 I'd like to thank the organizers for the opportunity to give this talk on a PD, probably the only PD in this conference perhaps. It's about quasi-periodic systems, infinite dimensional, and reducibility. And since you may not be all familiar with this, I should start by giving you some background. So what we are interested in is in a linear differential equation. It is a constant matrix, and we have a non-constant one, which is time-dependent. So in this, so this is my system, linear differential equation. It's an N by N system, so X is in CN, A is an N by N matrix, possibly real or in a complex. F is a mapping from a d-dimensional torus into the space of N by N matrices, and the elements on the torus is called phi. Omega here is a vector in Rd, and one can assume without restriction that it is rationally independent. So there is no integral relations between the components of omega, except the trivial one. Now, so this is now an ordinary differential equation. This one I will only consider analytic functions, and this is perturbation theory, so it will be small. So we have a linear autonomous system of differential equations. We have a small time-dependent perturbation, and this time dependence is periodic or quasi-periodic. Dot, by the way, here is d dt. So when omega, when d is one, then this is just a periodic one. Basically we know everything about this system, so I will restrict myself to d bigger than two. So this is the kind of difference from zero, thanks a lot. There is another way to write this, as you all know. You can transform it into an autonomous system, and then it takes the form A plus F of phi times X, phi dot equals omega, and now we get an autonomous system, a vector field, vector field on Cn cross Td, which is trivial in the direction, in the Td directions, and which has some non-trivial behavior in the fiber direction. So this is a skew system, or also called the co-cycle, quasi-periodic linear. So the most important notion for this kind of system in the perturbative setting is what is called reducibility. So the statement is the following that this co-cycle is reducible if and only if there exists a mapping from Td into the space of invertible n by n matrices, say analytic. There exists a constant matrix B, such that the transformation, mapping X, phi, into Y equals C phi times X and phi. So it's simply a linear transformation in the fibers so that this transformation transforms, conjugates the co-cycle here to a constant co-cycle, Y dot equals B of Y, phi dot equals omega. So conjugate the system, time-dependent system to a constant coefficient system for which we basically know everything. So let me call this one. So this is the notion of reducibility. It's basically a constant coefficient system under a linear change of variable. This, another way to write it, is to say that one is equivalent to two, that is that this partial differential equation is fulfilled. So here d omega is the directional derivative of each component in the direction of omega. This is a partial differential equation on the torus. If you only know this guy, then it is slightly non-linear because the unknown here are the transformation, capital C, and the transformed system. So slightly non-linear PDE on the torus. Another way to write it is to say, look at the fundamental solution. This is a linear differential equation. So it has a fundamental solution for each phi. It's a linear differential equation. So it has a fundamental solution, a matrix solution which gives you all the solution. And this one has, this fundamental matrix which has a decomposition in the following form. So these are the three equivalent ways to formulate the reducibility of a system. And it's an easy algebra to verify that these are in fact equivalent. You could, of course, construct different variants of this. You could relax the smoothing condition. This is an analytic system. It's natural to ask for an analytic change of variable. But you could, of course, ask for something less. You could relax the periodicity of C. Here we require that C is defined on the torus. That means that it's a function in D variables and it's one periodic in each variable. You could just as well require that it's two periodic or why not three periodic? I mean that would be a result that is equally interesting. So these are a relaxing condition. This is not, perhaps, not extremely interesting. This is important. You could also require more. You could require algebraic constraints. For example, if our system is real, you could ask for a decomposition into real matrices. If it has trace zero, then the fundamental solution has determinant one. You could ask for a decomposition into matrices of determinant one. This question was essentially solved by Clair Chavaudry in 2010 where she proved that if I have a system which belongs to some matrix group, symplectic group, orthogonal group, and you can make a factorization of the fundamental solution in this way in complex matrices, then you can also factorize it in the group. There is a small price to pay for this factorization, that is that the transformation will not be defined on the torus but on the double covering on the torus. So there is a small price to pay for this. Okay. So let me tell you something about the results which we know. So we have quite a lot of results for these kinds of system and most of them are for omega verifying some diophantane condition. So let me remind you what it is. It means that k omega scalar product is not only different from zero, but it is bounded from below by a condition of this type. And f from td into glcn is analytic, so it extends to a certain neighborhood of the torus, so on which we measure into by the sup norm on a complex neighborhood, f of phi for complex values of phi. And now the general theorem that we have is the following. It says that in under this condition, in the space of all systems bounded by some constant, which depends on the arithmetic and on the smoothness, one, almost all co-cycles are reducible. All co-cycles are almost reducible and three, maybe we should take it since I talk about A, so let me have glmc here to get the matrix. And the generic co-cycle af is non-reducible. So this is the general picture that we have in the perturbative regime for co-cycles which has diophantane frequencies and which are analytic. It's not a theorem, but it has been proven for many classes of systems and there is nothing really, there is good reason to think about this as being true, but it has not been proved in this general. So what does almost all mean and what does almost reducibility mean? Well, almost all means that you look on in this class, you look on generic one-parameter families. And then for generic one-parameter families of such a system, you prove that the system is reducible for almost all parameter values. While almost reducibility, what does it mean? It means the following. So you start with the co-cycle which is of size epsilon, it's small, and then it says that for any epsilon prime, there exists a transformation C prime from TD into GL into the space of invertible matrices. There exists a B epsilon prime, a constant one, such that, how should I formulate it? The transformation X phi is mapped into Y C epsilon prime phi times X phi. This transformation conjugates the co-cycle into two, to a new co-cycle which is not exactly constant, but which is arbitrarily close to constant coefficient, so G epsilon prime in some weaker norm is smaller than epsilon prime. So this is the notion of almost reducibility. You cannot say that you can conjugate your system to a constant co-cycle, but you can conjugate it to an arbitrarily small perturbation. And epsilon prime here is any epsilon prime. So it's not just epsilon square, epsilon to the power 3, but it's epsilon to the power 1 over epsilon to the power 1 over epsilon to the power 1 over epsilon, whatever you like. So this is the notion of almost reducibility. Now these guys here are in general not close to the identity. So they become big. So this tells you that it may not have a limit when you let epsilon prime goes to zero. It may not have a limit when epsilon prime goes to zero. However, the first statement here says that for good one parameter families, it will indeed converge to something. And the third statement says that it will not always converge. So this is the thing which we always have. This is the thing which we have generically in a measurable sense. And this is the thing which we have generically in a topological sense. These kinds of systems, if you think about this one as integrable system, this one as non-integrable system, then they are very, very, they have a very, very complicated structure. And it's hard to distinguish one from the other one. There's some sort of moral of this theorem. Okay, good. So this one is verified in many cases, for many systems, but not in this generality, which I have formulated. That's why I've written it as a theorem between cetacean model. But for many families, this has been verified, this kind of structure. Now, there is an open question here, which I'd like to mention. And for these, at least for the cases where we have verified this, we have verified this in the analytic setting. And I have no idea if this statement is true in C infinity, not in generality or not for special families. So the question of C infinity co-cycles is quite open. We have a similar kind of structure. And if you go to finite differentiability with M sufficiently large, then it's, of course, even still more open. So this is the kind of structure. So there is a lot of questions that remains to solve here, even in the perturbative setting. One may ask global questions. This is a local question. And it seems that this is very much a perturbative statement. There is one thing which is important here, and that is to note, is that almost reducibility is incompatible with non-uniform hyperbolicity. A system that is almost reducible cannot be non-uniformly hyperbolic. If it's hyperbolic, then it's uniformly hyperbolic. And since we think that there are large classes of systems which are non-uniformly hyperbolic, big systems, it seems that this is a local, very much a local phenomenon. But there exists some recent results starting from 2000, essentially for dimension D equals 2, and for discrete systems in dimension D equals 1, where there are some global results. And these are due to Crickoryan, Fayad, Avila, and you, I think, who has also done contributions. So there are some global results, but they are very much restricted. Now, I will not talk about global results. I would like to talk about infinite dimensional systems. If you do this, go through the proofs of this kind of stuff, you'll see the following that this constant here depends very much on the dimension of the system. And when you like to increase the dimension, then this constant will decrease, at least that's the one you get. So there is no obvious induction to infinite dimensional systems. And each infinite dimensional system raises a lot of problems. So let me give you, I would want to give you two examples where we have been able to prove something for infinite dimensional systems. And the first one is the linear Schrodinger equation, quasi-periodic Schrodinger. So this is a PD, and it looks like this. So you have U, T of X, you take the time derivative of it, and you have perhaps minus, I don't remember, U, T of X plus a potential, which depends on time, quasi-periodically. So X here, now X belongs to the torus, it's periodic boundary conditions. V is a function, X belongs to Tn. V is a function from T product torus into R. Here the variables are phi, here the variables are X, and we assume that it's analytic, and this is perturbation theory, so it will be small. Omega here will be our parameters. So they will lie, say, in the unit ball in Rd. So these are our parameters. We don't claim anything for a particular equation like that, but we like to prove something for a large set of parameters. Now, this is a co-cycle in infinite dimension. The way to see this is that you write it in Fourier modes. So Ux can be expanded in Fourier series, and if you write out this equation for each Fourier coefficient, for the Fourier coefficient, then you will see that this guy is equivalent to a co-cycle, which has the following form. A here is a diagonal matrix. So A belongs to GLC, the power CN, and it's a diagonal matrix which has A squared on the diagonal, and F is a mapping from CD into the space of infinite dimensional matrices and its matrix components. And this one is simply the Fourier coefficient of the potential V here. It's Fourier coefficient with respect to the X variables. So this is an Hermitian matrix. So this is what we get. Of course, this is completely formal. However, it's not easy, difficult to prove that star omega has a global flow on the space Ls of CN cross Td, where Ls, L2 of S is the space L2 with weights. So the notion is that U hat belongs to Ls if and only if U hat A squared and then I take A plus 1, I guess, to the power 2S is finite. So on this particular manifold, then this is a well-defined dynamical system which has a well-defined global flow, which is analytic or whatever. Moreover, the flow preserves the L2 unit ball. So if you take U in L2S of CN and you restrict it to the L2 unit ball, then this flow preserves this because these are Hermitian. So this one is anti-Hermitian and the flow is unitary. So this is an example here now of an infinite-dimensional quasi-periodic co-cycle. This one depends on... And the question is, is it reducible or not? Or can you prove anything? And there is a theorem about this. I'd like to mention theorem by Cuxin and myself from 2008, which says the following that if V analytic function is sufficiently small, then there exists a subset of good frequencies. And this is a large subset. So Lebesgue unit ball minus this set goes to zero when the potential becomes smaller and smaller. Such that for all frequencies in this good set, the co-cycle V is reducible. And it is reducible to what? To a system which... what are the variables? Let me call it V hat. So it's reducible to a constant coefficient system. Now, this is infinite-dimensional. So constant co-cycles may be pretty complicated. But this one is pretty good, where B, first of all, is close to A. A was here. B is her mission. And moreover, it is block diagonal, in the sense that B, the matrix elements, are zero unless norm of A equals norm of B. So this infinite-dimensional system decouples into a direct product of finite-dimensional systems. I forgot the I here. So this guy has only purely imaginary eigenvalues. And the conclusion of all this is that corollary, all solutions of these co-cycles are almost periodic. This is, of course, not for all omega, but for a large set of frequencies of omega. So this is what we can prove. This has other consequences. So let me mention another one, which is known as localization. So what you are interested is in the dynamics of this orbit. So here is the lattice, C of n. Here is my initial value, T equals 0. And when you take your dynamics of this, this flow will, this wave packet will evolve in time. So, of course, we start with something that lies in L2S. And what do we know? So the question is, what can we say about this for other times? So what we know is that U hat of T stays in the space L2S. We have a global flow here for each T. What we know is that if I measure the L0 norm, the L2 norm of this, then it does not change. This is because the flow preserves this unit ball. So the question is, what happens now in the higher, the weighted norms? And what you see here is of course that what we have here is a weighted L2 norm. So if this wave packet travels away to infinity, then of course these modes will be measured in the S norm by higher and higher weights. So it will increase. On the other hand, if this wave stays in a bounded region, then the S norm will remain. So the S norm describes part of the dynamics of this orbit. And the corollary here is then corollary is that U hat of T in the S norm is bounded by some constant times S times the zero norm for all T. So this is also a consequence of this reducibility statement. Now this should be compared with a slightly weaker result, but under more general conditions by Bergen, who proved I think in 98 that if omega verifies some diophantane condition, so for almost all omega, the growth is at most logarithmic. Growth is at most logarithmic. So this is a more general statement. It holds for almost all omega, but it gives not a bound like this, but it gives some kind of condition here. And we also constructed examples of V and omega and solutions U hat such that the supremum of this is infinity. Yeah, it's the standard diophantane condition. So this tells us that it's not always bounded. It's almost always in omega bounded by this, and this tells us what happens for a large depth of omega. Now there is a natural question that you may raise in this connection, and that is the following. Since this flow preserves this, then a natural question is, let me put it over here, existed V and omega such that the flow of the co-cycle V omega on the unit ball, which is a sort of unit sphere, such that the flow on this base is transitive, and you could even ask for minimum. So there exist examples where things may go to infinity, but this is a natural question, and this is motivated by something which we know for SO3R and SNR. So this is an open question, and I will be willing to bet a small amount of money on the fact that the answer should be yes. Okay, so let me tell you now something about the wave equation, linear co-cycle periodic wave equation. Or since we have a parameter, well, let me give you the equation. So we are looking at a second order equation. I never remember if I think it should be plus zero. So this is a linear equation, second order. V again is a mapping from quasi-periodic time and periodic space into R, analytic and small. Omega is an element in Rd, and now I will assume that it's fixed. On the other hand, M here is a parameter which belongs to some interval, for example this one, and it's usually in this connection called the mass parameter. So this is an equation that depends on M, and let me ask, for the Schrodinger equation, the signs have no big importance, but they have importance for the wave equation, so let's just check that I got the sign right, plus, plus, minus, plus, that's fine, yeah. So what can we say about the solutions of this equation? Well, one way is to write it as a co-cycle. So you start with phase variables u, and you add v, that is time derivative of u, you go over to Fourier modes, and then you do a linear change of variable, and introduce the variable c. Here is a linear change of variable, linear, linear change of variable, basically you take whatever. And in these new variables, you can write this system as a co-cycle. So it is psi, maybe I should call them hat, but we remember that they are Fourier coefficients. Psi hat, eta hat, time derivative is i times the symplectic matrix 01, 1, 0, acting on, well, we have this matrix here, plus f of phi times psi hat, eta hat, and then we have phi dot equals omega. So what is lambda? Lambda is a diagonal matrix which has on the diagonal the eigenvalues a squared plus m. It's an infinite dimensional matrix. The use of these variables is that the main part here now becomes diagonal. f is a mapping from td into gl, cc of n, and it is a symmetric matrix. This is symmetric, this is symmetric, this is Hamiltonian, so it's a symplectic co-cycle. Now, once again, this is completely formal, but this one has a global symplectic flow on LS2 times CN cross LS2 times CN. This is fibers, this is the phase space cross td. I forgot to say something about the Schrodinger. What is S? And S here must be bigger than some constant that depends on the dimension, which is, I think, d minus 1 over 2, because you need to apply Sobolev's lemma in this case. So it has a global symplectic flow, and moreover, that preserves the subspace eta hat equals psi hat complex conjugate. And the reason why you have this reality condition in this complex system is that it comes from a real system, and we have introduced here complex variables to diagonalize the link. So this is now a well-defined co-cycle, and you may ask, what are the dynamics of this one? Let me give you a theorem, which is quite recent, which says that under, assume a certain diaphanthine condition on omega. I will not specify it, but this is, besides the standard diaphanthine condition, you put on also a quadratic diaphanthine condition on omega. It's something which is fulfilled for almost all omega, so the standard diaphanthine condition plus a quadratic diaphanthine condition. I have no time really to give it to you. Then if the potential is sufficiently small, and sufficiently small here, depends on omega. This diaphanthine condition, then there exists a subset into 0, 1, which is a big subset, which goes to 1 when the potential goes to 0, such that for all m in this, for all good masses, the co-cycle is almost reducible. Now, almost reducible means that the solution, can I have two more minutes? Yeah. Means that the solution can be approximated by almost periodic solutions. And it also means that you have some growth condition on the sobolev norm. Now, what are these approximations and these growth conditions? Well, in general, what comes out of it is, it depends on the estimates that you have. And I will not give you the estimates because it will only bore you, but you have a transformation, which transforms this to, say, epsilon prime close to constant coefficient. You like to know something, how big is this transformation? And you like to know something about the reduced system. How is it diagonal? Is it block diagonal? So, besides this qualitative picture, what you need is to have some estimates. And what kind of estimates do you have? Well, I will not give them, but I will give you a corollary once you can prove. So, if you look at the solutions in this norm and how they grow, then these are bounded by some constant, log t plus 1 times c hat omega s plus eta hat s and this is for all t. So, you get some kind of growth condition. Now, I did this last year and I present this on a conference in Rome and the enthusiasm for this growth condition was very, very weak. I mean, Italians are hard to... So, I worked harder. And in fact, a bit harder, I improved the estimates a little and what you can prove is this. And now I invite you all to admire this beautiful log log estimate of the growth of the sublet norm. This is pretty constant. Well, the exponent, I think it's s. It depends on s and I think it's s, it may be 2s or something like that. So, this is what you can use almost reducibility to. Don't get growth conditions, but you get pretty close. So, with good estimates, I mean, the moral of this is with good estimates, almost reducibility is almost as good as reducibility. The final part of my talk will now be to give some ideas of the proof of this but since my time is running out, I think I will spare you from this. So, thanks for your attention.