0$. Nevertheless, in both experiments, we never see a Lyapunov exponent below $\log(\lambda/2)$. }}} \vspace{1.5cm} 19) {\bf Diffusion and Sinai's (H1) conjecture}. \\ A discrete Legendre transform brings the Standard map into the Hamiltonian form $T:(x,y) \mapsto (x+y + f(x), y+ f(x) )$ which is a map on the cylinder, the cotangent bundle $T^* \TT = \TT \times \RR$ of the circle. Let $A \subset \TT^2$ be a measurable $T$-invariant set of positive Lebesgue measure. If $(x_j,y_j)=T^j(x,y)$ is an orbit in the universal cover $\RR^{2d}$, then $X_j=y_j-y_{j-1}=f(x_j)=\psi(x_j,y_j)$ are random variables on the probability space $\Omega$ equipped with the normalized Lebesgue measure. They have the mean ${\rm E}[X_j]=\int_{\TT^{2}} f(x_j) \; dx dy =0$ and the variance ${\rm Var}[X_j]=\int_{\TT^{2}} \psi(x,y)^2 \; dx dy$. Interesting is the growth rate of $S_n^2=(\sum_{j=0}^{n-1} X_j)^2 = (y_n-y_0)^2$. Using translational invariance ${\rm E}[X_j X_l] = {\rm E}[X_{j+1},X_{l+1}]$, the variance of $S_n$ is $$ {\rm Var}[S_n]= {\rm E}[ (\sum_{j=0}^{n-1} X_j)^2] = n {\rm Var}[X] + \sum_{j=1}^{n-1} (n-j) {\rm E}[X_0 X_j] = n {\rm Var}[X] + \sum_{j=1}^{n-1} (n-j) \hat{\mu}_{\psi}(j) \; , $$ where $\mu_{\psi}$ is the spectral measure of $\psi \in L^2(\Omega), \psi(x,y) =f(x)$ with respect to the unitary Koopman operator $g \mapsto g(\tilde{T})$, where $\tilde{T}$ is the map $T$ induced on $\Omega$. The Fourier transform of $\mu_{\psi}$ is $\hat{\mu}_{\psi}(j) = {\rm E}[X_0 X_j]={\rm Cov}[X_0,X_j]$, a correlation function. Let $\beta$ be the infimum over all real numbers for which $\limsup_{n \to \infty} n^{-\beta} \sum_{j=1}^{n-1} (n-j) \hat{\mu}_{\psi}(j)$ is finite. If $\beta=1$, then $S_n$ behaves like a random walk (a case, where the $X_j$ are independent) and $D = \int_{\TT^{2}} V'(x)^2 \; dx + \limsup_{n \to \infty} n^{-1} \sum_{j=1}^n (n-j) \hat{\mu}_{\psi}(j)$ is the diffusion constant. \\ It is a conjecture of Sinai (H1) on page 144 in \cite{Sinai94} that there exists a set $\Omega$ of positive Lebesgue measure for which $\beta=1$ if $\lambda$ is larger then $\lambda_{crit}$, where the last homotopically nontrivial KAM torus disappears. If $\hat{\mu}_{\psi}(n)$ would decay fast enough, then $D = \int_{\TT^{2}} V'(x)^2 \; dx + \sum_{j=0}^{\infty} \hat{\mu}_{\psi}(j)$. First numerical experiments were done by Chirikov and Hizanidis \cite{ReWh80}. Numerically, the Standard is reported to show such Brownian diffusive behavior for large $\lambda$ \cite{Lichtenberg,Leb98}. The fact of having positive Lyapunov exponents on a set of positive measure makes it plausible that the random variables $X_0,X_n$ get decorrelated for $n \to \infty$. When computing numerically the first few hundred Fourier coefficients for smaller $g$ (like $g=5$) and checking with the Wiener theorem, we got the impression that $\mu_{f}$ still has some atoms (which prevents decorrelation) if $\Omega = \TT^2$. Indeed, the presence of elliptic islands could be responsible for an almost periodic component in the Fourier transform of $\mu_{f}$. In numerical experiments, one can get rid of this discrete part of the spectrum by adding stochastic noise \cite{Lichtenberg}. \\ In any case, the (H1) conjecture of Sinai would be settled for $\lambda>\lambda_0$ if one could show a fast enough decay of correlation of the spectral measure of $\psi(x,y)=f(x)$ on a mixing component of the Pesin set $\Omega$. No exponential decay is necessary. It is enough to establish a power law decay of correlation $\hat{\mu}_{\psi}(j) = O(j^{-2})$. This finite differentiability condition for the spectral measure $\mu_{\psi}$ is reasonable since the dynamics on $\Omega$ is conjugated to a Markov chain. We even expect many spectral measures to be realanalytic leading to exponential decay of correlations. \\ 20) {\bf Dissipative Standard maps}. \\ The Lyapunov exponent of dissipative Standard maps numerically often satisfy the lower bound of the conservative case. The Lyapunov exponent can drop however to zero. In order that the proof carries over to the dissipative case, it appears however, that the random variables $x_j$ need a smooth distribution $\mu$ sufficiently close to the uniform distribution. Results for Henon maps lead to the expectation that for many parameters, there exist invariant SRB measures for dissipative Standard maps like $$ T_b: (x,y) \mapsto (x+y + \lambda \sin(x), b(y + \lambda \sin(x))) \; , $$ with $b<1$. Results like in \cite{Cal91,Mor94} support this. Viana conjectured that in general a map with nonzero Lyapunov exponents almost everywhere in the phase space has an SRB measure \cite{Via98}. If this conjecture is true, it indicates that for most values of $b$, an SRB measure should exist. The estimates of Lyapunov exponents using spectral methods could even carry over to the case $b=0$, where on each invariant set $y=\alpha$ we get the one-dimensional Arnold family $x \mapsto x+\alpha + \lambda \sin(x)$. If such a circle map has a smooth invariant measure sufficiently close to the Lebesgue measure, the Lyapunov exponent with respect to this measure should be $\geq \log(\lambda/2) - C(\lambda)$ for large $\lambda$. \\ 21) {\bf More general stability of positive metric entropy?} \\ The stability, we have established holds for estimates of the entropy which are done using subharmonicity rsp. the Jensen formula. This result provokes the question, whether, in general, positive metric entropy is an open property in the realanalytic category: is it true that for any realanalytic, measure preserving diffeomorphism $T$ on the torus with positive metric entropy, there exists a Banach space of realanalytic measure-preserving maps such that an open neighborhood of $T$ has positive metric entropy? \\ A question related to this stability problem is whether the Riesz measure $dk$ of the $w$-parameterization of the operator $L$ for a general twist map has the property that the potential $\int \log|w-w'| \; dk(w')$ does not fluctuate too much around its mean on the unit circle $\{ |w|=1 \}$. This could be used to establish the stability if the entropy is comparable to upper bounds of the entropy like in the Standard map case. \\ 22) {\bf The Herman spectrum}. \\ One can also look at different analytic parameterizations of the cocycle. The Herman spectrum of a cocycle $A$ is the set of complex numbers $z$, such that $R(z) A$ is not uniformly hyperbolic, where $R(z)=\left( \begin{array}{cc} \frac{z+z^{-1}}{2} & \frac{z-z^{-1}}{2i} \\ -\frac{z-z^{-1}}{2i} & \frac{z+z^{-1}}{2} \\ \end{array} \right) \in SL(2,\CC)$ which has the property that $R(e^{i \alpha}) = R(\alpha) \in SO(2,\RR)$. It is a subset of $\{|z|=1\}$ because for $|z| \neq 1$, one can find a strict coinvariant cone field in $\CC P^1$. There is a measure $\mu$ supported by the Herman spectrum and a harmonic function $g$ such that the Lyapunov exponent $\mu(z) = \mu(z R(z) A)$ satisfies $$ \mu(z) = \int \log|z-z'| \; d\mu(z) + g(z) \; . $$ This abstract Thouless formula \cite{Her83} follows from Riesz theorem and $\mu$ is the Riesz measure of the subharmonic function $\mu(z)$. The proof of Theorem~\ref{Igeneral} shows $\log(\mu(\beta)) > \log( \cos(\beta) \lambda/2)-O(1/\lambda)$ for all $\beta$. The Lyapunov exponent is realanalytic and positive outside the Herman spectrum. At such points, the directional derivative with respect to variations of the angle $\beta$ in $z = r e^{i\beta}$ can be computed using a formula of Ruelle \cite{Rue79} for the Fr\'echet derivative of Lyapunov exponents on the open set of uniformly hyperbolic cocycles. One gets $\frac{d}{d\beta} \mu(\beta)= \int_{\TT^2} \cot(\omega(x,y)) \; dm(x)$, where $\omega(x)$ is the angle between the stable and unstable directions $m^+,m^-$ at the point $(x,y)$ (see \cite{Kni93diss}). This formula shows that the Lyapunov exponent can change a lot if the stable and unstable manifolds are close. In the case $A(x)=\left( \begin{array}{cc} c & b(x) \\ 0 & c^{-1} \\ \end{array} \right)$ with constant $|c| \neq 1$, one obtains $$ \frac{d}{d\beta} \mu(A(\beta))=\int_{\TT^2} \cot(\omega(x,y)) \; dx dy = \frac{c}{1-c^2} \int b \; dx dy \; .$$ If $A$ is the cocycle of a Standard map in Hamiltonian form like in 19), there is a spectral gap containing $\beta=-\pi/4$ in the Herman spectrum, because $ R(-\pi/4) dT_{\lambda}= \left( \begin{array} {cc} 2^{-1/2} & 0 \\ 2^{-1/2}+2^{1/2}\lambda \cdot \cos(x) & 2^{1/2} \\ \end{array} \right) $. We compute $\frac{d}{d\beta} A(-\pi/4) = 1$. The point $z=e^{i 0}$ is in the Herman spectrum because it follows from \cite{Mat68,Rue85} (see also \cite{Kni93diss}) that $z=1$ is outside the Herman spectrum if and only if the map $T_{\lambda f}$ is Anosov. The rotation number of Ruelle \cite{Rue85} defined for lifts of $A$ into the universal cover of ${\rm SL}(2,\RR)$ plays the role of the integrated density of states in the case of the Schroedinger spectrum. It is uniquely defined up to a multiple of $2 \pi$ if one fixes $\rho(A(0))=0$. The rotation number is constant on an interval $I$ if and only if $I$ is not in the Herman spectrum \cite{Kni93diss}. Furthermore, $\rho(\beta) = \int_0^{\beta} \; d\mu(\alpha)$, showing that $\mu$ is an 'integrated density of states'. From the stable and unstable direction fields, one can construct (nonselfadjoint) random Jacoby operators $L(z)$ having those direction fields $m^{\pm}(z)$ as Titchmarsh-Weyl functions (with energy $E=0$). They are analytic in $z$ outside the Herman spectrum. Call a subset of the unit circle a part of the 'absolutely continuous spectrum' of $L(z)$ if there, $m^+(z)= \overline{m^-}(z)$. This absolutely continuous spectrum is a subset of the Herman spectrum. An adaption of Kotani theory \cite{Cycon} shows that the 'absolutely continuous spectrum' is the essential closure of the set, where the Lyapunov exponent is zero and that in the ergodic case, the existence of some absolutely continuous spectrum implies that $T$ is 'deterministic' (see \cite{Fur63,Led86,KoSi88}). We know therefore for the Standard map that there is no absolutely continuous Herman spectrum for almost all $(x,y)$ in the Pesin region if $\lambda$ is large. We call $z$ an 'eigenvalue' of $L(z)(x,y)$ if $L(z)(x,y)$ has an eigenvalue $E=0$. The set of these 'eigenvalues' forms a 'discrete spectrum' of $L(z)(x,y)$ which is a subset of the Herman spectrum and which we expect to be nonempty for Lebesgue almost all $(x,y)$ in the Pesin region. \\ 24) {\bf Quasiconformality}. \\ The map $$ z \mapsto G_{n}(z)=\int_{\TT^{1}} \log|z^n A^n_{E}(z,y)| \; dy $$ is not conformal for small $|z|$ in general. For realanalytic $T$ near a linear automorphism $(x,y) \mapsto (x+y,y)$ it is quasiconformal for all $n$. Question: is there an open set of realanalytic, measure-preserving maps $T$ for which $G_n$ are quasiconformal? If yes, it would be important to estimate the complex dilation $\kappa(z)=\overline{\partial} G_{n}(z)/\partial G_n(z)$ and to estimate the Lyapunov exponent in terms of $\kappa$. For $\kappa=0$, one has conformality and the subharmonic estimates of Herman apply. \\ 25) {\bf The Aubry duality transform}. \\ The Aubry duality transform is an involutive map on a class of operators. It provides in the Mathieu case an elegant way to estimate the Lyapunov exponent. The transform can be defined in more general situations: Let $T_k$ be a cyclic interval exchange transformation on $\TT$ and let $L_{\lambda \cos,T_k}$ be the corresponding Schr\"odinger operator. The density of states of the random operator $L$ is the same as the density of states of the operator $$ (L u)_n(\theta) = u_{n+1}(\theta) + u_{n-1}(\theta) + \lambda \cos( T^n \theta) u_n(\theta) $$ on the Hilbert space $H = L^2(\TT \times \ZZ)$. There is a piecewise smooth potential $V$ such that $\cos(T^n(\theta)) = V(\theta + n \alpha)$, where $\alpha=1/k$. Let $V(\theta) = \sum_n V_n \exp(i n \theta)$ be its Fourier series. The duality transform (see \cite{Gor+97}) $$ (U u)_m(\eta) = \sum_{n \in \ZZ} \int_{\TT} e^{-(\eta+2\pi m \alpha)n} e^{-i m \theta} u_n(\theta) \; d\theta $$ satisfies $ \tau U = (\lambda/2) U \sigma$, where $\tau u_n(\theta) = u_{n+1}(\theta)$ and $\sigma u_n(\theta) = \exp( i (\theta+n\alpha)) u_n(\theta)$. The operator $L$ can be written as $L = \tau+\tau^* + \lambda \sum_n V_n \sigma^n$ so that $U^* L U = \sigma+\sigma^* + \lambda/2 \sum_n V_n (\tau^n+(\tau^n)^*)$. % Note that $\tau^n \neq Id$. Let $\epsilon=2/\lambda$. We are interested in the density of states $dk_{\epsilon}$ of $L_{\epsilon} = (2/\lambda) U^* L U = \epsilon (\sigma + \sigma^*) + \sum_{n} v_n (\tau^n + \tau_n^*)$ and $f(\epsilon)= \log(\det(L_{\epsilon}))=\int \log|E| \; dk_{\epsilon}(E)$ which satisfies $f(0)=0$. Because $\log(\det(L_{\lambda \cos,T_k})) \geq \log(\lambda/2) + f(2/\lambda)$, we want to estimate $f(\epsilon)$ from below for small $\epsilon$. If $T_k(x)=x+1/k$, where $f(\epsilon)$ is the Lyapunov exponent of a symplectic transfer cocycle, we have $\int \log|E| \; dk_{\epsilon}(E) \geq 0$. In general, the Thouless formula just becomes $\log(\det(L_{\epsilon})) = \lim_{N \to \infty} \sum_{k=1}^{2N} \int_{\TT} \log(\lambda_j(x,\epsilon)) \; dx$, where $\lambda_j(\epsilon,x)$ are the eigenvalues of the truncated $N \times N$ matrix $L_{\epsilon,N}(x)$. While $f(0)=0$ and a perturbation lemma of Lidskii (\cite{Simon79}) assures that $|\lambda_j(\epsilon,x)-\lambda_j(\epsilon,0)| \leq \epsilon$, this does not allow us to estimate $f(\epsilon)$ from below. The perturbation problem to estimate $f(\epsilon)$ seems still difficult and it is not clear, whether the Aubry transform, which transformed the perturbation problem $V \mapsto V + \epsilon \Delta$ into a perturbation $\Delta \mapsto \Delta + \epsilon V$ has made things simpler. \\ %There are %examples of limit periodic operators, where the $f(\epsilon)=-\infty$ for %all $\epsilon \to 0$ while $f(0)=0$. \\ 26) {\bf Lower bound for topological entropy?} \\ The topological entropy of $T \in \Xcal$ on $\TT^2$ is bounded below by $\log({\rm sp}(T_*))$, where ${\rm sp}(T_*)$ is the spectral radius of $T_*: H_*(\TT^2) \mapsto H_*(\TT^2)$ \cite{MiPr77}. Can the metric entropy of a (smooth) $T \in \Xcal$ with respect to the invariant Lebesgue measure become smaller than ${\rm sp}(T_*)$? \\ 27) {\bf Conjectures on the Standard map}. \\ The following of conjectures about Standard maps have still to be settled. \\ {\bf I)} In \cite{Chi79}, the entropy of the Standard map $T_{\lambda \sin}$ was measured $\geq \log(\lambda/2)$. Chirikov formulated the possibility that the elliptic islands might cover arbitrary large regions of the phase space. \\ \begin{center} \fbox{ \parbox{12cm}{ {\bf I \cite{Kni93diss}:} The Kolmogorov-Sinai entropy of the Chirikov Standard map is $\geq \log(\lambda/2)$. \\ }} \end{center} \vspace{0.5cm} {\bf II)} \cite{Spe89} introduced the problem of determining the spectrum of random operators $(L(x,y) u)_n = u_{n+1} - 2 u_n + u_{n-1} + \lambda \cos(x_n) u_n$, where $T^n(x,y)=(x_n(x,y),y_n(x,y))$ is the orbit starting at $(x,y)$. \\ \begin{center} \fbox{ \parbox{12cm}{ {\bf II \cite{Spe89}:} The operator $L(x,y)$ has some point spectrum for a set $(x,y)$ of positive Lebesgue measure if $\lambda$ is large enough. \\ }} \end{center} \vspace{0.5cm} {\bf III)} \cite{Car91} asked whether there exists a set of parameters $\lambda$ with full density at $\infty$ for which the Chirikov Standard map has no elliptic islands. We know that we have for large $\lambda$ an open dense set of parameters with no ergodicity and positive entropy. \\ \begin{center} \fbox{ \parbox{12cm}{ {\bf III \cite{Car91}:} There are parameters $\lambda$ with full density at $\infty$ for which the Chirikov Standard map $T_{\lambda \sin}$ is ergodic. \\ }} \end{center} \vspace{0.5cm} Heuristic arguments in \cite{GiLa99} predict that the Lebesgue measure of the set of parameters $\lambda \in [r,r+1]$ which lead to nonergodic Standard maps $T_{\lambda \sin}$ is of the order $O(1/r)$. \\ {\bf IV)} The following problem is related to III): \\ \begin{center} \fbox{ \parbox{12cm}{ {\bf IV \cite{Sin96}:} For large $\lambda$, the Standard map has (in some Baire topology of maps) a neighborhood in which a residual set of $f$'s gives ergodic maps $T_f$.\\ }} \end{center} \vspace{0.5cm} {\bf V)} In the textbook \cite{Sinai94} p. 144, conjecture (H2) contained as a second part the statement that the entropy grows to infinity for $\lambda \to \infty$. While we solved this part of the problem here, the first part of the conjecture is still open: \\ \begin{center} \fbox{ \parbox{12cm}{ {\bf V \cite{Sinai94}:} the entropy of the Chirikov Standard map is positive for all $\lambda>0$. \\ }} \end{center} \vspace{1.0cm} One should be able to prove that for any real-analytic non-constant periodic $f$, there exists $\lambda_0>0$ such that the Standard map $T_{\lambda f}$ has positive Kolmogorov-Sinai entropy for all $\lambda>\lambda_0$ and that for any real-analytic, non-constant, periodic map $f: \TT^d \to \RR^N$ and every symmetric, constant matrix $E \in GL(N,\ZZ)$, all the Lyapunov exponents of the symplectic map $T_{Ex+\lambda f}$ are nonzero on a set of positive Lebesgue measure for large enough $|\lambda|$. All these properties are expected to be stable with respect to realanalytic perturbations of the map $T_{E x+\lambda f}$, and hold for a fixed cocycle for all $T \in \Ycal$. \section*{Appendix A: Computation of $A(w),B(w),G(w),G(0)$.} The calculation of the transfer cocycle $A(w)$ as well as its representation $B(w)$ in the 2 particle Fermionic space is straightforward but unpleasant to do by hand. We go through the calculation with a computer even so help from a machine is not necessary in principle. In order to be as independent of the algebraic manipulation tool as possible (here we use Mathematica \cite{Wolfram91}), we use only ASCII symbols (i.e. no indices or Greek symbols, even so the software would allow this). So, we write $an=a_n, am=a_{n-1}, ak=a_{n-2}, al=a_{n-3}, g=\lambda/2$. \\ The code for this computation is available online at \\ http://www.ma.utexas.edu/ $\; \tilde{}$ knill/Twist/Mathematica/appendix.m\\ We first compute the $4 \times 4$ transfer matrix $A(E,w)$ as a $2 \times 2$ matrix of $2 \times 2$ matrices. (The variable $EE$ is used because $E$ is already reserved for $e=2.718...$.) $I2=\{\{1,0\},\{0,1\}\}$ is the $2 \times 2$ identity matrix. \begin{verbatim} Bn= {{bm, am},{cm,bn}}; Ak= {{ 1 , 0},{ak, 1}}; Ck= {{ 1 ,ck},{0 , 1}}; N2= {{ 0 , 0},{0 , 0}}; I2= {{ 1 , 0},{0 , 1}}; AA={{(EE*I2-Bn).Inverse[Ak], -Ck}, {Inverse[Ak] , N2}} \end{verbatim} We rewrite this as a $4 \times 4$ matrix. (The actual input really starts now). \begin{verbatim} A={{EE-bm+am*ak , -am , -1, -ck}, {-cm-ak*(EE-bn) ,EE-bn ,0 , -1 }, {1 , 0 ,0 , 0 }, {-ak , 1 ,0 , 0 }}; \end{verbatim} We plug in the coefficients. We need the transfer matrix only in the case $EE=0$. The entries are meromorphic functions in the variable $w$ and analytic in $E1=E_1,E2=E_2$. We use again the notation $sn =s_n, zn=z_n$ etc. With $L^{(i)} = \tau + \tau^* +V^{(i)} + E_j$, with $V^{(1)}_n = g (w^{-1} z_n + w z_n^{-1})$, $V^{(2)}_n = g (s_n + s_n^{-1})$, $\tau u_n = u_{n+1}, \tau^* u_n = u_{n-1}$, we get the entries of the transfer cocycle of $L_{T,E_1} L_{S,w,E_2}$: \begin{verbatim} EE=0; ak = g*( sk + sk^(-1) + w^(-1)*zm + w*zm^(-1) ) + E1+E2; am = g*( sm + sm^(-1) + w^(-1)*zn + w*zn^(-1) ) +E1+E2; bn = g^2*(sn + sn^(-1) )*( w^(-1)*zn + w*zn^(-1) )+2.0+E1*E2+ E2*g*( sn + sn^(-1) ) + E1*g*( w^(-1) zn + w*zn^(-1) ); bm = g^2*(sm + sm^(-1) )*( w^(-1)*zm + w*zm^(-1) )+2.0+E1*E2+ E2*g*(sm + sm^(-1) ) + E1*g*( w^(-1) zm + w*zm^(-1) ); ck = g*( sk + sk^(-1) + w^(-1)*zl + w*zl^(-1) ) + E1+E2; cm = g*( sm + sm^(-1) + w^(-1)*zk + w*zk^(-1) ) + E1+E2; cn = g*( sn + sn^(-1) + w^(-1)*zm + w*zm^(-1) ) +E1+E2; \end{verbatim} (With $A[[1,1]], \dots, A[[4,4]]$ one can access at this point the entries of $A$). \\ The variable $A$ represents a $4 \times 4$ transfer matrix $A(E_1,w)$ which has as entries rational functions in the variable $w$. We represent now this matrix $A$ as a matrix $B$ in $\wedge^2 \CC^4 = \CC^6$. \begin{verbatim} b[m_,n_]:=Module[{ u={{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}} }, {i,j}=u[[m]]; {k,l}=u[[n]]; A[[i,k]]*A[[j,l]]-A[[i,l]]*A[[j,k]]]; B=Table[b[m,n],{m,6},{n,6}]; \end{verbatim} (With $B[[1,1]], \dots, B[[6,6]]$ one can access at this point the (pretty messy) entries of $B$). \\ To compute the polynomial matrix entries at $w=0$, we conjugate with a diagonal matrix $D$ (which is written as $DD$ because again the symbol $D$ is already reserved for the differential operator) and multiply with $w^2$ and $sn, sm$, which both does not affect the Lyapunov exponent when $|s_n|=1,|w|=1$. We get now the matrix $G(w)$ and after setting $w=0$, we obtain the matrix $G(0)$. \begin{verbatim} Clear[w]; DD={{w,0,0,0,0,0}, {0,1,0,0,0,0}, {0,0,1,0,0,0}, {0,0,0,1,0,0}, {0,0,0,0,1,0}, {0,0,0,0,0,1}}; G=Expand[sn*sm*w^2*DD.B.Inverse[DD]]; w=0; G0=Simplify[Chop[G]] \end{verbatim} As expected, ${\rm G0}=G(0)$ does not depend on $E_2$ (which we have put zero anyway in the proof in section~(7)). We cut now away the nilpotent part (the additional Chop is used to eliminate zero terms like $10^{-16}$ which are zero but which were not recognized to be so). \begin{verbatim} F= Chop[{{G0[[1,1]],G0[[1,3]]},{G0[[3,1]],G0[[3,3]]}}] \end{verbatim} Now, we introduce a new variable $z$ which links $z_n$ with $s_n$. \begin{verbatim} Clear[z]; sm=z*zm; sn=z*zn; sl=z*zl;sk=z*zk; \end{verbatim} We are interested in what is left when one puts $z=0$: \begin{verbatim} z=0; F \end{verbatim} The cocycle $F(0)$ has the Lyapunov exponent $\log(g^4)= 2 \log(\lambda/2)$ and the function $z \mapsto \mu(F(z))$ is subharmonic. \\ \section*{Appendix B: determinant of a product of operators} This appendix was added later to answer a question from early readers of the May 27) preprint. \\ Linking the initial conditions of the orbits restricts the dynamics to a diagonal like set $Y_R$ of initial conditions in the product system $T \times S: \TT^2 \to \TT^2$ which is not $T \times S$ invariant. \begin{lemma} Let $T,S \in \Ycal$. Let $R \in \Ycal$ be an additional measure preserving transformation. Let $L$ be a random Jacobi operator over $(\TT^2,T,dx dy)$ and let $K$ be a random Jacobi operator over $(\TT^2,S,dx dy)$. Let $Y=Y_R \subset \TT^4$ be the graph $Y_R = \{ ((x,y),R(x,y)) \; | \; (x,y) \in \TT^2 \} \subset \TT^4$. Let $m$ be the measure on $Y$ which is the pull back of the projection $\pi_1: \TT^4 \mapsto \TT^2, \pi_1(x_1,y_1,x_2,y_2) = (x_1,y_1)$. \\ The arithmetic mean $\mu_2(x,y)$ of the two largest Lyapunov exponents of the $4 \times 4$ transfer cocycle of $L K$ exists for $m$-almost all $(x,y,R(x,y)) \in Y$. The map $(x,y) \mapsto \mu_2(x,y)$ is measureable. \\ The integral of $\mu_2(x,y)$ over $Y_R$ with measure $m$ is the same as the integral over $\TT^4$ with the Lebesgue measure and therefore does not depend on $R$. \end{lemma} \begin{proof} For Lebesgue almost all $(x,y) \in \TT^2$, there is an ergodic $T$-invariant measure $m_{(x,y)}$ for which $(x,y)$ is generic (see e.g. \cite{DGS} or \cite{Furstenberg}.) This means that for all $f \in C(\TT^2)$, the Cesaro averages $n^{-1} \sum_{k=0}^{n-1} f(T^k(x,y)) \mapsto \int_{\TT^2} f(x',y') \; dm_{x,y}(x',y')$). \\ Because $L$ becomes so an ergodic operator over the dynamical system $(\TT^2,T,m_{(x,y)})$, we can so define $\log \det L(x,y)$ for almost every $(x,y) \in \TT^2$ and it is equal to the pointwise Lyapunov exponent $\mu(x,y)$ of the transfer cocycle of $L$. \\ Let $K$ is the second additional random operator over $S \in \Ycal$. For almost all $(x,y) \in \TT^2$, and almost all $(u,v) \in \TT^2$, one has from the Fuglede-Kadison theory $$ \log \det (L(x,y) K(u,v)) = \log \det L(x,y) + \log \det K(u,v) \; . $$ (The operator $L(x,y) K(u,v)$ is a higher order random difference operator determined by $(x_n,y_n,u_n,v_n)$, where $(x_n,y_n)=T^n(x,y)$ and $(u_n,v_n)=S^n(u,v)$. Lebesgue almost all points $(x,y,u,v) \in \TT^4$ are generic with respect to $T \times S$.) \\ Especially, if $R: \TT^2 \mapsto \TT^2$ is a measure preserving map then, for almost all $(x,y)$, the relation \begin{equation} \label{pointwiseproductformula} \log \det (L(x,y) K(R(x,y))) = \log \det L(x,y) + \log \det K(R(x,y)) \end{equation} holds. \\ Its value is equal to the arithmetic mean $\mu_2(x,y)$ of the two largest Lyapunov exponents of the $4 \times 4$ transfer cocycle of $L(x,y) K(R(x,y))$. \\ Therefore, using (\ref{pointwiseproductformula}), the definition of $m$ and the measure preserving property of $R$, we have \begin{eqnarray*} \int_Y \log \det (L(x_1,y_1) K(x_2,y_2)) \; dm &=& \int_{\TT^2} \log \det(L(x,y) K(R(x,y))) \; dx dy \\ &=& \int_{\TT^2} \log \det(L(x,y)) \; dx dy + \int_{\TT^2} \log \det(K(R(x,y))) \; dx dy \\ &=& \int_{\TT^2} \log \det(L(x,y)) \; dx dy + \int_{\TT^2} \log \det(K(x,y)) \; dx dy \\ &=& \int_{\TT^4} \log \det(L)(x_1,y_1)) \; dx_1 dy_1 + \log \det (K(x_2,y_2)) \; dx_2 dy_2 \\ &=& \int_{\TT^4} \log \det (L(x_1,y_1) K(x_2,y_2)) \; dx_1 dy_1 dx_2 dy_2 \; . \end{eqnarray*} The right hand side is independent of $R$. \end{proof} We have shown with this lemma that the value of the averaged Lyapunov exponent can be obtained by integrating the pointwise Lyapunov exponents over a diagonal $Y_R$ without affecting the value of the Lyapunov exponent and that the value does not depend on $R$. \\ {\bf Acknowledgments.} I thankfully acknowledge the support of the Swiss National Science foundation for funding my research at UT. It is a pleasure to thank the mathematical physics group at the University of Texas at Austin for their hospitality, especially Raphael de la Llave for arranging my visit. Slightly weaker results than in this paper were announced at the sectional AMS meeting in Tucson in November 1998. Thanks to Hans Koch and Marek Rychlik and Maciej Wojtkowski for listening in Tucson to an extended exposition of my purely complex analytic but not yet fully successful approach. The proof contained in this paper has first been presented at the national AMS meeting in San Antonio in January 1999 and then at the joint AMS-SMM international meeting in Denton in May 1999. A question from readers of the May 27 preprint at a conference in Edinburgh (e-mail question of R. de la Llave from May 31) and from M. Herman in June 11 (e-mail) and June 28 (fax) helped to improve the May 27 preprint of this paper. 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