\AA_{k,\BB_{k,t}}$) of algebraic multiplicity $m$ if and only $1$ is an eigenvalue of multiplicity $m$ for $\MM_{k,\ell}(z)$ (on $\AA_{k,\BB_{k,t}}$), and in particular, for all $\ell$, setting $R=\widetilde R^{1/n}$, $$ V_{k,t}=\{ |z| < 1/ R \mid 1/z \notin \sp \MM_k \} = \{ |z| < 1/ R \mid 1 \notin \sp \MM_{k,\ell}(z) \}\, .\tag{3.7} $$ (Just use that $\MM_k$ commutes with each $\exp(z^j \MM_k^j/j)$.) One can use the same basis of generalised eigenvectors for both eigenspaces, and $\Pi_k$ commutes with each $\MM_{k,\ell}$. (See e.g. [GGK] for analogous results in the case when a power of $\MM_k$ is Hilbert-Schmidt.) We shall work with the regularised kneading operators (for suitably large $\ell$) $$ \DD_{k,\ell}(z)= \NN_{k,\ell}(z) (\Id- \MM_{k,\ell}(z))^{-1} \SS_k \, . \tag{3.9} $$ We explain next why Lemma 1 and a modified version of Theorem ~2 hold for the $\DD_{k,\ell}(z)$. The replacement of $z\MM_k$ and $z\NN_k$ by $\NN_{k,\ell}(z)$, $\MM_{k,\ell}(z)$ does not cause any problems in Baillif's [Bai, Ba2] proof of Lemma~ 1, since $\MM_{k,\ell}(z)$ is just an entire series with coefficient transfer operators (acting on $\AA_{k,L^q(K')}$). However, since \thetag{2.12} implies $$ \Det^\flat(\Id-\MM_{k,\ell}(z))=\Det^\flat(\Id-z\MM_k) \exp\biggl (\sum_{j=1}^{\ell-1}{z^j\over j} \tr^\flat \MM_k^j\biggr )\, , $$ the first equality in Theorem~2 must be replaced by $$ \Det^\#(\Id-z \MM)\, \exp\biggl ( \sum_{j=1}^{\ell-1} {z^j\over j} \tr^\#\MM^j\biggr ) =\prod_{k=0}^{n-1} \Det^\flat(\Id+\DD_{k,\ell}(z))^{(-1)^{k+1}} \, .\tag{3.10} $$ The additional factor $\exp\sum_{j=1}^{\ell-1} {z^j\over j} \tr^\#\MM^j$ is clearly an entire and non vanishing function of $z$ for each $\ell$. The final useful property of the regularised kneading determinants is: \proclaim{Corollary of Lemma 5} Assume Axioms 1 and 2. For each $z\in V_{k,t}$ (see \thetag{3.7}) and all $0\le k\le n-1$, $1 < t < \infty$, the essential spectral radius of $\DD_{k,L}(z)$ on $\AA_{k+1,\BB_{k+1,t}}$ goes to zero exponentially fast as $L\to \infty$, in particular it is $<1$ for large~ $L$. \endproclaim \demo{Proof of the Corollary} Let $\widetilde \Pi_k$ be a projector onto the finite-dimensional space $\{ \psi \in (\Id-\Pi_k)(\AA_{k,\BB_{k,t}}) \mid d_k\psi \in \hbox{Im}\, \Pi_{k+1}\}$. Use $\Pi_k\MM_{k,L}(z)= \MM_{k,L}(z)\Pi_k$ to get the decomposition $$ \eqalign{ &\DD_{k,L}(z)=\NN_{k,L}(z) (\Id-\widetilde \Pi_k)(\Id- \MM_{k,L}(z))^{-1} (\Id-\Pi_k) \SS_k\cr &\quad+\NN_{k,L}(z) \widetilde \Pi_k(\Id- \MM_{k,L}(z))^{-1} (\Id-\Pi_k) \SS_k +\NN_{k,L}(z) (\Id- \MM_{k,L}(z))^{-1}\Pi_k \SS_k } $$ into an operator of arbitrarily small spectral radius (by $|z|< 1/R$ and Lemma~5) and a finite rank operator (since $z\in V_{k,t}$). \qed \enddemo \smallskip {\bf Operators $\DD^{(r)}_{k,L}(z)$ } \smallskip We next introduce auxiliary operators $\DD_{k,L}^{(r)}(z)$ on $\AA_{k+1, L^2(K')}$, extending a one-dimensional construction of Ruelle [Ru4]. Their iterates will be trace-class, and their traces will coincide with the formal flat traces of iterates of the $\DD_{k,L}(z)$ in the sense of power series. This will allow us to prove the following crucial lemma: \proclaim{Lemma 6 (Meromorphic extension of $\Det^\flat(\Id+ \DD_{k,L}(z))$)} Assume Axioms 1--2 and~3. Set $\BB_k=\BB_{k,2}$ for $k=0, \ldots, n$. For $k=0, \ldots, n-1$ and all large enough ~ $L$: \noindent (1) $\Det^\flat(\Id+ \DD_{k,L}(z))$ extends holomorphically to $$V_k=\{ |z| < 1/R \mid 1/z \notin \sp ({\MM_k}|_{\AA_{k,\BB_{k}}}) \} =\{ |z| < 1/ R \mid 1 \notin \sp (\MM_{k,\ell}(z)|_{\AA_{k,\BB_{k}}}) \}\, .$$ \noindent (2) If $|z| < 1/R$ and $1/z \in \sp({\MM_k}|_{\AA_{k,\BB_{k }}})$, then $\Det^\flat(\Id+\DD_{k,L}(z))$ is meromorphic at $z$ with a pole of order at most the algebraic multiplicity of the eigenvalue. \endproclaim \demo{Proof of Lemma 6} Recall $\sigma^{(r)}_k(x,y)$ and $\SS^{(r)}_k$ from Axiom~1, and that $\sigma^{(r)}_k(\cdot,y) \in \AA_{k,\BB_{k,t}}$, for $1 \le t < n/(n-1)$ and $y\in K$. Set: $$ \eqalign{ \DD_{k,L}^{(r)}(z)=\upsilon_k (\widetilde \JJ_{k})^{-1} \NN_{k,L}(z) (\Id-\MM_{k,L}(z))^{-1} \SS^{(r)}_k\, . } $$ Axioms 1 and 2 first imply that $z \mapsto \DD_{k,L}^{(r)}(z)$ is a well-defined map (taking values in the space of bounded linear operators on $\AA_{k+1, L^2(K')}$), holomorphic in $V_k$, and meromorphic in the disc of radius $1/\widetilde R$, with possible poles at the inverse eigenvalues of $\MM_k$ on $\AA_{k,\BB_{k,2 }}$ (the order of the pole being at most the algebraic multiplicity of the eigenvalue). We use here that, by Axiom~1(2-3), $\NN_{k,L}(z)$ is a $z$-entire function, bounded from $\AA_{k, \BB_{k,t}}$ to $\AA_{k+1, \BB_{k+1,t}}$ for all $1 n/2$ the operator $(\DD_{k,L}^{(r)}(z))^\ell$ is Hilbert-Schmidt on $\AA_{k+1, L^2(K')}$. In particular, the regularised determinant of order $[n/2]+1$, $$ \Det^{\hbox{reg}}_{[n/2]+1}(\Id+\DD_{k,L}^{(r)}(z))= \exp -\sum_{\ell = [n/2]+1}^\infty {z^\ell\over \ell} \tr^\flat (\DD_{k,L}^{(r)}(z))^\ell \, , $$ is holomorphic on $V_k$. Indeed, $(\Id-\MM_{k, L}(z))^{-1}$ is holomorphic in $V_k$ as a bounded operator on $k$-forms with coefficients in $\BB_{k,t}$, all $t$, and in particular for all values of $t$ between $t_0 < n/(n-1)$ and $t_{[n/2]+1}\ge 2$ which appear along the successive iterations. By \thetag{3.3} $$ \DD^{(r)}_{k,L}(z)= (\widetilde \JJ_{k})^{-1}\NN_{k,L}(z) (\Id-\MM_{k,L}(z))^{-1}\SS_k \widetilde \JJ_{k} \, . \tag{3.12} $$ Recall that the wave front set of (the Schwartz kernel of) a pseudo-differential operator is included in $\{(x,x,\xi,-\xi)\mid \xi \ne 0\}$ and that composition with a pseudodifferential operator does not enlarge the wave-front set (see e.g. [AG]). Hence, using the assumptions on $\widetilde \JJ_k$, $(\widetilde \JJ_k)^{-1}$, a convolution with a $C^\infty$ mollifier, the transversality property \thetag{2.1} to restrict to the diagonal, and Fubini (see Section 4 of [Bai]), the following equalities between formal power series hold $$ \Det^\flat (\Id+\DD_{k,L}^{(r)}(z)) =\Det^\flat (\Id+\DD_{k,L}(z)) \, , \, \, \forall \, k=0, \ldots n-1 \, .\tag{3.13} $$ \smallskip To show that the full flat determinant $\Det^\flat(\Id+\DD_{k,L}(z))$ is holomorphic in $V_k$, we shall use Axiom 3 to see that the power series for $\tr^\flat (\DD_{k,L}(z))^\ell$ for each $1\le \ell \le [n/2]$ is holomorphic on $V_k$ (its exponential is thus holomorphic and nonvanishing). We consider the (hardest) case, i.e., $\ell=1$, leaving higher iterates to the reader. Also, we assume for simplicity that $k=0$ (dealing with $k$-forms only introduces notational complications). \smallskip We first show that the flat trace of $(\widetilde \JJ_0)^{-1}\NN_{0,L}(z)(\Id-\MM_{0,L}(z))^{-1} (\Id-\Pi_0) \SS^{(r)}_0$ is holomorphic in the disc of radius $1/R$. Let $\TT$ denote the restriction to the diagonal $x=y$ in $\real^n \times \real ^n$. For each $\EE < \EE_0$, let $W_\EE$ be the set of $(x,y)\in \real^n \times \real ^n$ with $\|x-y\| < \EE$ and let $\chi^\EE(x-y)+(1-\chi^\EE(x-y))$ be a $C^\infty$ radial partition of unity of $\real^n\times \real^n $ subordinated to $W_\EE$ in the sense that $\chi_\EE(u)$ depends only on $\|u\|$, $\chi^\EE(u)=1$ if $\|u\| \le \EE$, and $\chi^\EE(u)=0$ if $\|u\| \ge 2\EE$. Let $\delta_\epsilon$ be $C^\infty$ mollifiers (converging to the Dirac mass in $\real^n$ as $\epsilon\to 0$), and write $\sigma^{\epsilon,<\EE}$ for the convolution of $\delta_\epsilon$ with $\chi^\EE\sigma_0$ and $\sigma^{\epsilon,>\EE}$ for the convolution of $\delta_\epsilon$ with $(1-\chi^\EE)\sigma_0$ Using the identity $(\Id-\MM_{k,L}(z))^{-1} = \exp - [\sum_{i=0}^{L-1} z^i \MM^i_k ]\cdot (1- z\MM_k)^{-1}$, it suffices to show that there is $C$ so that for all $j$ there is $\EE$ so that for each $\epsilon>0$, all $|z|< 1/R$, $$ \eqalign{ &\int_x | \TT [ (\delta_\epsilon \star( \widetilde \JJ^{-1}_{0}\NN_{0,L}(z) ( \exp (- \sum_{i=0}^{L-1} z^i \MM^i_0) \MM_{0}^j (\Id-\Pi_0) \widetilde \JJ_{x,0}(\chi_K \sigma^{\epsilon,>\EE})(x,y))] | dx \cr &\qquad + \int_x \TT | [ (\delta_\epsilon \star( \NN_{0,L}(z) ( \exp (- \sum_{i=0}^{L-1} z^i \MM^i_0) \MM_{0}^j (\Id-\Pi_0) \sigma^{\epsilon,<\EE})(x,y))]| dx\le C R^j\, .} $$ The entire series $\NN_{0,L}(z) \exp (- \sum_{i=0}^{L-1} z^i \MM^i_0)=\sum_{q\ge 1}z^q \QQ_q $ has coefficients $$ \QQ_q = \sum_{q_1+q_2=q} \kappa_{q_1, q_2} \NN_0^{(q_1)} \MM_0^{q_2}: \AA_{0,\BB_0} \to \AA_{1,\BB_{1}}\, , \quad \kappa_{q_1, q_2}\in \real \, , $$ so that for any $\widehat R$ there is $C$ so that $|\kappa_{q_1, q_2}|< C \widehat R^{q_1+q_2}$. We may write $\NN_0^{(q_1)} \MM_0^{q_2}\varphi= \sum_{\vec \omega \in \Omega^{q_1+q_2}} h_{q_1,q_2,\vec \omega} \wedge (\psi_{\vec \omega}^{q_1+q_2})^*\varphi$ where the $h_{q_1,q_2,\vec \omega}$ are one-forms with $C^{r-1}$ coefficients supported in the domains of the $\psi_{\vec \omega}$. Write $\sigma^{\epsilon,<,>\EE}(x,y)=\sum_m \tau^{\epsilon,<,>\EE}_m(x-y)$. Fix $j$ and consider the (hypothetical) case $q_2=q$. The modulus of the contribution of $ \MM_0^{q_2} \MM^j_0$ may be bounded by the sum over $m$ of $$ \align & \biggl | \int \TT[ \delta_\epsilon \star \bigl ( \MM_{x,0}^{j+q} ( \tau_m^{\epsilon,<\EE})(x- y) \bigr ) ]\, dx \biggr | + \biggl | \int \TT[ \delta_\epsilon \star\bigl (\MM_{x,0}^{j+q} \Pi_{x,0} ( \tau_m^{\epsilon,<\EE})(x- y) \bigr )]\, dx \biggr | \cr &\quad + \biggl | \int \TT[ \delta_\epsilon \star\bigl ( \widetilde \JJ_0^{-1}\MM_{x,0}^{j+q}(\Id-\Pi_{x,0}) \widetilde \JJ_0 (1_K \tau_m^{\epsilon,>\EE})(x- y) \bigr ) ]\, dx \biggr | \, .\tag{3.15} \endalign $$ Recall that for $\EE < \EE_0$ as in Axiom~3, $V_{\vec \omega,\EE}$ is the set of $x$ so that $\|\psi^{j+q}_{\vec \omega}(x)-x\| < \EE$. By Axiom 3, writing $\Phi_{\vec \omega}$ for the inverse of $\psi_{\vec \omega}^{j +q}- \Id$ on $V_{\vec \omega,\EE}$, and performing the corresponding change of variable, the first term in the above sum is bounded by $$ \align & \biggl | \sum_{\vec \omega} \int_{(\psi_{\vec \omega}^{ j+q}- \Id) V_{\vec \omega,\EE}} { g^{(j+q)}_{\vec \omega}(u+\Phi_{\vec \omega}(u)) \over |\det (D\psi_{\vec \omega}^{j+q}(\Phi_{\vec \omega} (u)) -\Id ) |} ( \tau_m^{\epsilon,<\EE})(u) \, du \biggr | \cr &\quad\le \sum_{\vec \omega} \sup_{V_{\vec \omega,\EE}} { |g^{(j+q)}_{\vec \omega}\circ \psi^{j+q}_{\vec \omega} | \over |\det (D\psi_{\vec \omega}^{j+q}(x) -\Id ) |} \cdot \int |\tau^{\epsilon,<\EE}_m(u)|\, du %\cr %&\quad \le C \exp(\eta (q+j)) \EE \, . \endalign $$ %where $\eta>0$ can be taken arbitrarily small. %could probably be improved by using a telescoping trick Ru90 or Hay, or %the fact that 1/det is either exponentially small or exponentialy %close to 1, or some expansion for g(u+Phi(u)) or %(at least for the small powers of traces, but probably not %for Plemelj Smithies) alternating sum %or Schwarz inequality + oscillatory large j By the consequences of Axiom~1, for all $x$, $y$ in $K$, we have $\widetilde \JJ_{0} ((1_K \tau_m^{\epsilon,<\EE})(\cdot- y) )(x)= \tilde \upsilon_k \widetilde \JJ_{0}((1_K \tau_m^{\epsilon,<\EE})(x- \cdot) )(y)$ for $\tilde \upsilon_k\in \{+1, -1\}$. Hence, for the second term, use \thetag{3.4} to get (in the case where all eigenvalues are semisimple) $$ \align &\biggl | \int \TT[\delta_\epsilon \star \sum_s \lambda_s^{j+q} \varphi_s(x) \nu_s( \widetilde \JJ_{y,0}^{-1} \widetilde \JJ_{0,\cdot} (1_K \tau_m^{\epsilon,<\EE})(\cdot- y) ) ]\, dx \biggr | \cr &\le \biggl | \int \TT[\delta_\epsilon \star \sum_s \lambda_s^{j+q} \varphi_s(x) \widetilde \JJ_{y,0}^{-1} (\nu_s( \widetilde \JJ_{0,\cdot}( 1_K \tau_m^{\epsilon,<\EE})(\cdot- y) ) ) ]\, dx \biggr |\cr &\le \sum_s \bigl | \int_K \lambda_s^{j+q} \varphi_s(x) \widetilde \JJ_{x,0}^{-1} (\nu_s(\widetilde \JJ_{0,\cdot}(1_K \tau_m^{\epsilon,<\EE})(\cdot- x) ) ) \, dx \bigr |\cr \allowdisplaybreak &= \sum_s \bigl | \int_K \lambda_s^{j+q} \widetilde \JJ_{0}^{-1}( \varphi_s ) \nu_s(\widetilde \JJ_{0,\cdot}( 1_K \chi^\EE\cdot \tau_m^\epsilon)(\cdot- x) ) \, dx \bigr | \cr \allowdisplaybreak &\le C \sum_s \sup| \widetilde \JJ_{0}^{-1}( \varphi_s )|\cdot \sup_x \|\widetilde \JJ_{0,\cdot}(1_K \tau_m^{\epsilon,<\EE})(\cdot- x) ) \|_{\BB_{0,t}} \cr &\le C \sup_{x\in K} \|( \tau_m^{\epsilon,<\EE})(\cdot- x) \|_{L^t(K)} \le C \sup_x (\int |\tau_m^{\epsilon,<\EE}(u-x)|^t \, du)^{1/t} \le C \EE \, , \endalign $$ taking $t>1$ close to $1$ and using $|\lambda_s|\le 1$. Nilpotent contributions produce polynomial growth in $q+j$ which gives a bound $C \exp(\eta (q+j)) \EE$ for arbitrarily small $\eta$. If $a(x,y)$ is continuous then $\int_K |a(x,x)| \, dx \le \hbox{Vol}(K) \sup_{y}\sup_x |a(x,y)|$. Recall also that if $\sup_{t\to \infty} \|b(\cdot,y)\|_{L^t(K)} < B(y)$, then $b(\cdot,y)$ is in $L^\infty(K)$ and $\sup_K|b(\cdot,y)| \EE})(x- y))) | \cr &\le C \sup_y \sup_{t\to \infty} \| \MM_{0}^{j+q}(\Id-\Pi_{0}) \widetilde \JJ _0 (1_K\tau_m^{\epsilon,>\EE})(\cdot- y)\|_{\BB_{0,t}}\cr \allowdisplaybreak &\le C \widetilde R^{j +q} \sup_{y, t} \bigl (\|\widetilde \JJ_0(1_K \tau_m^{\epsilon,>\EE})(\cdot- y)\|_{\BB_{0,t}} + \| \Pi_0 \widetilde \JJ_0(1_K \tau_m^{\epsilon,>\EE})(\cdot- y)\|_{\BB_{0,t}} \bigr )\cr \allowdisplaybreak &\le C \widetilde R^{j +q}\sup_y \sup_{t\to \infty} \biggl (\|\tau_m^{\epsilon,>\EE}(\cdot- y)\|_{L^t(K)}\cr &\qquad \qquad \qquad\qquad+ \sum_s |\nu_s[\widetilde \JJ_0(1_K \tau_m^{\epsilon,>\EE})](\cdot- y)| \cdot \| \widetilde \JJ_0^{-1} \varphi_s\|_{L^t(K)} \biggr ) \cr &\le C \widetilde R^{j+q} \bigl ( \sup_{\|u\|\ge \EE} | \tau^{\epsilon,>\EE}_m(u)| + C \sup_y \|1_K \tau_m^{\epsilon,>\EE}(\cdot- y)\|_{L^{t_0}} \bigr ) \cr &\le C\widetilde R ^{j+q} ( \EE^{-(n-1)} + C) \, , \cr \endalign $$ for some $1 < t_0 < n/(n-1)$. Since $\|\NN^{(q_1)}_0\|\le C^{q_1}$, $\Omega$ is finite, and $ \|h_{q_1,q_2,\vec \omega}\|\le C^{q_1+q_2}$, the case $q_1 \ge 1$ follows from the bounds on $\kappa_{q_1,q_2}$. Choosing $\EE=\widetilde R^{j/n}$ gives claim (1) (because the projection $\Pi_0$ has finite rank, see also the proof of Lemma~8 below). \smallskip If $|z| < 1/R$ is in the spectrum of $\MM_k$ on $\BB_{k}$, the argument with mollifiers described above can be used to invoke the ordinary Plemelj-Smithies formula as in Lemma~4.4.2 of [Go], to see that the determinant $\Det^\flat(\Id+\DD_{k,L}^{(r)}(z))$ has at most a pole of order the algebraic multiplicity of the eigenvalue at $z$, proving claim (2). \qed \enddemo \smallskip We next relate the zeroes of the analytic continuation of the formal determinant $\Det^\flat(\Id+\DD_{k,L}(z))$, to the presence of an eigenvalue $-1$ for the operators $\DD_{k,L}(z)$: \proclaim{Lemma 7 (Zeroes of $\Det^\flat(\Id+ \DD_{k,L}(z))$)} Assume Axioms 1, 2 and 3. For $k=0, \ldots, n-1$ and all large enough $L$, $\DD_{k,L}(z)$ extends holomorphically on $V_k$ to a family of operators on $\AA_{k+1,\BB_{k+1 }}$. The essential spectral radius of each $\DD_{k,L}(z)$ is $< 1$. If $\Det^\flat(\Id+ \DD_{k,L}(z))=0$ for $z \in V_k$, then $-1$ is an eigenvalue of $\DD_{k,L}(z)$ on $\AA_{k+1,\BB_{k+1 }}$. \endproclaim \proclaim{Sublemma} Let $A$ be a compact subset of $\real^n$. Let $\widetilde \LL$ be a Hilbert-Schmidt operator on $L^2(A)$ written in kernel form $ \widetilde \LL\varphi(x) =\int \widetilde \KK_{xy} \varphi(y) \, dy $. Let $\LL$ be a bounded operator acting on a Banach space $\BB$ of distributions over $A$ containing $C^\infty(A)$. Assume that for small $\epsilon>0$ there are $C^\infty$ kernels $\KK_{\epsilon,xy}$ and $\widetilde \KK_{\epsilon,xy}$ so that the Fredholm determinants of the associated operators coincide $$ \Det (\Id-\lambda \LL_\epsilon)= \Det (\Id-\lambda \widetilde \LL_\epsilon) \, , \forall \epsilon \, , $$ and assume also $ \lim_{\epsilon\to 0} \| \LL_\epsilon - \LL \|_{\BB\to \BB}=0$, $\lim_{\epsilon \to 0} |\widetilde \KK_{\epsilon,xy} - \widetilde \KK_{xy} |_{L^2(K\times K)}=0 $. Then, for all $\lambda$ with $1/|\lambda| < \rho_{ess}( \LL)$, writing $$\Det_2^{\hbox{reg}}(\Id-\lambda \widetilde \LL)=\exp -\sum_{\ell =2}^\infty {\lambda^\ell\over \ell} \tr \widetilde \LL^\ell \, , $$ we have $ \Det_2^{\hbox{reg}}(\Id-\lambda \widetilde \LL)=0 \Longrightarrow \lambda^{-1} \hbox{ is an eigenvalue of } \LL \hbox{ on } \BB $. \endproclaim \demo{Proof of the Sublemma} The statements in this proof hold for all $1/|\lambda| < \rho_{ess} (\LL)$ (uniformly in any compact subset). $\Det(\Id-\lambda \LL_\epsilon)= \Det(\Id-\lambda \widetilde \LL_\epsilon)$ vanishes if and only if $1/\lambda$ is an eigenvalue of $\LL_\epsilon$ (on $L^2(A)$, or equivalently on $C^\infty(A)$, using that the image of an element of $L^2(A)$ by an operator with $C^\infty$ kernel is $C^\infty$), using the l.h.s., if and only if $1/\lambda$ is an eigenvalue of $\widetilde \LL_\epsilon$ (on $L^2(A)$ or equivalently on $C^\infty(A)$) using the r.h.s. The convergence of the kernels $\widetilde \KK_{\epsilon,xy}$ implies both that $\Det_2^{\hbox{reg}}(\Id-\lambda \widetilde \LL_\epsilon)$ converges to $\Det_2^{\hbox{reg}}(\Id-\lambda \widetilde \LL)$ (which is entire in $\lambda$), and that $\widetilde \LL_\epsilon$ on $L^2(A)$ converges to the compact operator $\widetilde \LL$ on $L^2(A)$. Thus every zero $\lambda_0$ of $\Det_2^{\hbox{reg}}(\Id-\lambda \widetilde \LL)$ is a limit of $\lambda_\epsilon$ so that $1/\lambda_\epsilon$ is an eigenvalue of $\widetilde \LL_\epsilon$ (on $L^2(A)$). (Indeed, such a zero corresponds to $1/\lambda_0$ being an eigenvalue of $\widetilde \LL$.) By the first observation, $1/\lambda_\epsilon$ is an eigenvalue of $\LL_\epsilon$ (on $C^\infty(A)$, so that the eigenfunction is in $\BB$). If $|\lambda_0| < 1/\rho_{ess} (\LL)$, since $\LL_\epsilon$ converges to $\LL$ in operator norm on $\BB$, if $1/\lambda_\epsilon$ is an eigenvalue of $\LL_\epsilon$ for all $\epsilon$ then $\lambda_0$ is an eigenvalue of $\LL$ (for an eigenvector which is the $\BB$-norm limit of $\varphi_\epsilon\in C^\infty(A)$). \qed \enddemo \demo{Proof of Lemma 7} We have shown that the left-hand-side of \thetag{3.13} extends holomorphically to $V_k$. By definition and the Axioms the operator $\DD_{k,L}(z)$ extends holomorphically to $V_k$ in the sense of bounded operators on $\AA_{k+1, \BB_{k+1 }}$. By the corollary of Lemma ~5, it is enough to show the claim about the zeroes of the determinant. Taking iterates to get Hilbert-Schmidt operators on $\AA_{k+1,L^2(K')}$, one can apply the sublemma to show that if $z \in V_k$ is such that $\Det^\flat(\Id+\DD_{k,L}(z))= \Det^\flat(\Id+\DD^{(r)}_{k,L}(z))=0$ then $-1$ is an eigenvalue of $\DD_{k,L}(z)$ on $\AA_{k+1, \BB_{k+1 }}$. More precisely, for fixed $z\in V_k$, set $ \LL = (\DD_{k,L}(z))^m \, , \quad \tilde \LL = (\DD_{k,L}^{(r)}(z))^m $. Then, to construct the smooth kernels required by the assumptions of the sublemma, we (again) smoothen $\DD_{k,L}(z)$ by pre and post-convolution with a $C^\infty$ mollifier $\delta_\epsilon$, writing $\DD^\epsilon_{k}(z)= \delta_\epsilon \DD_{k,L}(z)\delta_\epsilon$ for the new operator. If we mollify $\DD^{(r)}_{k,L}(z)$ similarly, the equality \thetag{3.13} between determinants remains true, and the kernels of $\DD^{\epsilon,(r)}_{k}(z)$ converge as $\epsilon \to 0$, in the $L^2(K\times K)$ topology. To see that $\DD^\epsilon_{k}(z)$ converges to $\DD_{k,L}(z)$, in the sense of operators on $\AA_{k+1, \BB_{k+1 }}$, use that $\delta_\epsilon$ converges to $\Id$ in the $\AA_{k+1,\BB_{k+1 }}$ topology. \qed \enddemo \smallskip {\bf A modification of the homotopy operators} \smallskip We next explain how to exploit Lemmas 6--7. We first discuss the effect of making a suitable finite-rank perturbation of $\SS$ in the definition of the kneading operators: \proclaim{Lemma 8 (Perturbing the homotopy operators)} Assume Axioms 1 and 2. Let $\SS'$ be a finite-rank perturbation of $\SS$ (i.e., $\SS'_k-\SS_k$ is finite-rank from $\AA_{k+1, \BB_{k+1}}$ to $\AA_{k,\BB_k}$), so that $d\SS'+\SS' d =\Id$ and $(\SS')^2 =0$. Then the statements of Lemma ~1 and the modified equality \thetag{3.10} from Theorem~2 remain true for $\SS'$ and for the perturbed kneading operators $\DD'_{k,L}(z)$ defined by $$ \DD'_{k,L}(z)= \NN_{k,L}(z) (\Id- \MM_{k,L}(z))^{-1} \SS'_k \, , k=0, \ldots, n-1 \, . \tag{3.16} $$ \endproclaim \demo{Proof of Lemma 8} Since $d\SS'+\SS'd =\Id$, Baillif's proof extends relatively straightforwardly to the modified convolution operators $\SS'$. Indeed, we may use the decomposition $$ \DD_{k,L}'(z)=\DD_{k,L}(z)+ ( \DD'_{k,L}(z)-\DD_{k,L}(z)) =\DD_{k,L}(z)+\FF_{k,L}(z)\, , $$ into a power series with coefficients operators whose Schwartz kernel is well-behaved along the diagonal (in particular the flat trace exists, and Fubini is allowed, so that all desired commutations hold [Bai]), summed with a power series whose coefficients are finite-rank operators. These finite-rank operators act on $\AA_{k+1, \BB_{k+1 }}$ and each coefficient of $\DD_{k,L}(z)$ is bounded on this space. Hence, we may define the formal traces and determinants (as power series only) by setting $$ \tr^\flat (\DD_{k,L}(z)+\FF_{k,L}(z))^m = \tr^\flat \DD_{k,L}(z)^m + \tr^\flat \FF_{m,k,L}(z)\, , $$ where $\FF_{m,k,L}(z)$ is a power series with coefficients finite-rank operators on $\AA_{k+1, \BB_{k+1 }}$, for which the flat trace is just defined to be the sum of eigenvalues. In fact, since $\DD'_{k,L}(z)-\DD_{k,L}(z)$ starts with a precomposition by $\chi_K(\SS'_k-\SS_k)$, which is finite-rank from $\AA_{k+1, \BB_{k+1 }}$ to $\AA_{k, \BB_{k }}$, the resolvent factor $(\Id-\MM_{k,L}(z))^{-1}$ is well defined if $z\in V_k$. Thus, the power series $\tr^\flat \FF_{m,k,L}(z)$ defines in fact a holomorphic function in $V_k$. (This argument will be useful in the proof of Lemma~10 below.) \qed \enddemo %The additional assumption of Lemma 9 can be obtained %by perturbing the $g_\omega$ and $\psi_\omega$ (see the proof of %Theorem~13). \proclaim{Lemma 9 (Adapted homotopy operators $\SS'$)} Assume Axioms ~1 and~ 2, and suppose that all eigenvalues of modulus $> R$ of the $\MM_k$ on $\AA_{k,\BB_k}$($k \le n-1$) are simple, and that the corresponding eigenvectors $\varphi_k$ do {\it not} belong to $\Ker d=\Imm d$ or to $\Ker \NN_k$. Then there are finite-rank operators $$ \eqalign{ &\FF_{k}: \AA_{k+1, \BB_{k+1}}\to \{ \varphi \in \AA_{k} \mid \chi_K \varphi \in \AA_{k, \BB_{k }} \} \, , \quad 0 \le k \le n-1 \, ,\cr & \hbox{so that the perturbed operators }\qquad \SS'_k = \SS_k +\FF_k\, ,} $$ satisfy $\SS' \SS'=0$, $\SS'd+ d\SS'=1$ and, additionally, if $0 \le k \le \min(s,n-1)$ and $1/z$ with $|z|> R$ is an eigenvalue of $\MM_k$ on $\AA_{k, \BB_{k }}$ with corresponding (simple) fixed vector $\varphi_k(z)$ for $\MM_{k,\ell}(z)$ and dual fixed vector $\nu_k(z)$, with $\nu_k(z)(\varphi_k(z))=1$, then for large enough $\ell$ $$ |\nu_k (z) (\chi_K \SS'_k \NN_{k,\ell}\varphi_k(z))| > 0 \, . $$ \endproclaim \demo{Proof of Lemma 9} For each $k$ we consider the finite set of ``bad pairs'' $(z, \varphi_k)$, with $z=z_{k,i}\in \complex$ and $\varphi_{k,i}$ an fixed vector such that $\nu_{k,i} (z) (\chi_K \SS_k \NN_{k,\ell} \varphi_{k,i}(z))= 0$. We write the argument assuming that this set is either empty or a singleton $\{(z, \varphi_k)\}$ for each $k$ in order to simplify notation. Our assumptions imply that $\upsilon_k=\nu_k(\SS_k d \varphi_k)\ne 0$ and $\NN_{k,\ell} (z) \varphi_k(z)\ne 0$ for all $\ell$. For each $k$ so that a bad pair $(z,\varphi_k)$ exists, set $\varphi'_{k}=\varphi_k$ and let $\nu'_{k}$, $\nu'_{k+1}$ be continuous functionals on $\AA_{k , \BB_{k }}$, $\AA_{k+1, \BB_{k+1}}$, respectively, which satisfy $\nu'_k \circ d = 0$, $\nu'_{k+1} \circ d = 0$, and $$ \alpha_{k,\ell}=\nu'_{k+1}(\NN_{k,\ell} \varphi_k) =-\nu'_{k+1}(\MM_{k+1,\ell}d \varphi_k)\ne 0 \, , \forall \ell \, , \quad \beta_k=\nu'_k(\varphi_k)\ne 0\, . $$ If there is no $\varphi_k$ we set $\varphi'_{k}=0$ and $\nu'_{k+1}=0$ and allow $\alpha_k$ and $\beta_{k,\ell}$ to vanish. We put, for complex $\epsilon$ of small modulus, and all $k$ $$ \FF'_{k} (\varphi)= -\epsilon d \varphi'_{k-1} \cdot \nu'_{k}(\SS_k \varphi) +\epsilon \varphi'_{k} \cdot \nu'_{k+1}(\varphi)\, . $$ We have $d\FF'_{k-1}=-\FF'_k d$ so that $d(\SS+\FF')+(\SS+\FF')d=\Id$. Thus, setting $\FF=\FF'-d\SS\FF'-d\FF'\SS-d\FF'\FF'$ we have $d\SS'+\SS'd=\Id$ and $\SS'\SS'=0$. Finally, for each $k$ with a bad pair, we find for uncountably many small values of $\epsilon$ and all large enough $\ell$ $$ \eqalign { \nu_k((\SS'_{k}\NN_{k,\ell}) \varphi_k)&= \epsilon \biggl (\nu_k(\SS_k d \varphi_{k}) \cdot \nu'_{k+1} (\NN_{k,\ell}(z)\varphi_k) \cr &\quad- \nu_k(d \varphi_{k -1})\cdot \bigl [\nu'_{k}(\SS_k \NN_{k,\ell}(z)\varphi_k) +\epsilon \nu'_k(\varphi_k)\nu'_{k+1} (\NN_{k,\ell}(z)\varphi_k) \bigr ] \biggr )\cr &= \epsilon \bigl (\upsilon_k \alpha_{k,\ell}+ \nu_k(d\varphi_{k-1}) (\nu'_{k}(\SS_k \NN_{k,\ell}(z)\varphi_k)+\epsilon \beta_k \alpha_{k,\ell}) \bigr )\ne 0 \, . } $$ By taking $\epsilon$ small enough we may ensure that no new bad pairs are created. \qed \enddemo \noindent Replacing $\SS$ by the finite rank perturbation $\SS'$, we get a stronger version of Lemmas~6--7: \proclaim{Lemma 10 (Meromorphic extension -- guaranteeing poles)} Under the assumptions of Lemma ~9, and up to taking a larger value of $L$, for each $k=0, \ldots, n-1$, Axiom~3 implies: \noindent (1) The power series $\Det^\flat(\Id+ \DD'_{k,L}(z))$ defines a holomorphic function in $V_k$. \noindent (2) If $|z_0| < 1/R$ is such that $1/z_0$ is a simple eigenvalue of $\MM_k|_{\AA_{k, \BB_{k }}}$, then $\Det^\flat(\Id+\DD'_k(z))$ is meromorphic at $z_0$, with a pole of order exactly one. \noindent (3) $\DD'_{k,L}(z)$ extends holomorphically on $V_k$ to a family of operators on $\AA_{k+1, \BB_{k+1}}$, each such $\DD'_{k,L}(z)$ has essential spectral radius strictly less than $1$, and $\Det^\flat(\Id+ \DD'_{k,L}(z))=0$ if and only if $\DD_{k,L}'(z)$ has an eigenvalue $-1$ on $\AA_{k+1,\BB_{k+1 }}$. \endproclaim \demo{Proof of Lemma 10} From the proof of Lemma~6, $\DD^{(r)}_k(z)$ has its $[n/2]+1$th iterate trace-class on $\AA_{k+1, L^2(K')}$ for all $z \in V_k$. Using Axioms~1--2, for each $z\in V_k$, %the operator $$ \eqalign {\DD^{(r)'}_{k,L}(z) &=\DD^{(r)}_{k,L}(z)+ (\widetilde \JJ_{k})^{-1} \NN_{k,L}(z)(\Id-\MM_{k,L}(z))^{-1} (\SS'_k-\SS_k)\widetilde \JJ_{k} } $$ is such that its $[n/2]+1$th iterate is a finite-rank perturbation on $\AA_{k+1, L^2(K')}$ of $\DD^{(r)}_{k,L}(z)$. The sum of the $L^2(K')$ (or, equivalently, flat) trace of $(\DD^{(r)}_{k,L}(z))^{[n/2]+1}$ and the flat trace of a composition (in any order) of $j$ factors of the finite-rank term $[\DD^{(r)'}_{k,L}(z)-\DD^{(r)}_{k,L}(z)]^{[n/2]+1}$ with $([n/2]+1-j)$ factors $\DD^{(r)}_{k,L}(z)$ is holomorphic in $V_k$. Its power series is equal to the sum of the formal flat trace of $\DD_{k,L}^{[n/2]+1}(z)$ with the trace of a finite rank operator which has the same trace as $(\DD'_{k,L}(z)-\DD_{k,L}(z))^{[n/2]+1}$. (Recall that $\DD'_{k,L}(z)-\DD_{k,L}(z)$ starts with a precomposition by $\chi_K(\SS'_k-\SS_k)$ which is finite-rank from $\AA_{k+1, \BB_{k+1}}$ to $\AA_{k, \BB_{k}}$.) Therefore $\tr^\flat (\DD^{(r)'}_{k,L}(z))^{[n/2]+1}=\tr^\flat (\DD'_{k,L}(z))^{[n/2]+1}$, which yields the first claim of Lemma~10. Next, we may apply the Sublemma, essentially as in Lemma~7, to see that the zeroes of the determinant $\Det^\flat(\Id+ \DD'_{k,L}(z))$ correspond to eigenvalues $-1$, using $$ \eqalign{ \DD^{\epsilon, (r')}_{k,L}(z)&= \DD^{\epsilon, (r)}_{k,L}(z) + \delta_\epsilon (\widetilde \JJ_{k})^{-1}\NN_{k,L}(z)(\Id-\MM_{k,L}(z))^{-1} (\SS'_k-\SS_k)\widetilde \JJ_{k}\delta_\epsilon \cr \DD^{\epsilon'}_{k,L}(z)&= \DD^\epsilon_{k,L}(z) + \delta_\epsilon \NN_{k,L}(z)(\Id-\MM_{k,L}(z))^{-1} (\SS'_k-\SS_k)\delta_\epsilon\, , } $$ which both have a finite-rank (in $L^2(\real^n)$) second term. This gives Lemma~10,\therosteritem{3}. If $1/z_0$ is a simple eigenvalue for $\MM_k$ with $z_0\in V_k$ then one proves, like in Lemma~ 6, that $\Det^\flat(\Id+\DD'_{k,L}(z))$ is meromorphic at $z_0$. with a pole of order at most one. Finally, we shall prove that $\Det^\flat(\Id+\DD'_{k,L}(z))$ does not have a removable singularity at $z_0$ showing that the pole has order exactly one i.e., claim \therosteritem{2}. For this, we use a spectral decomposition of $\MM_{k,\ell}(z)$ for the simple eigenvalue $\lambda_z$ on $\AA_{k,\BB_{k}}$, for $z$ close to $z_0$ (with $\lambda_{z_0}=1$): $$ (\Id-\MM_{k,\ell}(z))^{-1} = {\varphi_{k}(z) \over \lambda_z-1} \cdot \nu_k(z) + \RR_k(z)\, , \tag{3.17} $$ with $\lambda_z$ holomorphic in $z$; $\RR_k(z): \AA_{k,\BB_{k}} \to \AA_{k,\BB_{k}}$ depending holomorphically on $z$ at $z_0$; $\RR_k (z) \varphi_k(z) = \nu_k(z) \RR_k(z) = 0$; the eigenvector $\varphi_k(z) \in \AA_{k,\BB_{k}}$ of unit norm depending holomorphically on $z$ with $\varphi_k(z_0) =\varphi_k \in \AA_{k,\BB_{k}}$ so that $\MM_k(z_0) \varphi_k =\varphi_k$; and $\nu_k(z)$ unit-norm linear functionals on $\AA_{k,\BB_{k}}$, depending holomorphically on $z$, with $\nu_k(z)(\varphi_k(z)) = 1$. By Lemma 9, $|\nu_k(z_0) (\chi_K \SS'_{k} \NN_{k,\ell}(z_0) \varphi_k(z_0)) | > 0$ if $\ell$ is large enough. It is then easy to see that $ (z-z_0)\Det^\flat(\Id+\DD'_{k,L}(z)) $ does not vanish at $z_0$. \qed \enddemo %\pagebreak \smallskip {\bf Using the new homotopy operators} \smallskip Making use of the homotopy operators from Lemma~9, we state the final ingredients needed in our main result. The eigenvalues of modulus larger than $R$ of the operators $d \SS \MM_k$ produce zeroes of the flat kneading determinants (Lemma~11). To show that such eigenvalues do not contribute to the zeroes and poles of the sharp determinant (except when $\MM_k \tilde\varphi_k=d\SS\MM_k\tilde \varphi_k$, in particular if $k=n$), we shall prove that they are not intrinsic. More precisely, in Lemma~12 we construct perturbed homotopy operators $\SS''$ which cause these eigenvalues to vary. \proclaim{Lemma 11 (Zeroes of the flat kneading determinants)} Under the assumptions of Lemma~9, and up to taking a larger value of $L$, for all $k \in \{1, \ldots, n\}$ and $|z| < 1/R$, \noindent (1) If $\Det^\flat(\Id+ \DD'_{k-1,L}(z))= 0$ then $1/z$ is an eigenvalue of $d\SS'_{k-1}\MM_k$ acting on $\AA_{k,\BB_k}$ or $k\ge 2$ and $1/z$ is an eigenvalue of $d\SS'_{k-2}\MM_{k-1}$ acting on $\AA_{k-1,\BB_{k-1}}$, in either case, the geometric multiplicity of the eigenvalue is at least equal to the order of the zero. \noindent (2) If $1/z$ (with $|z| < 1/R$) is an eigenvalue of $\MM_n$ acting on $\AA_{n,\BB_n}$ and is not an eigenvalue of $\MM_{n-1}$ acting on $\AA_{n-1,\BB_{n-1}}$, then $\Det^\flat(\Id+ \DD'_{n-1,L}(z))= 0$ and the order of the zero is at least the algebraic multiplicity of the eigenvalue. \noindent (3) If $k \ge 2$ and $1/z$ (with $|z| < 1/R$) is an eigenvalue of $d\SS'_{k-2}\MM_{k-1}$ acting on $\AA_{k-1,\BB_{k-1}}$ then $\Det^\flat(\Id+ \DD'_{k-1,L}(z))= 0$ and the order of the zero is at least the geometric multiplicity of the eigenvalue (which is thus finite and the eigenvalue isolated). \endproclaim \demo{Proof of Lemma 11} By Lemma~10, the assumption $\Det^\flat(\Id+ \DD'_{k-1,L}(z))= 0$ implies that $1/z \notin \sp(\MM_{k-1})$ and there is a nonzero $\varphi\in \AA_{k,\BB_{k}}$ with $\DD'_{k-1,L}(z)\varphi=-\varphi$. It follows that $\varphi$ is {\it not} in the kernel of $\SS'_{k-1}$. Writing $\varphi=d\SS'_{k-1}\varphi + \SS'_k d\varphi=\varphi_1+\varphi_2$ (with $\SS'_{k-1}\varphi_1\ne 0$), our assumption implies $$ \cases d\SS'_{k-1}\varphi_1 + d \SS'_{k-1}\NN_{k,L}(z)(\Id-\MM_{k-1,L}(z))^{-1} \SS'_{k-1} \varphi_1=0&\cr \varphi_2=\SS'_k d\varphi_2 = -\SS'_k d \NN_{k,L}(z) (\Id-\MM_{k-1,L}(z))^{-1} \SS'_{k-1} \varphi_1 \, .&\cr \endcases \tag{3.18} $$ The first equality in \thetag{3.18} is equivalent with $$ \eqalign { 0&=\left ( d+ (d\MM_{k-1,L}(z) -d\SS'_{k-1} \MM_{k,L}(z) d) (\Id-\MM_{k-1,L}(z))^{-1} \right )\SS'_{k-1} \varphi_1\cr &=(\Id-d\SS'_{k-1} \MM_{k,L}(z)) d (\Id-\MM_{k-1,L}(z))^{-1} \SS'_{k-1} \varphi_1 \, .} $$ Note that $d (\Id-\MM_{k-1,L}(z))^{-1} \SS'_{k-1} \varphi_1$ is in $\AA_{k, \BB_k}$ by the boundedness of the $d$ operator in Axiom ~1. Then it is not very difficult to check (see the proof of Lemma~5) that the bounds on the norm of $\NN_{k-1}^{(j)}$ translate into exponentially decaying bounds for $j\mapsto \|\NN_{k-1,L}(z)^{(j)}\|$. Since $\chi_K\SS'_{k-1}\varphi_1\in \AA_{k-1, \BB_{k-1}}$ does not vanish, $(\Id-\MM_{k-1,L}(z))^{-1} \SS'_{k-1} \varphi_1\ne 0$. If $$ \hat \phi=d(\Id-\MM_{k-1,L}(z))^{-1} \SS'_{k-1} \varphi_1\ne 0 \, ,\tag{3.19} $$ then we are done. Indeed, $\hat \phi\in \Imm d\cap \AA_{k,\BB_k}$ would then be a nonzero fixed vector for $d\SS'_{k-1} \MM_{k,L}(z)$. Since the essential spectral radius of $d\SS'_{k-1} \MM_{k,L}(z)$ is smaller than one, is an easy algebraic exercise to see that the fixed vectors $d\SS'_{k-1} \MM_{k,L}(z)$ (if $|z|< 1/R$) are in bijection with the eigenvectors of $d\SS'_{k-1} \MM_{k}$ for the eigenvalue $1/z$ so that we are in the first case of the first claim. Note also at this point that if there exists $\hat\phi \in \AA_{n,\BB_n}$ with $d\SS'_{n-1}\MM_{n,L}(z) \hat\phi=\hat\phi$ then $\varphi_1:= d(\Id-\MM_{n-1,L}(z)) \SS'_{n-1} \hat\phi\in \AA_{n,\BB_n}$ would satisfy the first identity in \thetag{3.18}, while $\varphi_2$ may be defined by the second equality of \thetag{3.18}. In this case $\Det^\flat(\Id+ \DD'_{n-1,L}(z))= 0$ by Lemma~10. Let us assume that \thetag{3.19} does not hold. Writing $\upsilon=(\Id-\MM_{k-1,L}(z))^{-1} \SS'_{k-1}\varphi_1$ we would then have $ \upsilon \in \AA_{k-1, \BB_{k-1}}$ and $d\upsilon=0$ and, applying $\SS'_{k-1} d$ to both sides of $\SS'_{k-1}\varphi_1=d\SS'_{k-2}\upsilon-\MM_{k-1,L}(z) \upsilon$, $$ \SS' _{k-1}\varphi_1=-\SS'_{k-1} d \MM_{k-1,L} (z)\upsilon\, , $$ since $\SS'_{k-1}\varphi_1=(\Id-\MM_{k-1,L}(z)) \upsilon$, we find $(\Id-\MM_{k-1,L}(z))\upsilon=-\SS'_{k-1}d \MM_{k-1,L}(z)\upsilon$, contradicting $1/z \notin \sp \MM_{k-1}$ if $k=1$, and if $k\ge 2$ implying $$ d\SS'_{k-2}\upsilon=\upsilon= (\Id-\SS'_{k-1} d)\MM_{k-1,L} (z)\upsilon= d \SS'_{k-2} \MM_{k-1,L} (z)\upsilon \, . \, $$ For the converse (i.e., the last claim in Lemma~11), we see that if $\upsilon= d \SS'_{k-2} \MM_{k-1,L} (z)\upsilon$, we may take $\varphi_1=-d\MM_{k-1,L}\upsilon$ since then $(\Id-\MM_{k-1,L})^{-1}\SS'_{k-1}\varphi_1=-\upsilon$ so that $d(\Id-\MM_{k-1,L})^{-1}\SS'_{k-1}\varphi_1=0$. \qed \enddemo \smallskip \proclaim{Lemma 12 (Modified homotopy operators $\SS''$: perturbing nonintrinsic eigenvalues)} Assume that for some $ 2 \le k \le n-1$ there are $0\ne \tilde \varphi_k \in \AA_{k, \BB_{k }}$ and $z \in \complex$ with $|1/z| > R$ such that $$ \tilde \varphi_k = {1 \over z}d \SS'_{k-1} \MM_k\tilde \varphi_k \, , $$ as an isolated eigenvalue of finite geometric multiplicity. Then, there are $z'\ne z$, arbitrarily close to $z$, and two rank-one operators of arbitrarily small norm $$ \FF'_{\ell}: \AA_{\ell+1, \BB_{\ell+1}}\to \{ \varphi \in \AA_{\ell} \mid \chi_K \varphi \in \AA_{\ell, \BB_{\ell }} \} \, , \, \ell=k-1, k\, , $$ so that the perturbed operators $ \SS''_{k-1}=\SS'_{k-1}+\FF'_{k-1}$, %\, ,\quad $\SS''_{k}=\SS'_{k}+\FF'_{k}$, % \, ,\quad $\SS''_\ell = \SS'_\ell$, % \, , \quad $\ell\notin \{ k-1 , k\} $, still satisfy $\SS'' \SS''=0$, $\SS''d + d\SS''=1$ and, additionally, $1/z'$ is an eigenvalue of $d \SS''_{k-1} \MM_k\varphi$ on $\AA_{k, \BB_{k }}$ while $d \SS''_{\ell} =d\SS'_\ell$ for all $\ell\ne k-1$. \endproclaim \demo{Proof of Lemma 12} We take $\varphi'_{k-1} := \SS'_{k-1}\MM_k \tilde \varphi_k\in \AA_{k-1, \BB_{k-1}}$. By our assumptions, $d\varphi'_{k-1}\ne 0$. Let then $\nu'_{k}$ be a unit-norm continuous functional on $\AA_{k, \BB_{k}}$ which satisfies $$ \nu'_k \circ d = 0 \, , \quad \alpha:=\nu'_k(\MM_k \tilde \varphi_k)\ne 0 \, . $$ (Take $\nu'_k (\varphi)= \nu'_{k+1}(d\varphi)$ with $\nu'_{k+1}$ continuous on $\AA_{k+1, \BB_{k+1}}$ and $\nu'_{k+1}(d\MM_k \tilde \varphi_k)\ne 0$.) We set, for small complex $\epsilon$, $$ \eqalign { \FF'_{k-1} (\varphi)= \epsilon \varphi'_{k-1} \cdot \nu'_{k}(\varphi) \, , \quad \FF'_{k} (\varphi)= -\epsilon d \varphi'_{k} \cdot \nu'_{k}(\SS_k \varphi)\, . } $$ We have $d\FF'_{k-1}=-\FF'_k d$, $\FF'_{k-1}d=0$ and $d \FF'_k=0$ so that $d\SS''+\SS''d=\Id$. Also, $\FF'_k \SS'_{k+1}=0$, and $\SS'_{k-1}\FF'_k + \FF'_{k-1} \SS'_k + \FF'_{k-1}\FF'_k =0$, guaranteeing $\SS''\SS''=0$. Clearly, $d\FF'_k =0$ so that $d\SS''_k=d\SS'_k$. Finally, $$ z d \SS''_{k-1} \MM_k \tilde \varphi_k= \tilde \varphi_k + \epsilon z \cdot (d\SS'_{k-1}\MM_k \tilde \varphi_k) \cdot \nu'_k (\MM_k \tilde \varphi_k) = (1+ \alpha \epsilon) \tilde \varphi_k \, . \hbox{\qed} $$ \enddemo %\smallskip \pagebreak {\bf Main Result} \smallskip \proclaim{Theorem 13} Let $\psi_\omega$, $g_\omega$ satisfy the assumptions of Section ~2, let $\BB_{k,t}$ and $R$ satisfy Axioms~ 1 and ~2--3. Then $\Det^\#(\Id-z\MM)$ is meromorphic in the disc $\{ |z| < 1/ R\}$. The order of $z$ as a zero/pole of $\Det^\#(\Id-z\MM)$ coincides with the sum of the algebraic multiplicity of $1/z$ as an eigenvalue of the $\MM_{2k}$ on $\AA_{2k, \BB_{2k}}$, for $0 \le 2k \le n$, minus the sum of the algebraic multiplicity of $1/z$ as an eigenvalue of the $\MM_{2k+1}$ on $\AA_{2k+1, \BB_{2k+1}}$ for $1 \le 2k+1 \le n$. \endproclaim \demo{Proof of Theorem 13} Perturbing our family, we may assume that all eigenvalues of moduli $> R$ of the $\MM_k$, $k=0, \ldots, n-1$, are simple, and none of their eigenvectors belong to $\Ker d$ or $\Ker \NN_k$. We may also assume that the eigenvalues of $\MM_n$ are simple. We apply Lemma ~9 to construct adapted homotopy operators $\SS'$ and let $\DD'_{k,\ell}(z)$ be the kneading operators from \thetag{3.16}, for large enough $\ell\ge L$. We consider the modified equality \thetag{3.10} from Theorem~2 for the perturbed family and the adapted homotopy operators, using Lemma~10 to view it as an alternated product of meromorphic functions in the disc of radius $1/R$. (Later in the proof we shall use further finite rank perturbations of the $\SS'$ given by Lemma ~12 and thus satisfying the assumptions of Lemma~8.) By Lemma 10, the determinant $\Det^\flat (\Id+\DD'_{2k,\ell}(z))$ for $0\le 2k\le n-1$ (which appears in the denominator) has poles in the disc of radius $1/R$ only at the inverse eigenvalues of $\MM_{2k}$, with order exactly one. We have the same statement for the $\Det^\flat (\Id+\DD'_{2k+1,\ell}(z))$ for $1\le 2k+1\le n-1$, which appear in the numerator. Lemma 11 also says that if $\Det^\flat (\Id+\DD'_{k-1,\ell}(z))$ for $1\le k \le n$ (in the denominator for even $k-1$ and in the numerator for odd $k-1$) vanishes in this disc then $1/z$ is an eigenvalue of $d \SS'_{k-1}\MM_{k}$ or $k \ge 2$ and $1/z$ is an eigenvalue of $d \SS'_{k-2}\MM_{k-1}$. Also, whenever $1/z$ is an eigenvalue of $d \SS'_{k-2}\MM_{k-1}$ then $\Det^\flat (\Id+\DD'_{k-1,\ell}(z))=0$, and if $1/z$ is an eigenvalue of $d\SS \MM_n=\MM_n$ then $\Det^\flat (\Id+\DD'_{n-1,\ell}(z))=0$. To finish the proof, we will show that the zeroes of the flat determinants must cancel in the alternated product, except of course for the zeroes of $\Det^\flat (\Id+\DD'_{n-1,\ell}(z))$ (in the denominator if $n-1$ is even, in the numerator if $n-1$ is odd) corresponding to $1/z$ being an eigenvalue of $\MM_n$. Assume for a contradiction that $\Det^\#(1-\hat z\MM)= 0$ to order $D$ strictly larger than the value claimed in Theorem~13, due (at least in part) to a factor $\Det^\flat (\Id+\DD'_{\hat k-1,\ell}(\hat z))=0$ vanishing to order $\hat D\ge 1$ for some odd $0\le \hat k-1\le n-1$. (The case of poles and even $k-1$ is dealt similarly.) Lemma~11 says that $1/\hat z$ is then either an eigenvalue of $d\SS'_{k-2}\MM_{\hat k-1}$ (since $\hat k\ge 2$) or an eigenvalue of $d\SS'_{\hat k-1}\MM_{\hat k}$ (and $\hat k < n$). If $1/\hat z$ is an eigenvalue of $d\SS'_{k-2}\MM_{\hat k-1}$ (the other case is left to the reader), we may use Lemma~12 to perturb this eigenvalue to some $1/\hat z'\ne 1/\hat z$. Note that this does not modify the $\Det^\flat (\Id+\DD'_{k-1,\ell}( z))$ except for $k= \hat k$ and $k=\hat k+1$. (For both of these determinants, only the set of zeroes may change, and only one of the determinants is in the numerator.) The perturbation $\DD''_{\hat k-1,\ell}(z)$ of $\DD'_{\hat k-1,\ell}(z)$ may be made as small as desired in (finite-rank) operator norm in a neighbourhood of $\hat z$ by taking small enough $\epsilon\ne 0$ in Lemma ~12. Since $\hat z\ne \hat z'$, the continuous dependence of the regularised determinant on the operator together with Rouch\'e's Theorem guarantee that the order of $\hat z$ as a zero of $\Det^\flat (\Id+\DD''_{\hat k-1,\ell}(z))$ is strictly smaller than $\hat D$. Iterating this procedure (or the procedure associated to the other case) at most $\hat D$ times, we get that $\Det^\flat (\Id+\DD'''_{\hat k-1,\ell}(\hat z))\ne 0$, while none of the zeroes of the other kneading determinants (in the numerator) or the poles (in the denominator or in fact also the numerator) have been altered. Then, either the order of $\hat z$ as a zero of $\Det^\#(1- z\MM)$ is strictly smaller than $D$, a contradiction, or $\Det^\flat (\Id+\DD'''_{k-1,\ell}(\hat z))=0$ for some odd $k-1\ne \hat k-1$. In the second case, we may proceed as above to ensure $\Det^\flat (\Id+\DD''''_{k-1,\ell}(\hat z))\ne 0$ and eventually obtain the contradiction that the order of $\hat z$ as a zero of $\Det^\#(1- z\MM)$ is strictly smaller than $D$. \qed \enddemo \noindent A corollary of the proof of Theorem~13 is (we do not have to replace $\widetilde R$ by $\widetilde R^{1/n}$): \proclaim{Theorem 14} Assume Axioms~1 and 2. For each large enough $L$, the following alternated product of regularised determinants extends holomorphically to $\{ |z|< 1/\widetilde R\mid 1/z \notin \cup_k \sp(\MM_k)\}$: $$ \prod_{k=0}^{n-1} \Det^{\hbox{reg}}_{[n/2]+1}(\Id+\DD_{k,L}^{(r)}(z)) ^{(-1)^{k+1}} \, . \tag{3.20} $$ \endproclaim %should be checked once more (order poles etc: e.g. is %the proof of Lemma 10 (2) ok to show order of singularity % of regularised determinant is AT LEAST %one without assuming Axiom 3? Although $ \Det^{\hbox{reg}}_{[n/2]+1}(\Id+\DD_{k,L}^{(r)}(z)) =\exp -\sum_{\ell = [n/2]+1}^\infty {z^\ell\over \ell}\tr^\flat (\DD_{k,L}(z))^\ell $, it does not seem easy to relate \thetag{3.20} to a dynamical zeta function. \smallskip \head 4. Application to expanding maps on compact manifolds \endhead In this section, we apply Theorem~13 from Section~ 3 to $C^{r}$ (locally) expanding endomorphisms on compact manifolds and $C^r$ weights, giving a new (and completely different) proof of a result of Ruelle [Ru3], avoiding Markov structures. We do not recover the full strength of his statement except if the dynamics is $C^\infty$ and if the weight is the inverse Jacobian (with respect to Lebesgue). \smallskip Let us start by stating precisely this result. Let $M$ be a $C^{r}$ ($r\ge 1$) compact manifold of dimension $n\ge 2$, and let $f:M\to M$ be $C^{r}$ and (locally) uniformly expanding, that is, there is $\theta <1$ such that $\|D_a f(\xi)\|\ge\theta^{-1}\|\xi\|$ for all $\xi$ in the tangent space $T_a M$ at $a$, where $\|\cdot\|$ is the Euclidean norm on $T_a M$. Let $g:M\to\complex$ be $C^r$. The transfer operator $\LL_0=\LL_{f,g}$, acting on the Banach space of $C^r$ functions $M\to\complex$, is given by $$ \LL_0\phi(a)=\sum_{b:f(b)=a}g(b)\phi(b) =\sum_j g_j(a) \phi(\psi_j (a)) \, , \tag{4.1} $$ where the $\psi_j$ are the finitely many local inverse branches of $f$. Similarly, we can introduce operators acting on Banach spaces of $k$-forms on $M$ with $C^m$ coefficients, for $0\le m \le r-1$, putting $ \LL_k\phi=\sum_{j}g_j \cdot (\psi_j^*) \phi $. An expanding map is transversal, and the Lefschetz numbers of the inverse map at the periodic orbits are all positive, so that the Ruelle zeta function associated to $f$,~ $g$ $$ \zeta_{f,g}(z)=\exp \sum_{m=1} ^\infty{z^m\over m} \sum_{a \in \Fix f^m} \prod_{\ell=0} ^{m-1} g(f^\ell(a)) $$ can be viewed as a Lefschetz-Ruelle zeta function. Ruelle proved: %, Thm. 1.3, Cor. 1.5--1.6 \proclaim{Theorem (Ruelle [Ru3])} Let $P\in \real$ be the topological pressure of $\log|g|$ and $f$. \noindent (1) The spectral radius of $\LL_0$ acting on $C^m$ is at most $e^P$ while its the essential spectral radius is at most $\theta^{m}e^P$ for $0\le m \le r$. The spectral radius of $\LL_k$ acting on forms with $C^m$ coefficients is at most $\theta^{k}e^P$ while its the essential spectral radius is at most $\theta^{m+k}e^P$, for $1 \le k\le n$ and $0\le m \le r-1$. \noindent (2) The power series $\zeta_{f,g}(z)$ defines a meromorphic function in the disc of radius $\theta^{-r}e^{-P}$. In this disc, the order of $z$ as a zero/pole of $\zeta_{f,g}(z)$ coincides with the sum of the algebraic multiplicity of $1/z$ as an eigenvalue of the $\LL_{2k+1}$ acting on $C^m$ for any $r-(2k+1)\le m \le r-1$ and $1 \le 2k+1 \le n$, minus the sum of the algebraic multiplicity of $1/z$ as an eigenvalue of the $\LL_{2k}$ acting on $C^m$ for $r-2k\le m \le r-1$ and $0 \le 2k \le n$ (for $k=0$ one can also let $\LL_0$ act on $C^r$). \endproclaim {\it We will recover Ruelle's result if the system is in fact $nr$ times differentiable, with the same estimates if $g(y)=1/| \det Df (y)|$, (i.e., the weight giving rise to the absolutely continuous invariant measure), and only a weaker result, replacing $e^P$ by $\theta^{-n+1} e^P$, in the case of an arbitrary smooth weight.} There is certainly room for improvement here. \smallskip Once the first claim of the above theorem is established (see [GuLa] for better estimates), Ruelle [Ru3] associates to each $\LL_k$ a Fredholm-like (flat) dynamical determinant $$ d_k(z)=\Det^\flat (\Id - z\LL_k)= \exp -\sum_{m=1} ^\infty{z^m\over m} \sum_{a \in \Fix f^m} \prod_{\ell=0} ^{m-1} { g(f^\ell(a))\cdot \hbox{Tr}\, \Lambda^k (D_a f^{-m} ) \over \det(\Id - D_a f^{-m}) }\, ,\tag{4.2} $$ and proves that $d_k(z)$ is holomorphic in the disc of radius $\theta^{-k-r}e^{-P}$, where its zeroes correspond exactly the the inverse eigenvalues of $\LL_k$ (outside of the disc of radius $\theta^{k+r}e^{P}$). Writing $\zeta_{f,g}(z)$ as an alternated product of the $d_k(z)$ gives the second claim. The present approach does not allow us to analyze the independent factors $d_k(z)$. However, since the spectral radii of the operators $\LL_k$ are strictly decreasing, the annulus $e^{-P}\le|z|<\theta^{-1}e^{-P}$, e.g., only contains inverse eigenvalues of $\LL_0$ acting on $C^r(M)$. \smallskip {\bf From a manifold to $\real^n$ -- Equivalent models} \smallskip We wish to associate to $f$ and $g$ data $\{\psi_\omega,g_\omega\}_{\omega\in\Omega}$ in such a way that the operators $\LL_k$ and $\MM_k$ are conjugated. Their spectra on suitable spaces will thus coincide. For this, first choose a $C^r$ atlas $\{V_j\}$ for $M$ such that $f|_{V_j}\to f(V_j)$ is a diffeomorphism. Choose a $C^r$ partition of the unity $\{\chi_j\}$, where each $\chi_j$ is supported in $V_j$. Denote by $\psi_j:f(V_j)\to V_j$ the inverse map of $f|_{V_j}$. Since $f(V_j)$ is not necessarily contained in some $V_i$, we refine the cover $V_j$ by putting $V_{ji}=f(V_i)\cap V_j$ and set $\psi_{ji}=\psi_j|_{V_{ji}}:V_{ji}\to\psi_j(V_{ji})\subset V_i$. We choose for each $j$ a $C^r$ partition of the unity $\{\chi_{ji}\}$ on $V_j$ such that each $\chi_{ji}$ is supported in $V_{ji}$. Finally, we set $ g_{ji}(a)=\chi_i(\psi_j(a))\cdot\tilde{\chi}_{ji}(a)\cdot g_j(a) $ (note that $g_{ji}$ is compactly supported in $V_{ji}$), and we may rewrite our operator as $$ \LL_0\phi(a)=\sum_{j,i} g_{ji}(a)\cdot\phi(\psi_{ji}(a)) \, . \tag{4.3} $$ The operators $\LL_k$ acting on $k$-forms on $M$ are similarly defined (replacing the composition with $\psi_{ji}$ by the pullback). We next choose charts $\pi_j:U_j\to V_j$, where the $U_j$ are bounded and two-by-two disjoint open subsets of $\real^n$. We denote by $\pi$ the map $\pi|_{U_j}=\pi_j$ and set $U_{ji}=\pi_j^{-1}(V_{ji})$. Let now $\varphi$ be a form in $U=\cup_j U_j$. We set $(\tau\varphi)(a)=\sum_i\chi_i(a)\cdot(\pi_i^{-1})^*\varphi(a)$. Then, $\tau\pi^*$ is the identity on $C^r$ forms in $M$ (in particular $\pi^*$ is injective). We now define $\MM_k$ acting on $k$-forms in $U\subset\real^n$ as $\MM_k=\pi^*\LL_k\tau$, that is $$ \MM_k\varphi(x)=\sum_{ji}g_{ji}(\pi(x)) \chi_i(\psi_{ji}\circ\pi(x))\cdot(\pi_i^{-1}\circ\psi_{ji}\circ\pi)^*\varphi(x) \, . \tag{4.4} $$ It is easy to choose $\Omega$, $\psi_\omega$ and $g_\omega$ in order to view $\MM_k$ as an operator of the form \thetag{2.3}, with $g_\omega$ compactly supported in $U_\omega$. \smallskip {\bf Sobolev spaces} \smallskip We shall work with the Bessel potential $\JJ_1$, defined on $L^{q}(\real^n)$ by (see [St, Chapter ~V.3], $\Gamma(\cdot)$ denotes the Euler gamma function): $$ \JJ_1 (\varphi)(x)={1 \over \Gamma(1/2)\sqrt{4\pi}} \int_{\real^n} \int_0^\infty e^{-\pi |y|^2/t} e^{-t/4\pi} t^{(-n+1)/2} {dt \over t} \, \varphi(x-y) \, dy \, . $$ Write $\JJ_\ell$ for the $\ell$th iterate of $\JJ_1$. It is a well-known (see [St, V.3.3-3.4]) and important result in the theory of Sobolev spaces that for all $1 < q < n$ and each $\ell \ge 1$, $\JJ_\ell$ is an isomorphism from the Sobolev space $L^q_m(\real^n)=W^{m,q}(\real^n)$ to the Sobolev space $L^q_{m+\ell}(\real^n)$ if $1< q < \infty$, $m$ is a nonnegative integer, and $\ell \ge 1$ [St, V.3.2-3.3]. In particular, $\JJ_1^{-1} \varphi\in L^q_{m-1}(\real^n)$ if $\varphi \in L^q_m$ and $m \ge 1$. We take $\widetilde \JJ_k=\JJ_{r-k}$. \smallskip We define the Sobolev spaces $W^{m,p}(M)$ [Ad] of the manifold $M$ using our chart $\pi:U\to M$ (by compactness, other chart systems yield equivalent norms): $$ W^{m,p}(M):=\{ \phi:M\to\complex\, |\, \phi\circ\pi \in W^{m,p}(U)\}. $$ For $\phi\in\AA_{k,W^{m,p}(M)}$, set $||\phi||_{\AA_{k,W^{m,p}(M)}}:= ||\pi^*u||_{\AA_{k,W^{m,p}(U)}}$. Clearly, $\LL_k$ is bounded on $\AA_{k,W^{m,p}(M)}$ for all $00$ and a sequence $\{\rho_\ell\}$ of positive real numbers so that for all $\ell$ and $\varphi\in W^{m,t}(K')$, $$ ||\MM_k^\ell\varphi||_{m,t}\le \rho_\ell ||\varphi||_{m-1,t} + C\cdot\theta^{\ell m }||(\MM_k^+)^{\ell }||_t\cdot ||\varphi||_{m,t}\, . \tag{4.5} $$ \endproclaim \demo{Proof of Lemma 15} The norm $\|\cdot \|_{m,t}$ on $W^{m,t}(K')$ is given by $ \left( \sum_{|\alpha|\le m} \|D^\alpha \varphi \|_t^t\right)^{1/t} $, where $\|\cdot \|_t$ is the $L^t(K')$ norm, $\alpha=(\alpha_1,\dots,\alpha_p)$ is a multi-index of size $|\alpha|=p$ and $D^\alpha=\partial_{\alpha_1}\cdots\partial_{\alpha_p}$. We begin with $\ell=1$ and $k=0$. Let $\alpha$ be a multi-index with $|\alpha|= m$. The Leibnitz rule gives $D^\alpha \MM_0\varphi=\sum_\omega g_\omega D^\alpha(\varphi\circ\psi_\omega) + H_\alpha\varphi$, where $H_\alpha\varphi$ is a sum of terms $$ D^{\beta}g_\omega\cdot D^{\beta'} \left ( \varphi \circ \psi_\omega\right )\, , \tag{4.6} $$ with $|\beta|\le |\alpha|$, $|\beta'|\le |\alpha|-1$. The data being at least $C^m$, all terms \thetag{4.6} are bounded by a constant multiple of $||\varphi||_{m-1,t}$. Now, $D^\alpha(\varphi \circ\psi_\omega)$ is a sum of, on the one hand, terms involving derivatives of $\varphi$ of order $\le|\alpha|-1 $ (which are also bounded by a constant multiple of $||\varphi||_{m-1,t}$), and, on the other hand, terms of the form ($(\psi_\omega)_j$ denotes the $j$-th coordinate of $\psi_\omega$) $$ \sum_\omega g_\omega\cdot \bigl(\prod_{i=1}^m\partial_{\alpha_i} ((\psi_\omega)_{\gamma_i})\bigr)\cdot \left( D^\alpha \varphi\right)\circ\psi_\omega \, . \tag{4.7} $$ The $L^t$ norm to the power $t$ of \thetag{4.7} is $\le\theta^{mt}||\MM_0^+(D^\alpha \varphi)||_t^t$, by the contraction assumption. Since $\varphi\in W^{m,t}(K')$, $||D^\alpha\varphi||_t\le||\varphi||_{m,t}$. The number of terms \thetag{4.7} depends only on $m$ and $n$. The result for $\ell=1$ follows by summing over $\alpha$ with $|\alpha|\le m$ (if $|\alpha|\le m-1$, terms of the form \thetag{4.7} are also bounded by a constant multiple of $||\varphi||_{m-1,t}$). For $\ell>1$, just use that $\|(\MM_0^\ell)^+\|_t\le \|(\MM_0^+)^\ell\|_t$. The claims for $k\ge 1$ are obvious by the contraction property of the pullback. \qed \enddemo \proclaim{Lemma 16 (Axiom 2)} Let $1 < t < \infty$. \noindent (1) $\rho_{ess}(\MM_0|_{W^{m,t}(K')})\le \theta^{m} \rho(\MM^+_0|_{L^{t}(K')})$ for all $0\le m\le \hat r$. \noindent (2) For all $0\le m\le \hat r-1$ we have $\rho(\MM_k|_{\AA_{k,L^{t}(K')}})\le \theta^{k}\rho(\MM^+_0|_{L^{t}(K')})$ and $$ \rho(\MM_k|_{\AA_{k,W^{m,t}(K')}}) \le \theta^k \rho(\MM^+_0|_{L^{t}(K')}) \, , \,\, \rho_{ess}(\MM_k|_{\AA_{k,W^{m,t}(K')}})\le \theta^{m+k} \rho(\MM^+_0|_{L^{t}(K')})\, . $$ \noindent (3) $\rho(\MM^+_0|_{L^{t}(K')})\le 1 = e^P$ if $g_\omega = |\det D\psi_\omega|$. Otherwise $\rho(\MM^+_0|_{L^{t}(K')})\le \theta ^{-n/t} e^P$. The spaces $\BB_{k,t}=W^{\hat r-k,t}(K')$ for $k=0, \ldots, n$, together with the operators $\MM_k$, satisfy Axiom ~2 for $\widetilde R=\theta^{\hat r}$ if $g_\omega=|\det D\psi_\omega|$, and (after renormalising so that $e^{P(\log |g|}=1$) and $\widetilde R> \theta^{\hat r-(n-1)}$ otherwise (if $t$ is not much smaller than $n/(n-1)$). \endproclaim \demo{Proof of Lemma 16} For the first claim, since $W^{m-1,t}(K')$ is compactly embedded in $W^{m,t}(K')$ [Ad], we may combine Lemma~ 15 and the Hennion formula ([He]) to deduce $ \rho_{ess}(\MM_0|_{\AA_{0,W^{m,t}(K')}})\le \theta^m\rho(\MM^+_0|_{\AA_{0,L^t(K')}}) $. The inequalities in the second claim are then obvious, by the action of the pullback. To prove the third claim, it is more convenient to estimate the spectral radius of $\LL_0^+$ acting on $L^t(M)$ (see [MS] for an analogous bound in a more specific situation). The H\"older inequality for finite sums gives, for $1 < t' < t$ so that $t^{-1}+(t')^{-1}=1$, and writing $\LL_{0,t'}$ for the operator associated to $|\det (D\psi_\omega )|\cdot (|g_\omega|/|\det D\psi_\omega|)^{t'}$ $$ \align &\|(\LL^+_0)^\ell \varphi \|^t_{L^t(M)} = \int_{M} \bigl ( \sum_{\vec \omega \in\Omega^\ell} \prod_{j=1}^\ell |g_{\omega_j} \circ\psi^{j-1}_{\vec \omega}| \cdot |\varphi\circ \psi^\ell_{\vec \omega}| \bigr )^t \, dx\cr &\quad \le \int_{M} \biggl ( \sum_{\vec \omega \in\Omega^\ell} |\varphi\circ \psi^\ell_{\vec \omega}|^t \cdot|\det D\psi^\ell_{\vec \omega}| \biggr ) \biggl ( \sum_{\vec \omega \in\Omega^\ell} \bigl ( \prod_{j=1}^\ell |g_{\omega_j} \circ\psi^{j-1}_{\vec \omega}| \cdot |\det D\psi^\ell_{\vec \omega}|^{-1/t} \bigr )^{t'} \biggr )^{t/t'} \, dx \cr \allowdisplaybreak &\quad \le \bigl ( \int_{M} |\varphi|^t \, dx \bigr ) \cdot \sup_M \biggl ( \sum_{\vec \omega \in\Omega^\ell} |\det D\psi^\ell_{\vec \omega}| \cdot \bigl ( {\prod_{j=1}^\ell |g_{\omega_j} \circ\psi^{j-1}_{\vec \omega}| \over |\det D\psi^\ell_{\vec \omega}|} \bigr )^{t'} \biggr )^{t/t'} \, dx \cr &\quad \le \|\varphi\|^t_{L^t}\cdot \bigl ( \sup_M ((\LL_{0,t'})^\ell (1)) \bigr )^{t/t'}\, , \qquad\forall \ell \, , \forall \varphi \in L^t(M) \, . \tag{4.8} \endalign $$ Taking the $t$-th root of the above inequality implies $$ \rho(\MM_0|_{L^t(K')}) =\rho(\LL_0|_{L^t(M)}) \le 1=e^{P(\log g)} \hbox{ if } g_\omega=|\det D\psi_\omega| \, . $$ If $g_\omega\ne |\det D\psi_\omega|$, we get $$ \rho(\LL^+_0|_{L^t(M)}) \le \lim_{\ell \to \infty} \bigl (\sup_M \bigl ( \sum_{\vec \omega \in\Omega^\ell} |\det D\psi^\ell_{\vec \omega}|^{-1+1/t'} \cdot \prod_{j=1}^\ell |g_{\omega_j} \circ\psi^{j-1}_{\vec \omega}| \bigr ) \bigr )^{1/\ell} \le \theta^{-n/t}\cdot e^{P(\log |g|)}\, . $$ This immediately implies Axiom~2 for the $\MM_k$ and the claimed value of $\widetilde R$. \qed \enddemo The properties in Axiom 3 hold by uniform contraction and bounded distortion. \smallskip {\bf Completing the new proof of Ruelle's theorem} \smallskip The proofs of Lemmas~15--16 adapt to the $C^m$ setting, giving for all $f$, $g$ the well-known bound for all $0\le m \le \hat r-1$ ($0\le m\le \hat r$ if $k=0$) $$ \rho_{ess}(\LL_k|_{\AA_{k,C^{m}(M)}})= \rho_{ess}(\MM_k|_{\AA_{k,C^{m}(\overline{K'})}})\le \theta^{m+k}e^P \, . $$ Indeed, when proving the Lasota-Yorke inequality, we may work with $\MM_0$ acting on $C^0$ instead of $L^t$ and the extraneous $\theta^{-n/t}$ factor does not appear. 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