0$ so that \b \sup_{\kappa}\mid\mid Op_{\kappa}^{AW}(f)\mid\mid_{{\cal L}(\hn)}\leq C \e Theorem XIII.83 in [RS] then implies that the right hand side of (3.12) is well defined. To show (3.15) note simply that for all $\phi$ and $\psi$ in $\hn$ $$ <\phi,Op_{\kappa}^{AW}(f)\psi>_{(\kappa,N)} \leq\mid\mid\phi\mid\mid_{(\kappa,N)}\mid\mid\psi\mid\mid_{(\kappa,N)}\mid\mid f\mid\mid_{\infty} \int_{T^{(2)}}\frac{dqdp}{2\pi\hbar}. $$ Since, as is easily checked, $$ is continuous in $\kappa$, (3.15) follows. To prove (3.12) it suffices to compute $Op^{AW}(f)\psi$ for $\psi\in\so$. Using the periodicity of $f$ and Proposition 2.2 (iii), we have, writing $\psi_{\kappa}=S(\kappa)\psi$ \baa Op^{AW}(f)\psi&=&\bd\sum_{m,n}\ed\int_{0}^{a} \int_{0}^{b}\frac{dqdp}{2\pi\hbar}f(q,p)\mid q+na,p+mb,z>\\ & &\times\int_{0}^{\frac{2\pi}{a}}\int_{0}^{\frac{2\pi}{b}}dm(\kappa)_{(\kappa,N)}.\\ \eaa \noindent Since, as a simple calculation shows, $U(na,mb)S(\kappa)=S(\kappa)U(na,mb)$ and using (3.10), \baa \mid q+na,p+mb,z,\kappa>&=&S(\kappa)U(na,mb)e^{\frac{i}{2\hbar}(nap-mbq)}\mid q,p,z>\\ &=&(-1)^{Nmn}e^{-i(\kappa_{1}na-\kappa_{2}mb)} e^{\frac{i}{2\hbar}(nap-mbq)}\mid q,p,z,\kappa>. \eaa Then, one obtains using (2.6) \baa Op^{AW}(f)\psi&=&\int_{0}^{\frac{2\pi}{a}}\int_{0}^{\frac{2\pi}{b}}dm(\kappa) \int_{0}^{a}\int_{0}^{b}\frac{dqdp}{2\pi\hbar}f(q,p)\bd \sum_{m,n}\ed(-1)^{Nmn}e^{+i(\kappa_{1}na-\kappa_{2}mb)}\\ & &U(na,mb)\mid q,p,z>_{(\kappa,N)}\\ &=&\int_{0}^{\frac{2\pi}{a}}\int_{0}^{\frac{2\pi}{b}} dm(\kappa)Op_{\kappa}^{AW}(f)\psi_{\kappa}. \eaa This proves (3.12). Note that (iii) is a special case of (ii) for $f=1$ and that (iv) follows from (iii).\ \fin \vskip10pt \noindent{\bf Remark:\ } Using the resolution of the identity, one can define an isometric map $$ W(\kappa,z):\psi\in\hn\longrightarrow W(\kappa,z)\psi\in L(T^{(2)},\frac{dqdp}{2\pi\hbar}) $$ by $$ (W(\kappa,z)\psi)(q,p)=. $$ The image of $\hn$ under $W(\kappa,z)$ is a reproducing kernel subspace of $\L(T^{(2)},\frac {dqdp}{2\pi\hbar})$. In the language of geometric quantization, it is the space of square integrable polarized sections of a prequantum line bundle over $T^{(2)}$ determined by $\kappa$ and $N$ [DBDEG]. Here the polarization is K\"ahlerian and determined by the complex vector field $X_{z}=\partial_{q}+z\partial_{p}$ on $T^{(2)}$. The function $\mid<\cdot,\cdot,z,\kappa\mid\psi>\mid^{2}$ on $T^{(2)}$ is referred to as the ``Husimi distribution of $\psi$" in the physics literature. For the proof of the equipartition of the eigenfunctions we need to compare the Weyl and the anti-Wick quantizations as $N\rightarrow\infty$. The simple estimate needed is given by the following lemma. \begin{lem} For all $f\in C^{\infty}(T^{(2)})$ \b \mid\mid Op_{\kappa}^{W}(f)-Op_{\kappa}^{AW}(f)\mid\mid_{{\cal L}(\hn)}=O(N^{-1})\hskip15pt (N\rightarrow\infty). \e \end{lem} \pf Thanks to (3.2), (3.12) and Theorem XIII.83 in [RS] , we have $$ \sup_\kappa\mid\mid Op_{\kappa}^{W}(f)-Op_{\kappa}^{AW}(f)\mid\mid_{{\cal L}(\hn)}=\mid\mid Op^{W}(f)-Op^{AW}(f)\mid\mid_{{\cal L}(\L^{2}(\R))}. $$ It is a standard result [GL] [HMR] that $$ \mid\mid Op^{W}(f)-Op^{AW}(f)\mid\mid_{{\cal L}(\L^{2}(\R))}=O(N^{-1})\hskip15pt (N\rightarrow\infty). $$ Thus (3.16) is proved.\fin \vskip10pt We end this section with a semi-classical estimate on the Weyl quantization, crucial for the proof of the equipartition result in section 5. \begin{pro} For all $f\in C^{\infty}(T^{(2)})$ \b \frac{1}{N}TrOp_{\kappa}^{W}(f) \buildrel N\rightarrow\infty \over \longrightarrow \int_{0}^{a}\int_{0}^{b}\frac{dqdp}{ab}f(q,p). \e \end{pro} \pf Using the basis defined above, $$ \frac{1}{N}TrOp_{\kappa}^{W}(f) =\frac{1}{N}\sum_{j=0}^{N-1}\sum_{n,m} f_{n,m}. $$ By (2.10) and (2.11), we have $$ \frac{1}{N}TrOp_{\kappa}^{W}(f) =\frac{1}{N}\sum_{j=0}^{N-1}\sum_{l,n} \sum_{r=0}^{N-1}f_{n,lN+r}e^{i\frac{\pi}{N}(lN+r)n} e^{-il\kappa_{1}a}e^{i\frac{2\pi}{N}n(\frac{b\kappa_{2}}{2\pi}+j+r)} _{(\kappa,N)}. $$ Then, since $\{e_{j}^{\kappa}\}_{j=0}^{N-1}$ is an orthonormal basis, \baa \frac{1}{N}TrOp_{\kappa}^{W}(f) &=&\bd\sum_{l,n}\ed f_{n,lN}(-1)^{ln} e^{-il\kappa_{1}a}e^{i\frac{2\pi}{N}n\frac{b\kappa_{2}}{2\pi}} \frac{1}{N}\sum_{j=0}^{N-1}e^{i\frac{2\pi}{N}nj}\\ &=&f_{0,0}+\bd\sum_{l\in\Z^{*}}\ed f_{0,lN}e^{-il\kappa_{1}a}. \eaa The result follows using the regularity of $f$.\fin \sect{Translations and skew translations} In this section, we study two particular maps which, as said in the introduction, are undoubtedly the simplest ergodic transformations of the torus: the irrational translations and the skew translations. We will set $a=b=1$ for convenience. The translations are denoted by $\tau_{\alpha}$ where $\alpha=(\alpha_{1},\alpha_{2})\in T^{(2)}$. If $\alpha$ is such that $\alpha_{1}/\alpha_{2}\not\in\Q$, $\tau_{\alpha}$ is said to be irrational. They act on $T^{(2)}$ as $\tau_{\alpha}:(q,p)\in T^{(2)}\rightarrow(q+\alpha_{1},p+\alpha_{2})\in T^{(2)}$. The skew translations are denoted by $\Phi_{\beta}^{k}$ where $\beta\not\in\Q, k\in\Z$; $\Phi_{\beta}^{k}$ is defined as $\Phi_{\beta}^{k}=\tau_{(0,\beta)}\circ K$ with $K\in SL(2,\Z)$ of the form $K=\pmatrix{1 & k\cr 0 & 1}$. Both $\tau_{\alpha}$ and $\Phi_{\beta}^{k}$ are uniquely ergodic area-preserving maps [CFS], meaning that there exists a {\em unique} invariant probability measure for them (the Lebesgue measure). A well known and useful fact concerning uniquely ergodic maps $\Phi$ on a compact metric space $X$ is that for each continuous function $f$ on $X$ \b \frac{1}{T}\sum_{k=1}^{T}f\circ\Phi^{k} \buildrel T\rightarrow\infty \over \longrightarrow m(f)\hskip15pt in\;\;\;L^{\infty}(X) \e where $m$ is this unique invariant probability measure [M] [CFS]. We will see below how this permits a particularly simple proof of the equipartition property. We first need to show how to associate to each $\tau_{\alpha}$ and $\Phi_{\beta}^{k}$ a unitary operator $M_{\kappa}(\tau_{\alpha})$ and $M_{\kappa}(\Phi_{\beta}^{k})$ on $\hn$. We treat $\tau_{\alpha}$ first. Translations and skew translations were already quantized in [DBDEG] using ideas from geometric quantization. We proceed slightly differently here. A natural way to associate a unitary operator to $\tau_{\alpha}$ would have been to choose the translation operator $U(\alpha_1,\alpha_2)=e^{\frac{i}{\hbar}(\alpha_2 Q-\alpha_1 P)}$, restricted to $\hn$. But as we saw in section 2, no $\hn$ is preserved under these operators, unless $\alpha=(\frac{m}{N},\frac{n}{N})$ for some $n,m\in \N$. We therefore need a different approach to quantize irrational translations. Recall first that $$ U(\alpha_1,\alpha_2)=e^{i\frac{\alpha_1\alpha_2}{2\hbar}} U(\alpha_1,0)U(0,\alpha_2). $$ This suggests quantizing $\tau_{(\alpha_1,0)}$ and $\tau_{(0,\alpha_2)}$ separately yielding $M_{\kappa}(\tau_{(\alpha_1,0)})$ and $M_{\kappa}(\tau_{(0,\alpha_2)})$ and then defining \b M_{\kappa}(\tau_{\alpha})= e^{i\frac{\alpha_1\alpha_2}{2\hbar}} M_{\kappa}(\tau_{(\alpha_1,0)})M_{\kappa}(\tau_{(0,\alpha_2)}). \e This is how we will proceed. Recall now from (2.11) that $$ U(0,\frac{n}{N})e_{j}^{\kappa}=e^{i(\kappa_{2}+2\pi j)\frac{n}{N}}e_{j}^{\kappa}. $$ In other words, the $e_{j}^{\kappa}$, $j\in\{0,...,N-1\}$ are a basis of eigenvectors for the translations $U(0,\frac{n}{N})$. This suggests the definition \b M_{\kappa}(\tau_{(0,\alpha_2)})e_{j}^{\kappa}= e^{i(\kappa_{2}+2\pi j)\alpha_2}e_{j}^{\kappa}. \e $M_{\kappa}(\tau_{(\alpha_1,0)})$ is defined similarly by noting that $U(\frac{n}{N},0)$ is diagonal in the momentum representation. To establish the equipartition result, the last ingredient needed is an Egorov type theorem. \begin{pro} There exists a $G_{\delta}$ dense subset ${\cal D}$ of $T^{(2)}$ with the property that $\forall \alpha\in{\cal D}$, $\exists (N_{k})_{k\in\N}$ such that $\forall f\in C^{\infty}(T^{(2)})$, $\exists C$ so that $$ \mid\mid M_{\kappa}(\tau_{\alpha})^{*}Op_{\kappa}^{W}(f) M_{\kappa}(\tau_{\alpha}) -Op_{\kappa}^{W}(f\circ\tau_{\alpha})\mid\mid_{{\cal L}(\hn)}\leq\frac{C}{N_{k}}. $$ \end{pro} The proof is given below. We can now write and prove the equipartition result for the translations. \begin{th} Write $M_{\kappa}(\tau_{\alpha})\phi_{j}^{N}=\lambda_{j}^{N}\phi_{j}^{N}$ for the eigenfunctions and the eigenvalues of $M_{\kappa}(\tau_{\alpha})$, where $\alpha$ belongs to the $G_{\delta}$ dense set ${\cal D}$ of Proposition 4.1. For all $f\in C^{\infty}(T^{(2)})$ and for all maps $$ j:N_{k}\in\N\longrightarrow j(N_{k})\in\{1,...,N_{k}\}, $$ we have $$ \lim_{k\rightarrow\infty}<\phi_{j(N_{k})}^{N_{k}}, Op_{\kappa}^{W}(f)\phi_{j(N_{k})}^{N_{k}}>_{(\kappa,N)} =\int_{T^{(2)}}f\frac{dqdp}{ab}. $$ \end{th} \pf >From the unitarity of $M_{\kappa}(\tau_{\alpha})$, it follows that, $\forall\; T\in\N^{*}$\\ \vskip5pt $ \mid <\phi_{j(N_{k})}^{N_{k}},Op_{\kappa}^{W}(f) \phi_{j(N_{k})}^{N_{k}}>_{(\kappa,N)} -\int_{T^{(2)}}f\frac{dqdp}{ab}\mid $ \baa &=&\mid <\phi_{j(N_{k})}^{N_{k}},\frac{1}{T} \bd\sum_{l=1}^{T}\ed M_{\kappa}(\tau_{\alpha})^{*l}Op_{\kappa}^{W}(f)M_{\kappa}(\tau_{\alpha})^{l} \phi_{j(N_{k})}^{N_{k}}>_{(\kappa,N)}-\int_{T^{(2)}}f\frac{dqdp}{ab}\mid\\ &\leq&\mid <\phi_{j(N_{k})}^{N_{k}},Op_{\kappa}^{W}[\frac{1}{T}\bd\sum_{l=1}^{T}\ed f\circ\tau_{l\alpha}-\int_{T^{(2)}}f\frac{dqdp}{ab}] \phi_{j(N_{k})}^{N_{k}}>_{(\kappa,N)}\mid +\frac{C_{T}(f)}{N_{k}} \eaa where we used Proposition 4.1. Now using Lemma 3.9 and Lemma 3.8 (iv), we have\\ $ \mid <\phi_{j(N_{k})}^{N_{k}},Op_{\kappa}^{W}(f) \phi_{j(N_{k})}^{N_{k}}>_{(\kappa,N)} -\int_{T^{(2)}}f\frac{dqdp}{ab}\mid $ \baa &\leq&\mid <\phi_{j(N_{k})}^{N_{k}},Op_{\kappa}^{AW}[\frac{1}{T} \bd\sum_{l=1}^{T}\ed f\circ\tau_{l\alpha}-\int_{T^{(2)}}f\frac{dqdp}{ab}] \phi_{j(N_{k})}^{N_{k}}>_{(\kappa,N)}\mid +\frac{C'_{T}(f)}{N_{k}} \\ &\leq&\mid\mid\frac{1}{T}\bd\sum_{l=1}^{T}\ed f\circ\tau_{l\alpha}-\int_{T^{(2)}}f\frac{dqdp}{ab} \mid\mid_{\infty}+\frac{C'_{T}(f)}{N_{k}}. \eaa Using (4.1), the result follows.\fin \vskip10pt \noindent Comparing this to Theorem 1.1, we see this is better in that we can take $E(N_{k})=\{1,...,N_{k}\}$. Of course, the result is subject to the same flaw as the one pointed out in the introduction concerning [DEGI]: the sequence $N_{k}$ depends on $\alpha$. This is all the more unsatisfactory here since the translations $\tau_{\alpha}$ are uniquely ergodic so that there are no other invariant measures for eigenfunctions to concentrate on. The origin of the problem is to be found in the rather weak version of Egorov given by Proposition 4.1. It can probably not be improved considerably. If, for example $\alpha=(\alpha_1,0)$, $\alpha_1\not\in\Q$, then one can show $$ \limsup_{N\rightarrow\infty} \mid\mid M_{\kappa}(\tau_{\alpha})^{*}Op_{\kappa}^{W}(f)M_{\kappa}(\tau_{\alpha}) -Op_{\kappa}^{W}(f\circ\tau_{\alpha)})\mid\mid_{{\cal L}(\hn)}=2 $$ if $f(q,p)=e^{2\pi i(nq-mp)}$. We now turn to the proof of Proposition 4.1. \begin{lem} Let $\alpha, \alpha'\in T^{(2)}$. Then $\exists C>0$ so that $\forall N>0$ $$ \mid\mid M_{\kappa}(\tau_{\alpha})-M_{\kappa}(\tau_{\alpha'})\mid\mid_{{\cal L}(\hn)} \leq CN(\mid \alpha_1-\alpha'_1\mid+\mid \alpha_2-\alpha'_2\mid). $$ \end{lem} \pf Note that, for $\psi=\bd\sum_{j=0}^{N-1}\ed c_{j}e_{j}^{\kappa}$ one has, using (4.3) $$ \mid\mid M_{\kappa}(\tau_{(0,\alpha_{2})})-M_{\kappa}(\tau_{(0,\alpha'_{2})}) \psi\mid\mid_{(\kappa,N)} \leq C N \mid \alpha_{2}-\alpha'_{2}\mid \mid\mid \psi\mid\mid_{(\kappa,N)}. $$ This, together with a similar argument for $M_{\kappa}(\tau_{(\alpha_1,0)})$ and (4.2) yields the result.\fin \vskip10pt \noindent Note that this implies that the map $\alpha\rightarrow M_{\kappa}(\tau_{\alpha})$ is continuous. \vskip10pt \noindent {\bf Proof of Proposition 4.1:} The set ${\cal D}$ is defined as those $\alpha\in T^{(2)}$ for which $\exists (N_{k})_{k\in\N}$, so that $\mid\alpha_1-\frac{p_{1,k}}{N_{k}}\mid<\frac{1}{N_{k}^{2}}$ and $\mid\alpha_2-\frac{p_{2,k}}{N_{k}}\mid<\frac{1}{N_{k}^{2}}$ have simultaneous solutions for $p_{1,k}\in\Z$ and $p_{2,k}\in\Z$. This is easily seen to be a $G_{\delta}$ set. The proof follows immediately from Lemma 4.3 with $\alpha'=(\frac{p_{1,k}}{N_{k}},\frac{p_{2,k}}{N_{k}})$ and the observation that $\forall n,m\in\Z$ $$ U(\frac{n}{N},\frac{m}{N})^{*}Op^{W}(f)U(\frac{n}{N},\frac{m}{N}) =Op^{W}(f\circ\tau_{(\frac{n}{N},\frac{m}{N})}). $$ \fin \vskip10pt We end this section with a brief discussion of the skew translations. We will explain in section 6 how to quantize $K=\pmatrix{1 & k\cr 0 & 1}\in SL(2,\Z)$. Consequently, we define $$ M_{\kappa}(\Phi_{\beta}^{k})\equiv M_{\kappa}(\tau_{(0,\beta)})\circ M_{\kappa}(K). $$ Since the Egorov theorem is exact for $M_{\kappa}(K)$ (see Lemma 6.2), it suffices to use Lemma 4.3 as in the proof of Proposition 4.1 to get an Egorov theorem for all $(\beta,k)\in\R\times\Z$. This yields the desired equipartition result that we won't state explicitly. \sect{Equipartition of the eigenfunctions} In this section we prove Theorem 1.1. The hypotheses of the theorem will be assumed throughout. Using the results of section 3, the proof is easily adapted from the proof of Theorem 2.2 and 3.1 in [HMR]. We start with some preliminary work. The first lemma is a triviality that does not involve any knowledge on the dynamics $\Phi$ or on $M_{\kappa}(\Phi)$. It is in fact a direct consequence of Proposition 3.10. Its analogue in [Z] [CdV] [HMR] is not trivial at all and does involve the dynamics. \begin{lem} $\forall f\in C^{\infty}(T^{(2)})$ \b \overline{\mu}_{N}(f)\equiv\frac{1}{N}\sum_{j=1}^{N}\mu_{j}^{N}(f) \buildrel N\rightarrow\infty \over \longrightarrow\mu(f) \e where $\mu_{j}^{N}(f)\equiv $. \end{lem} {\bf Remarks:\ } \hspace{3.5in}\\ (i) It is understood that we choose a $\kappa$ for each $N$. We shall not indicate the $\kappa$ dependence of the $\mu_{j}^{N}$.\\ (ii) We can write $$ \mu_{j}^{N}(f)=\int_{T^{(2)}}f(q,p)d\mu_{j}^{N}(q,p) $$ where $d\mu_{j}^{N}(q,p)=\mid \mid^{2} \frac{dqdp}{2\pi\hbar}$. So the $\mu_{j}^{N}$ are in view of Lemma 3.8 positive probability measures on $T^{(2)}$, absolutely continuous with respect to the Liouville measure. \vskip10pt \pf \baa \mid\overline{\mu}_{N}(f)-\mu(f)\mid&=&\mid\frac{1}{N} TrOp_{\kappa}^{AW}(f)-\mu(f)\mid\\ &\leq&\frac{1}{N}\mid Tr(Op_{\kappa}^{AW}(f)-Op_{\kappa}^{W}(f))\mid +\mid\frac{1}{N}TrOp_{\kappa}^{W}(f)-\mu(f)\mid\\ &\leq&\mid\mid Op_{\kappa}^{AW}(f)-Op_{\kappa}^{W}(f)\mid\mid_{{\cal L}(\hn)}+ \mid\frac{1}{N}TrOp_{\kappa}^{W}(f)-\mu(f)\mid. \eaa The result follows from Lemma 3.9 and Proposition 3.10.\fin\\ The lemma states that on average the $$ converge to the desired quantity, a fact we already knew for the $ $ from Proposition 3.10. The core of Theorem 1.1 is hidden in the following proposition. To establish it, the dynamics $\Phi$ intervenes through its ergodicity and the Egorov theorem (i.e equation (1.1)). The proof is a direct adaptation from the one in [HMR]. \begin{pro} $\forall f\in C^{\infty}(T^{(2)})$ and $\forall\epsilon>0$ \b \bd\lim_{N\rightarrow\infty}\ed\frac{\#\{j;\mid\mu_{j}^{N}(f)-\mu(f) \mid<\epsilon\}}{N}=1 \e \end{pro} \pf Fix $\epsilon>0$. For $f\in C^{\infty}(T^{(2)})$ and for $p\in \N^{*}$ introduce the time average $$ f_{p}=\frac{1}{p}\bd\sum_{k=1}^{p}\ed f\circ A^{k}. $$ As in section 4, we will use four observations. First, $$ \mu_{j}^{N}(f)-\mu(f)=<\phi_{j}^{N},Op_{\kappa}^{AW}(f-\mu(f))\phi_{j}^{N}> $$ is close to $<\phi_{j}^{N},Op_{\kappa}^{W}(f-\mu(f))\phi_{j}^{N}>$ for large $N$ thanks to Lemma 3.9. Next, since the $\phi_{j}^{N}$ are eigenfunctions of $M_{\kappa}(\Phi)$ (stationary states), we have \b <\phi_{j}^{N},Op_{\kappa}^{W}(f-\mu(f))\phi_{j}^{N}>= \frac{1}{p}\sum_{k=1}^{p} <\phi_{j}^{N},M_{\kappa}(\Phi)^{* k}Op_{\kappa}^{W}(f-\mu(f))M_{\kappa}(\Phi)^{k}\phi_{j}^{N}> \e The Egorov hypothesis 1.1 then implies that, for large $N$, the right hand side of is close to $<\phi_{j}^{N},Op_{\kappa}^{W}(f_{p}-\mu(f))\phi_{j}^{N}>$. Finally, we will use ergodicity, i.e $f_{p}\rightarrow\mu(f)$ together with Lemma 5.1, to conclude that, for large $p$ and $N$, $<\phi_{j}^{N},Op_{\kappa}^{W}(f_{p}-\mu(f))\phi_{j}^{N}>$ is small, at least for "many" $j$. To make this into a proof, compute first $\forall p\in N^{*}$\\ $\mid<\phi_{j}^{N},Op_{\kappa}^{AW}(f)\phi_{j}^{N}>-\mu(f)\mid $ \ba &\leq&\mid<\phi_{j}^{N},Op_{\kappa}^{AW}(f-f_{p})\phi_{j}^{N}>\mid +\mid<\phi_{j}^{N},Op_{\kappa}^{AW}(f_{p}-\mu(f))\phi_{j}^{N}>\mid \nonumber \\ &\leq&\mid<\phi_{j}^{N},Op_{\kappa}^{AW}(f-f_{p})\phi_{j}^{N}>\mid +<\phi_{j}^{N},Op_{\kappa}^{AW}(\mid f_{p}-\mu(f)\mid)\phi_{j}^{N}>.\nonumber\\ & & \ea Note that we used that $Op_{\kappa}^{AW}(f_{p}-\mu(f))\leq Op_{\kappa}^{AW}(\mid f_{p}-\mu(f)\mid)$. This is crucial in the proof and does not hold for the Weyl-quantization. To control the first term on the right, we need Egorov and the invariance of the eigenfunctions under the quantum evolution. Ergodicity shows up only in the control of the second term. For the first term, Lemma 3.9 implies that $\forall p$ $\exists C_{p}>0$ so that $\forall N\in N^{*}$ $$ \mid<\phi_{j}^{N},Op_{\kappa}^{AW}(f-f_{p})\phi_{j}^{N}>\mid\leq \mid<\phi_{j}^{N},Op_{\kappa}^{W}(f-f_{p})\phi_{j}^{N}>\mid +\frac{C_{p}}{N}. $$ But since the $\phi_{j}^{N}$ are the eigenfunctions of $M_{\kappa}(\Phi)$, the Egorov hypothesis implies\\ $ \mid<\phi_{j}^{N},Op_{\kappa}^{W}(f-f_{p})\phi_{j}^{N}>\mid $ \baa &=&\mid\frac{1}{p}\sum_{k=1}^{p} <\phi_{j}^{N},M_{\kappa}(\Phi)^{* k}Op_{\kappa}^{W}(f)M_{\kappa}^{k}(\Phi)\phi_{j}^{N}> -<\phi_{j}^{N},Op_{\kappa}^{W}(f_{p})\phi_{j}^{N}>\mid\\ &\leq&\frac{C_{p}}{N}. \eaa Hence $\forall p$ $\exists C_{p}$ so that $\forall N$ \b \mid<\phi_{j}^{N},Op_{\kappa}^{AW}(f-f_{p})\phi_{j}^{N}>\mid \leq\frac{2C_{p}}{N}. \e This controls the first term of (5.4). For the second term, we will use the ergodicity of $\Phi$, Lemma 5.1 and the positivity of the measures $\mu_{j}^{N}$. First, ergodicity implies that $\forall \delta>0$ $\exists p(\delta)\in N^{*}$ so that $\forall p\geq p(\delta)$ $$ \int_{T^{(2)}}\mid f-f_{p}\mid d\mu<\epsilon\delta. $$ As a result, using Lemma 5.1, $\exists C'_{p}$ so that \baa \frac{1}{N}\bd\sum_{j=1}^{N}\ed<\phi_{j}^{N},Op_{\kappa}^{AW}(\mid f-f_{p}\mid)\phi_{j}^{N}> &\leq&\int_{T^{(2)}}\mid f-f_{p}\mid d\mu+\frac{C'_{p}}{N}\\ &\leq&\epsilon\delta+\frac{C'_{p}}{N}. \eaa Now fix $p=p(\delta)$ and pick $N$ large enough so that $\frac{C'_{p}}{N}<\epsilon\delta$. Then $$ \frac{1}{N}\bd\sum_{j=1}^{N}\ed<\phi_{j}^{N},Op_{\kappa}^{AW}(\mid f-f_{p}\mid)\phi_{j}^{N}> \leq 2\epsilon\delta. $$ So \baa \frac{\epsilon}{2} \frac{\#\{j:<\phi_{j}^{N},Op_{\kappa}^{AW}(\mid f-f_{p}\mid)\phi_{j}^{N}> >\frac{\epsilon}{2}\}}{N} &\leq& \frac{1}{N}\bd\sum_{j=1}^{N}\ed<\phi_{j}^{N},Op_{\kappa}^{AW}(\mid f-f_{p}\mid)\phi_{j}^{N}>\\ &\leq& 2\epsilon\delta. \eaa Hence, for $N$ large enough, and for $p=p(\delta)$, \b \frac{\#\{j:<\phi_{j}^{N},Op_{\kappa}^{AW}(\mid f-f_{p}\mid)\phi_{j}^{N}>\leq\frac{\epsilon}{2}\}}{N} \geq 1-4\delta. \e We can now summarize (5.4) (5.5) and (5.6) as follows. For $f, \epsilon, \delta$ fixed, there exists $p(\delta)$ so that, for $p=p(\delta)$ and $j$ in the set described in (5.6), we have, for all $N$ large enough, $$ \mid<\phi_{j}^{N},Op_{\kappa}^{AW}(f)\phi_{j}^{N}>-\mu(f)\mid\leq \epsilon. $$ Taking $\delta$ to zero the result follows.\fin \begin{cor} For all $f\in C^{\infty}(T^{(2)})$ for all $N$, there exists a subset $E(N,f)$ of $\{1,...,N\}$ satisfying \\ (i) $$ \lim_{N\rightarrow\infty}\frac{\# E(N,f)}{N}=1; $$ (ii) for all maps $j:N\in \N\longrightarrow j(N)\in E(N,f)$ $$ \lim_{N\rightarrow\infty} _{(\kappa,N)}=\mu(f), $$ and $$ \lim_{N\rightarrow\infty} _{(\kappa,N)}=\mu(f), $$ uniformly with respect to the sequence $j$. \end{cor} {\bf Remark:\ } This is essentially the statement of Theorem 1.1, except for the dependence of $E(N,f)$ on $f$. \vskip10pt \pf Let $$E_{k}(N,f) =\{j: \mid<\phi_{j}^{N},Op_{\kappa}^{AW}(f-\mu(f))\phi_{j}^{N}>\mid<\frac{1}{k} , \mid<\phi_{j}^{N},Op_{\kappa}^{W}(f-\mu(f))\phi_{j}^{N}>\mid<\frac{1}{k}\}. $$ >From Lemma 3.9 and Proposition 5.2 it follows immediately that $\exists N_{k}\in \N^{*}$ so that $\forall N\geq N_{k}$, $$ \frac{\# E_{k}(N,f)}{N}\geq1-\frac{1}{k}. $$ We can suppose without loss of generality that $\bd\lim_{k\rightarrow\infty}\ed N_{k}=+\infty$, and that $N_{k} _{(\kappa,N)} <\frac{1}{k}\leq\frac{1}{p} $$ and $$ \mid _{(\kappa,N)} <\frac{1}{k}\leq\frac{1}{p}. $$ This implies (ii).\fin \noindent{\bf Proof of Theorem 1.1:\ } \\ We have to get rid of the $f$-dependence in the corollary. This is done via a diagonalization procedure [Z1] [CdV] [HMR]. Consider the set $A$ of trigonometric polynomials with rational coefficients. It is dense in $C^{\infty}(T^{(2)})$ with respect to the $L^{\infty}$-norm. We write $A=\{f_{l}: l\in \N^{*}\}$ and $E_{l}(N)=E_{l}(N,f_{l})$ as defined in Corollary 5.3. We know from this corollary that \b \lim_{N\rightarrow\infty}\frac{\# E_{l}(N)}{N}=1 \e and \b \lim_{N\rightarrow\infty} _{(\kappa,N)} =\mu(f_{l}), \e \b \lim_{N\rightarrow\infty} _{(\kappa,N)} =\mu(f_{l}), \e uniformly for all $j:N\in \N\longrightarrow j(N)\in E_{l}(N)$. First, we want to find $\tilde{E}_{L}(N)$ so that (5.8) and (5.9) hold simultaneously for all $\{f_{l}: 1\leq l\leq L\}$, with $j(N)\in\tilde{E}_{L}(N)$ and so that moreover \b \lim_{N\rightarrow\infty}\frac{\# \tilde{E}_{L}(N)}{N}=1. \e For that purpose, define $\forall\; L\in \N^{*}$, \b \tilde{E}_{L}(N)=\bigcap_{l=1}^{L} E_{l}(N). \e To see (5.10) holds, remark that the simple set equality $(A\cap B)^{c}=A^{c}\cup B^{c}$ implies $$ N-\# \tilde{E}_{L}(N)\leq(N-\# E_{L}(N))+(N-\# \tilde{E}_{L-1}(N)) $$ so that $$ \frac{\# \tilde{E}_{L}(N)}{N}\geq \frac{\# E_{L}(N)}{N}+\frac{\# \tilde{E}_{L-1}(N)}{N}-1. $$ By recurrence on $L$ and using (5.7), (5.10) holds for all $L$. Since $\tilde{E}_{L}(N)\subset E_{l}(N)$ for $1\leq l\leq L$, (5.8) and (5.9) hold for all $1\leq l\leq L$, provided $j(N)\in \tilde{E}_{L}(N)$. Now, $\forall L$ $\exists N_{L}$ so that $\forall N\geq N_{L}$, \b \frac{\# \tilde{E}_{L}(N)}{N}\geq 1-\frac{1}{L}. \e We can suppose $N_{L} 0$. Then $\exists l$ so that $\mid\mid f-f_{l}\mid\mid_{\infty}<\epsilon$. Compute $$ \mid<\phi_{j},Op_{\kappa}^{AW}(f)\phi_{j}>-\mu(f)\mid\leq \mid<\phi_{j},Op_{\kappa}^{AW}(f_{l})\phi_{j}-\mu(f_{l})>\mid+2\epsilon $$ where we used Lemma 3.8. Since $E(N)\subset E_{l}(N)$ for $N\geq N_{l}$, it follows from (5.8) that $$ \lim_{N\rightarrow\infty}\mid<\phi_{j(N)}^{N}, Op_{\kappa}^{AW}(f)\phi_{j(N)}^{N}>-\mu(f)\mid<2\epsilon $$ provided $j(N)\in E(N)$. This proves (1.3); (1.2) follows upon using Lemma 3.9.\fin \sect{Quantization of the automorphisms of $T^{(2)}$} The results of section 3 allow us to quantize any globally Hamiltonian flow on $T^{(2)}$ as follows. If $H\in C^{\infty}(T^{(2)})$ is the classical Hamiltonian of the flow then the corresponding unitary one-parameter group on $\hn$ is $\exp(-i\frac{t}{\hbar}Op_{\kappa}^{W}(H))$. But for discrete maps such as the automorphisms of the torus, this approach does not work. The framework we set up nevertheless provides an easy quantization for them, as we now show. Any quadratic Hamiltonian $H(q,p)=\alpha q^{2}+\beta q p +\gamma p^{2}$ on $\R^{2}$ generates a linear flow $\Phi_{t}$: for all t, $\Phi_{t}\in SL(2,\R)$. The corresponding Weyl quantized operators generate a unitary one-parameter group on $\L^{2}(\R)$. Explicitly, one obtains a map $ A\in SL(2,\R)\longrightarrow M(A)\in \U(\L^{2}(\R)) $, where for $a_{2}\neq 0$ \b [M(A)\psi](x)=(\frac{i}{2\pi\hbar a_{2}})^{1/2}\int e^{\frac{i}{\hbar}S(x,y)}\psi(y)dy \e with $$ S(x,y)=\frac{1}{2}\frac{a_4}{a_2}x^2-\frac{1}{a_2}xy+ \frac{1}{2}\frac{a_1}{a_2}y^{2}, $$ where $A=\pmatrix{a_1 & a_2\cr a_3 & a_4}$. The $M(A)$ are uniquely determined up to a phase, which can be fixed by group-theoretic considerations that won't concern us here [F]. A crucial property of the $M(A)$ is: \b M(A)U(q,p)M(A)^{*}=U(A \pmatrix{q\cr p}). \e We shall refer to $M(A)$ as the quantum propagator associated to $A$. The subgroup of $SL(2,\R)$ which leaves $\Gamma$ (see section 2) invariant is made up of elements of the form \b A=\pmatrix{\alpha & \beta\frac{a}{b} \cr \gamma\frac{b}{a} & \delta} \e with $\alpha\delta-\beta\gamma=1$ and $(\alpha,\delta,\beta,\gamma)\in\Z^{4}$. It is diffeomorphic to $SL(2,\Z)$, and we shall refer to the $A$ in (6.3) as belonging to $SL(2,\Z)$ with some abuse of language. Any $A$ of the form (6.3) acts naturally on $T^{(2)}=\R^{2}/\Gamma$ as an area-preserving map. It is well known [AA] [CFS] that these transformations are globally hyperbolic dynamical systems and hence ergodic iff $\mid TrA\mid>2.$ We now show how to associate to any $A\in SL(2,\Z)$ a unitary operator on (some) $\hn$. Clearly these $A$ do not belong to one-parameter groups generated by a Hamiltonian. Hence we can not use Weyl-quantization as above. Instead, we simply study the restriction of $M(A)$ in (6.1) to $\hn$. We have \begin{lem} $$ M(A)\hn\subset{\cal H}_{N}(\kappa') $$ where \b \pmatrix{\kappa'_2\cr \kappa'_1}=A\pmatrix{\kappa_2\cr \kappa_1} +\pi N\pmatrix{\frac{\alpha\beta}{b}\cr \frac{\gamma\delta}{a}} +\pmatrix{mod(\frac{2\pi}{b})\cr mod(\frac{2\pi}{a})}. \e \end{lem} \pf Equation (6.2) implies $$ U(na,mb)M(A)=M(A)U(n'a,m'b) $$ where $\pmatrix{n'a\cr m'b}=A^{-1}\pmatrix{na\cr mb}$, so $\pmatrix{n' \cr m'}=\pmatrix{\delta & -\beta\cr -\gamma & \alpha}\pmatrix{n\cr m}$. Then, using (2.1)-(2.3), for all $\psi\in\hn$, we have that $$ U(na,mb)M(A)\psi=e^{i\pi Nn'm'}e^{-i(\kappa_{1}n'a-\kappa_{2}m'b)}M(A)\psi. $$ Introducing the canonical symplectic form on $T^{(2)}$: $\omega:=dq\wedge dp$, we remark that $\kappa_{1}n'a-\kappa_{2}m'b=\omega((n'a,m'b),(\kappa_{2},\kappa_{1}))$. As a result, since $detA^{-1}=1$ $$ \kappa_{1}n'a-\kappa_{2}m'b=\kappa''_{1}na-\kappa''_{2}mb $$ where $\pmatrix{\kappa''_2\cr \kappa''_1}=A\pmatrix{\kappa_2\cr \kappa_1}$ and $$ U(na,mb)M(A)\psi=e^{i\pi N[-\gamma\delta n^{2}-\alpha\beta m^{2}+(\alpha\delta+\beta\gamma)nm]} e^{-i(\kappa''_{1}na-\kappa''_{2}mb)}M(A)\psi. $$ Since $-\gamma\delta n^{2}-\alpha\beta m^{2}+(\alpha\delta+\beta\gamma)nm =\gamma\delta n+\alpha\beta m+mn \pmod{2}$, we have $$ U(na,mb)M(A)\psi=e^{i\pi Nnm}e^{-i[(\kappa''_{1}+\pi N\frac{\gamma\delta}{a})na- (\kappa''_{2}+\pi N\frac{\alpha\beta}{b})mb]}M(A)\psi. $$ Hence, since $\kappa_{1}$, $\kappa_{2}$ are defined $mod(\frac{2\pi}{a})$, $mod(\frac{2\pi}{b})$ (respectively) and the relation modulo is preserved by $A$, (6.4) follows.\fin \vskip10pt Given $A$ with $\mid TrA\mid>2$ there exists for each $N$ a $\kappa\in[0,\frac{2\pi}{a}[\times[0,\frac{2\pi}{b}[$ so that $\kappa'=\kappa$. This choice can be made independent of $N$ iff $A$ is of the form \b A=\pmatrix{even & odd\frac{a}{b}\cr odd\frac{b}{a} & even}\;\;\mbox{or}\;\; \pmatrix{odd & even\frac{a}{b}\cr even\frac{b}{a} & odd}, \e in which case one can take $\kappa=0$. This is the case studied in [HB]. Otherwise, there exists at least one choice for all even $N$ and at least one choice for odd $N$, different one from the other. The choices are unique if $TrA=3$. When $A$, $N$, $\kappa$ are fixed so that $M(A)\hn\subset\hn$, we shall write $M_{\kappa}(A)$ for the restriction of $M(A)$ to $\hn$; we shall refer to $M_{\kappa}(A)$ as the quantum propagator associated to the area-preserving map $A\in SL(2,\Z)$ on $T^{(2)}$. It remains to show $M_{\kappa}(A)$ is unitary. But this is obvious since it is the restriction of $M(A)$ to one fiber of the direct integral (2.8). We can now introduce the last ingredient that is needed in order to apply Theorem 1.1 to $\Phi=A$. \begin{lem} Let $f\in C^{\infty}(T^{(2)})$ then \b Op_{\kappa}^{W}(f\circ A)=M_{\kappa}(A)^{*}Op_{\kappa}^{W}(f)M_{\kappa}(A). \e \end{lem} \pf Since $Op^{W}(f\circ A)=M(A)^{*}Op^{W}(f)M(A)$ [F], (6.6) is obvious, because $M_{\kappa}(A)$ and $Op_{\kappa}^{W}(f)$ are simply obtained by restricting $M(A)$ and $Op^{W}(f)$ to $\hn$. \fin \vskip10pt \noindent Note that our proof of the equipartition result for $M_{\kappa}(A)$ does not use the explicit form of the $M_{\kappa}(A)$. Viewed as matrices on $\C^{N}$, the latter are indeed rather complicated [HB] [DE] [DEGI] unless $$ A=\pmatrix{2g & 1\cr 4g^{2}-1 & 2g},\;\;\;\;g\in\N, $$ which are the matrices studied in [HB] and for which the equipartition result is proven in [DEGI]. As a final remark, we point out that the Wigner function $W_{\kappa}(\phi_{j}^{N},\phi_{j}^{N})$ of any eigenfunction of $M_{\kappa}(A)$ is invariant i.e $$ W_{\kappa}(\phi_{j}^{N},\phi_{j}^{N})\circ A^{-1}=W_{\kappa}(\phi_{j}^{N},\phi_{j}^{N}). $$ This is a direct consequence of (3.7) and (6.6). This property is not shared by the polynomial Weyl symbol, since ${\cal I}_{N}$ is not invariant under $A$. \vskip20pt \noindent{\bf Acknowledgements:} The authors would like to thank Mirko Degli Esposti and Sandro Graffi for many helpful conversations and for explaining their work to them. Stephan De Bi\`evre would like to thank the Department of Mathematics of the University of Bologna, where part of this work was performed, for its hospitality. \vskip25pt \begin{thebibliography}{aAaAaA} \baselineskip=10pt \bibitem[AA]{aa} V.I.Arnold, A.Avez, {\em Ergodic problems in classical mechanics}, W.A.Benjamin, New York (1968). \bibitem[AB]{ab} O.Agam, N.Brenner, {\em Semi-classical Wigner functions for quantum maps on the torus}, Preprint Technion-Phys-94. \bibitem[BV]{bv} N.L. Balazs, A. Voros, {\em The quantized Baker's transformation}, Annals of Physics {\bf190} (1989), 1-31. \bibitem[CdV]{cdv} Y.Colin de Verdi\`ere, {\em Ergodicit\'e et fonctions propres du Laplacien}, Commun. Math. Phys {\bf 131} (1985), 493-520. \bibitem[CFS]{cfs} I.P.Cornfeld, S.V.Fomin and Ya.G.Sinai, {\em Ergodic theory}, Springer Verlag, Berlin, 1982. \bibitem[CR]{cr} M.Combescure, D.Robert, {\em Distribution of matrix elements and level spacings for classically chaotic systems}, Ann. Inst. H. Poincar\'e {\bf 61},4 (1994), 443-483. \bibitem[DB]{db} S.De Bi\`evre, {\em Oscillator eigenstates concentrated on classical trajectories}, J.Phys.A. {\bf 25} (1992), 3399-3418. \bibitem[DBDEG]{ddeg} S.De Bi\`evre, M.Degli Esposti, R.Giachetti, {\em Quantization of a class of piecewise affine transformations on the torus}, Preprint 1994. \bibitem[DBG]{dbg} S.De Bi\`evre, J.A.Gonzalez, {\em Coherent states on tori}, in ``Quantization and coherent states methods", Proceedings of the 11th Workshop on Geometrical Methods in Mathematical Physics-Bialystoc 1992, Eds. S.T.Ali, I.M.Ladanov, A.Odzijewicz (World Scientific Singapore 1993). \bibitem[DE]{de} M.Degli Esposti, {\em Quantization of the orientation preserving automorphisms of the torus}, Ann.Inst.H.Poincar\'e {\bf 58} (1993), 323-341. \bibitem[DEGI]{egi} M.Degli Esposti, S.Graffi, S.Isola, {\em Stochastic properties of the quantum Arnold cat in the classical limit}, Commun.Math.Phys {\bf 167} (1995), 471-509. \bibitem[FGBV]{fgbv} H.Figueroa, J.M.Garcia-Bondia, J.C.Varilly, {\em Moyal quantization with compact symmetry groups and noncommutative harmonic analysis}, J.Math.Phys.{\bf 31} (1990), 2664-2671. \bibitem[F]{fol} G.Folland, {\em Harmonic analysis in phase space}, Princeton University Press, Princeton 1988. \bibitem[GL]{gl} P.G\'erard, E.Leichtnam, {\em Ergodic properties of the eigenfunctions for the Dirichlet problem}, Duke.Math.J. {\bf 71},2 (1993), 559-607. \bibitem[HB]{bh} J.H.Hannay, M.V.Berry, {\em Quantization of linear maps-Fresnel diffraction by a periodic grating}, Physica D.{\bf 1} (1980), 267-291. \bibitem[HMR]{hmr} B.Helffer, A.Martinez, D.Robert, {\em Ergodicit\'e et limite semi-classique}, Commun.Math.Phys {\bf 131} (1985), 493-520. \bibitem[HOSW]{hosw} M.Hillery, R.F.O'Connel, M.O.Scully and E.P.Wigner, {\em Distribution functions in Physics: Fundamentals}, Phys.Rep.{\bf 106} (1984), 121-167. \bibitem[LV]{lv} P.Leboeuf, A.Voros, {\em Chaos-revealing multiplicative representation of quantum eigenstates}, J.Phys {\bf A23} (1990) 1765-1774. \bibitem[M]{m} R.Man\'e, {\em Ergodic theory and differentiable dynamics}, Springer Verlag, Berlin, 1987. \bibitem[Pe]{pe} A.Perelomov, {\em Generalized coherent states and their applications}, Springer, New York, 1985. \bibitem[Ro]{ro} D.Robert, {\em Autour de l'approximation semi-classique}, Birkh\"auser, 1987. \bibitem[RS]{rs} M.Reed, B.Simon, {\em Analysis of operators IV}, Academic Press, 1978. \bibitem[S]{s} M.Saraceno, {\em Classical structures in the quantized Baker transformation}, Annals of Physics {\bf 199} (1990), 37-60. \bibitem[Sc]{sc} A.Schnirelman, {\em Ergodic properties of eigenfunctions}, Usp. Math. Nauk {\bf29} (1974), 181-182. \bibitem[Si]{si} Ya.G.Sinai, {\em Topics in ergodic theory}, Princeton University Press, Princeton 1994. \bibitem[Z1]{z1} S.Zelditch, {\em Uniform distribution of the eigenfunctions on compact hyperbolic surfaces}, Duke.Math.J. {\bf 55} (1987), 919-941. \bibitem[Z2]{z2} S.Zelditch, {\em Quantum ergodicity of $C^{*}$ dynamical systems}, Preprint 1994. \end{thebibliography} \end{document}