 Okay, first of all, I would like to thank the organizer for giving me the opportunity to present my work here. So as Stefano said it, I'm going to talk about Integrability of Quentinus bundle and since also I'm given less than half an hour I'm just going to go through to the point of of this. So throughout the talk, we will denote M is an N plus M dimensional compact manifold. So as you see, there are just maybe two technical words that we need to define from the title, which is continuous bundle and what does it mean to say that a Quentinus bundle is integrable? So definition dimensional continuous Okay, I will really emphasize about the continuity is a continuous choice the continuous choice of dimensional linear spaces for every P in M So, I mean, I think most of you are familiar with this You you can think of like if M is one you have just a choice of a line Which you can think of it as also a vector field. So if M is two you have a choice of planes So we said it is continuous. So if the change of the point is continuous So I also want to define what it means to be integrable before we go to the point of an M dimensional Quentinus E is said to be integrable respectively unique, I will Just discuss a bit about this respectively uniquely if through every point There exists dimensional supermanifold. Okay, there exists respectively unique submanifolds everywhere tangent to E everywhere tangent to E Yes, so this is the problem that we are interested in here and As you may know or so it is a very classical problem in the study of many areas mathematics It has application like on PD partial differential equation on ordinary differential equation and particularly on dynamical system Where we are in and I will I will tell you about more a little bit later so and also as you may know this problem is a very Classical problem and also you might heard about the the well-known Frobenius theorem Which really gives you a necessary and sufficient condition for integrability of a bundle given regularity, so and and before I talk about the Like the Frobenius condition I would like to to to change a little bit of language because you know you if you have a bundle You can define it using using differential forms Okay, let's say if you are in a in a Three-dimensional space you have a distribution of plane. There is an orthogonal vector of it So you can choose a form sitting on that vector So that the kernel of that form defines exactly the bundle E and and you can choose a form Also has the same regularity as a as a bundle. So what I'm saying basically is that given And the dimensional is bundle E There exists there exists eta 1 Okay eta n since I am in a E is co-dimension n So I will have n linearly independent continuous yes One forms Is that you have exactly E is given by the intersection of the kernels of eta i's i equals 1 n Yeah, so what is the usual condition that we took that that that that is in the Frobenius theorem is what is called the involutivity Okay, so I will define what is that? invuletivity so from now on every time I'm taking a bundle I write it like this associated to N linear forms and the regularity of the form is exactly the regularity of the bundle itself so E equals intersection kernel of eta i's Is smooth By smoothness I mean just see one is enough Move there are E is involutive if For every If this is satisfied eta 1 wedge eta n wedge d eta i equals 0 for all i Yeah, this is also in most of the textbooks on differential geometry and many kind of textbook This is what is known as The invuletivity condition and the theorem of Frobenius is saying that E is Inter-global if and only if Is involutive so the problem that we are addressing here is to try to give a Version of the Frobenius theorem by relaxing the regularity assumption and we will see later What are the motivation of our study really why we want to relax this assumption? So we need to introduce another notion of invuletivity here Yeah, so we we introduced the notion of asymptotic invuletivity Yeah, yeah. Yeah. Thank you. That's the thing because this This quantity here you cannot talk about it when E is just continuous You cannot talk about extra derivative of these one forms. They don't make sense That makes sense. So what we say is that E equals kernel of Intersection is asymptotically invuletive there exists epsilon positive and a sequence eta k i i equals 1 till n k greater than 1 a sequence of differential c1 smooth One forms is that these two condition holds we have that these forms they approximate these guys in the C0 norm eta k i Is converging to eta i for all i this is just C0 convergence and the most important condition is that We have eta 1k wage eta lk wage d and k i the norm of this times the exponential of epsilon norm of d eta k i Is going to zero as k go to infinity for for every i One This is what we call asymptotic invuletive and as you can see if you have smoothness and invuletivity Is the equivalent to to this because you can take exactly your sequence to be the form that you started with And you have it automatically Think I'm going to move to okay, so we need another concept or so Which I will tell Why we need it which is a kind of regularity assumption which is exterior regularity Say that e of kernel of eta i is exterior Regular or so we have a sequence If there exists epsilon positive always and a sequence as there but Here the sequence that we have here in the sequence that we have in the other definition They might be different if the both both condition are satisfied Sequence of smooth One forms is that so that this condition is satisfied meaning Beta k i minus b minus eta i sorry times exponential of the epsilon d eta d beta K i is going to zero as k goes to infinity for every i You can see really this is this is kind of thing that the the rate of convergence should should really should dominate how this quantity can blow up because as you know beta k is a In some sense it might approximate this Continuous form so you can expect the d beta k to behave very badly meaning Exploding to infinity so what you really want here is this to be control Do to go to zero faster than this can blow to infinity and as also you can see if the starting Differential one form was smooth you have this is automatically satisfied because you take your beta to be the starting form And this is zero at the first Yes Now I'm in a position to state the main problem that I want to state which is a joint work with with Stefano Lozato Then naturally and myself so as you can as you expected is that II that II be an M-dimensional Continuous we have that if a is asymptotically invuletable Then it is integrable Maybe not uniquely though, but it is integrable Moreover if II is integrable and exterior regular then It is uniquely integrable so this can stand as a Generization of the Frobenus theorem for the continuous in the continuous case where we relax the regularity also I should I Should say that there are various version of generation of the Frobenus theorem recipient the literature mainly due to hot man and Simich slow but and smith so hot man deep hot man and I Mean there are many others. So the literature so hot man has a version of Frobenus theorem where He has he has a stronger version of us of invuletivity We should require we should require this quantity Deeta to exist for discontinues for discontinues form deeta to exist and also Simich has the organization to lip sheets to lip sheets continuous bundles like assuming lip sheets where Where you will check this condition almost everywhere. So both of of these generalization they They they apply to system where you have some regularity and our our case We found for instance in dynamica system an application where you don't you don't assume any regularity just continuity This continuity that's what I'm going to go first About to talk about the application on dynamica systems is Which is which is the class of system? This generalized a little bit what what Federico defined yesterday the annus of the thermo fisms So they belong to a larger class which is the class of the thermo fism that admits what is called a dominated splitting So I just recall that briefly What it means was this time to have a dominated splitting. Let's say let phi From M to M C to D fellow Morphism is said to have The dominated splitting splitting two conditions Was like we have Dm the tangent spacer of M at every point splits into two bundles E plus F namely and and they are invariant meaning that if I apply the differential d phi x e x equals e at phi of x and d phi x f x apply to is Phi of f of phi f of phi of x So you have seen this in the previous course, which is a typical case of An annus of annus of the thermo fism and there is another condition, which is why we call it dominated splitting that the supremum of D phi I'm assuming that I'm given a remanion metric that I'm using here d phi of x restricted to e x is less restricted less than the infinimum if I Fix so I should also tell you that this is not the most generous splitting like dominate splitting that you can have This is what is called absolute Dominate splitting you could require this to be point wise But for our purpose for the application of our term we will we will require it to be absolute for instance And as an example you can see that annus of the thermo fism They really have this they really have this splitting and the problem that we we addressing here is now the Integrability like as it exists integral like is e integrable meaning that we have Integral manifold of e which in the case where it is uniquely integrable will give a rise to what is called foliation and So we have this condition. Oh, yes. Yes So yes, this is the core norm. Yes. Thank you Yeah, so this is the minimum expansion that you can see on the on the F. Yeah Thank you. So this is a condition that we're requiring. So definition Okay, I'm assuming that I have a dominate splitting e is said to have Is said to grow at most linearly? there exists c d positive set that Define k restricted to e This norm is less or it was to ck plus d In an application of the version of our theorem said that so in this setting if he grows at most linearly then It is uniquely integrable uniquely integrable and I mean this linear growth is is a condition which which Which is I mean you can see that for the annus of the thermo fism We have linear growth because what the annus of different morphism you have this Define restricted to e Is less than one which is less than Define restricted to f So this is really a contraction. So I'll show me saying that you can allow a little bit of Expansion but not more than linearly and you you still have Lee you still have unique integrability so this also could stand as as a new proof of the of the well-known stable manifold theorem because the proof that e in this case of annus of different morphism is uniquely integrable is do is called a stable manifold theorem and and this also gives another variation of a proof and So as just the proofs lecture was saying if in the case of partial hyperbolic system Or so when you have quite as a metric on the When you have isometric story when you have isometric on the central bundle because if you remember just half an hour ago there were there were this type of splitting that is called partial hyperbolic that the previous speaker just defined and One condition. I mean which would imply this for instance is that you see you have isometric on the on the central on the central Bundle in that case you can integrate the join of ec plus ec plus es and if you suppose it also for the inverse map You can integrate these two uniquely and you take the intersection of the leaves you get you get exactly in variant Foliation tangent to the central bundle which which is also a Very well, I mean a problem for for the study of these systems So I think as I said before our theorem has also Applications on partial partial hyperbolic partial differential equations partial differential equation because you the original for a benefit I am was was for the study of of what is called a FAF system, which is a system of linear PDE So you can also apply this to get uniqueness Solution of linear PDE, but since I'm given half an hour and the urgency is about dynamical system So this is the the application that I wanted to tell you about so I think that's up here