 Okay. Thank you very much, Stefano. Well, I've been interested in minimal diffeomorphisms and homeomorphisms in dimension two since the very beginning of my PhD at IMPA. And in fact, I'm very interested in, essentially, in this question, in this problem. And my main motivation to study such systems comes from the lack of periodic points of minimal diffeomorphisms or more generally homomorphisms. You know, periodic points are extremely useful in dynamics and at certain extent, minimal diffeomorphisms are the simplest periodic point-free systems because when you have minimality, you have a certain form of homogeneity. So if you want to understand systems with no periodic points, maybe that's a good starting point. And the choice of dimension two, of course, comes from the fact that in dimension one, everything is very well understood. I mean the dynamics of minimal homeomorphisms after Poincaré. And dimension three already is too difficult. You will see that all the results I will mention so far we don't have any idea of how to attack the problem in higher dimensions. And in fact, the first problem I considered for my PhD thesis was the following one which can be considered as a very weak version of the CR-closing lemma. Essentially, the idea is if you have a minimal diffeomorphism, a CR diffeomorphism, if it is possible to destroy minimality by arbitrary small CR perturbations. Fortunately, I found some other problems because that one is still open. I mean, to tell you the truth, I think we are very far away from solving this. And you see dimension two, minimality, so far it's not clear how to attack this problem. And in that moment, in the beginning of my PhD, I had the naive idea, the naive project of trying to get some kind of classification of minimal diffeomorphisms in dimension two. And so if you are interested in finding some kind of, maybe not classification, but to find some common characteristic between minimal diffeomorphisms, let's start recalling very well-known examples of minimal diffeomorphisms. Of course, the first one are ergodic rotations, are the fundamental example. In fact, all the time what you do is, what you consider as another system is to compare the dynamics of the diffeomorphism that you are studying with respect to the rotation. So this, of course, is the fundamental example. The second kind of examples are time t of reparametrizations of minimal flows. So you start with a minimal linear flow. You consider a reparametrization. The reparametrization is still minimal, the flow. And so that means that you have a genetic set of times such that the time t is diffeomorphisms. You have an ecological condition here. And if this ecological equation does not admit continuous solutions, then you really get something new. I mean, something new means different to an ergodic rotation. And in fact, if you have that property, you'll get weak mixing, for instance. So it's the time t, when it is minimal, it's weak mixing, so you have something new, different to a rotation. The third kind of examples are the celebrated work of Furstenberg, which was the first example of a minimal diffeomorphism, which was not uniquely ergodic. Once again, you have a comological condition here. You ask this comological equation if there is no continuous solution, then automatically you know that the skew product, this is an skew product here, is minimal. And if this equation admits a measurable solution, then you have something which is metrically isomorphic to the rotation with alpha zero, which is not, of course, not ergodic. And so this was another kind of example. And the four kind of examples that we will consider here are irrational then twist. This one is completely different to the first three examples because the first three examples belong to the isotopic class of the identity. They are isotopic to the identity. And this fourth one, it is not. It's isotopic to, so right here, but it's isotopic to the linear map given by this matrix in t2. So these are probably the most well-known examples of minimal dephiomorphisms. And what they have in common, all these examples here is that they exhibit an invariant foliage. You can see in all these examples you have an invariant foliage. And this is, of course, extremely useful because when you have an invariant foliage you have a one-dimensional transferred structure. And so at certain extent you can reduce the dimension of the manifold. And so it's very desirable to have an invariant foliage. So the question is, okay, is it true that any dephiomorphism in t2, minimal dephiomorphism exhibit an invariant foliage. The answer is no. In 2009, with Andres Koropeki, we showed that if you take the smooth conjugacy class of rotations and you take the infinity closure, this is the typical space where Anosov-Katok method work, you show that a generic dephiomorphism exhibit no invariant foliage. And previously, Michel Hermann and Albert Fatih had shown that a generic dephiomorphism here in this space is minimal and uniquely algorithmic. So combining these two generic constructions you can show that there are a lot of dephiomorphisms with no even topological foliage. So at certain extent this is a bad news because, well, all the things are a little more complicated than you can think before. What happened in the isotopic class? Whoops. Stop it. Ah, sorry. What happened in the isotopic class of the twist? Is it true that there are minimal dephiomorphisms with no invariant foliage? Okay, we will return to this question here at the end of this talk. But the answer for question number one is no. So how you can study minimal dephiomorphisms? How you can continue the analysis of the dynamics of such systems? Well, let's concentrate in the isotopic class of the identity. In the isotopic class of the identity you can try to imitate the rotation theory of Poincaré. So not to denote the isotopic class of the identity. When you have a different homomorphism which is isotopic to the identity, any lift to the universal cover can be written as identity plus a CD periodic function. Okay, this is very classical. And in Poincaré showed that when you are in such a situation the integral with respect to any invariant measure is the same, it's always the same. This is the so-called rotation vector, rotation number, sorry. Hence if you have this for any invariant measure that means that this expression must converge uniformly to the rotation number. Okay, this is a situation in the one-dimensional setting. In higher dimension the situation changes a lot. In general you cannot expect to have a well-defined rotation vector but what you have is a rotation set. Again, you are considering the same expression than here, but the difference is that you can have different measures with different integrals when you integrate the periodic part of the cover. Okay? And so you will say that f is a pseudo-rotation when you have a well-defined rotation vector. The setting in higher dimension to study homeomorphisms which are isotopic to the identity. And so the question is, okay, what happened with minimum, yeah, sorry. For instance, a good example here is the following. Let's suppose this is the circle, 0, 1. Here in the middle you have any rotate. It's an SQ product. And here you put a completely different rotation. You press up here. And in this direction you put a north-south dynamics. And so for instance, in this case, you have every point here has rotation vector in the up direction, here in the down. And any other point here has exactly this rotation vector because you converge on S. So this is a very simple example when you have essentially two, only two rotation vectors that they are associated to ergodic. You have only here, if these are irrational rotations, you have only two ergodic invariant measures. And in this case, the rotation vector is vertical. This is over the, sorry, imagine this is the 0 and this is the rotation vector of this guy here and this is the rotation vector of this guy here. And the rotation set is a whole segment here, but you only have point with this rotation vector and with this rotation vector. I don't know it. Oh, okay. No, no. Yes, for instance, in... Not very much. In this case, when you have minimality and when you have unique ergodicity, you automatically have a pseudo rotation because you have only one invariant measure, you have only a vector here. And you can see that here... Well, I didn't say that, but genetic diffeomorphism here are weak mixing. So you have a pseudo rotation with a completely different dynamics comparing with the rotation, with the real rotation. Okay. So, yeah, in general, it is not to be a pseudo rotation. Well, it's not very close to be a rotation. So this is a pseudo rotation. So all the examples I mentioned, all four... No, sorry, the first three examples that were isotopic to the identity, all of these examples were pseudo rotations. Of course, the rotation, it is clear. The parametrization of the flow to show that it's a pseudo rotation is a nice exercise. A skew product of ergodic rotations of the circle are also pseudo rotations. This is a nice exercise. Not exercise. It's a problem. This was a lemma proven by Hermann in some paper of his. So, all these examples that I mentioned before are pseudo rotations. Okay. So, what else? What can we say about the rotation set of a minimal diffeomorphism, a minimal homeomorphism? You have this theorem of Frank's, which is very useful, because as you can see here, I said if you put an irrational rotation here and a rational rotation here, in this example you have no periodic points, even though you have a lot of rational points in the rotation set. So, in higher dimension, it is not true that any rational vector is realized by a periodic orbit. But the theorem of Frank's said that if the measure is topologically equivalent to Lebesgue, then if the rotation vector of the measure is rational, then you have a periodic point. So, that means that in the case of minimal diffeomorphisms, you know, any variant measure has no atoms and has no total support. So, that means, by Oksofius and Thornin, it is equivalent to Lebesgue. And so, that means that if you have a minimal diffeomorphism, then it cannot contain any rational point. In particular, we saw the rotation set is a compact convex set. So, it must have the rotation set has empty interior. So, essentially, you have two possibilities. You have just a point. So, you have pseudo rotation or you have a segment, but you cannot have no empty interior for the rotation. A segment like this. When the rotation set is not a point, the rotation set is a compact convex set. So, it can be a point. It can be a segment or it has no empty interior. But it cannot have no empty interior because of this property here. So, these are the two possibilities. And in the beginning of 1990s, Miserévic has conjectured that any minimal diffeomorphism, in fact, any periodic point-free homomorphism must be a pseudo rotation. Very recently, Artur Abila has constructed an example of a minimal diffeomorphism which is not a pseudo rotation. In fact, the rotation set is a segment of irrational slope. So, if we were trying to get some kind of classification of minimal diffeomorphism, things are getting worse and worse because we have no invariant foliation. The rotation set can be not a point, but maybe a segment. So, the situation is much more complicated. The good news is that these pathologies in a certain way cannot coexist. So, that means that, this is the first result I want to talk about, if you have a minimal homomorphism which is not a pseudo rotation, then you have something very similar to an invariant foliation. It is not exactly a foliation. It is a partition of the manifold in sets which, in the universal cover, they are closed and connected, and they have a well-defined homological direction. So, it's something like a foliation with leaves having probably some hairs. Maybe it's not something like this, but probably it's something like this, something here, maybe something even worse on this, but they are set in the universal cover that can be written as intersection, or less the intersection of strips having a well-defined homological direction. So, this is the first good news that you cannot have simultaneously a minimal homomorphism which is not a pseudo rotation or a invariant foliation. It is not a foliation, but it's something similar to that. And the second result is a work in progress with Patrice Le Calvet. And essentially, it's answered in a topological way. The second question I asked before, if you have a minimal homomorphism, which is isotopic to a dend twist, then automatically, sorry, automatically exhibit an invariant pseudo-foliation. Essentially, what you have is that any minimal homomorphism in the isotopic class of dend twist is automatically an extension of a circle minimal rotation, a topological extension. Once again, the fibers can be very weird. But you have something like a foliation, you have definitely a factor, you have a semi-conjugacy with the rotation. Okay? So, how you can study this problem of invariant foliation and so on? The key idea here is the rotational deviations. In the one-dimensional case, you have a very, very nice property, which is uniformly deviations with respect to the rotation number. That means that if you have a circle homomorphism, anyone, and you consider a lift, then the displacement of any point after n iterates, and if you compare this displacement with the rotation, yes, it can be different from zero, but it's always bounded by one. So that means that every point goes with essentially the same mean speed of the rotation. In higher dimension, this doesn't hold anymore. For instance, it's very instructive to see what happens in the Furzenberg case. In the Furzenberg case, in the horizontal direction, you have a rotation, a true rotation. So, of course, there is no deviation there. So that means that if you analyze the deviation with respect to the rotation vector here, and you analyze what happens in the horizontal direction, there is no deviation at all. But on the other hand, you have unbounded deviation in the vertical direction for every point. So this is a typical situation where in certain directions, it's okay. You have bounded deviation but unbounded deviation in others. And there is a folklore theorem, which is, in fact, it's an easy corollary of Gotchalk-Hedlum theorem, Gotchalk-Hedlum theorem that I will need later. If you have a homeomorphism, which is minimal, and you have a continuous function, a real continuous function, such that the birch of sum, sorry, and there exists a point x0 in x, such that the birch of sum of x0 is uniformly bounded, is uniformly bounded here, and exists m for every n. If you have this, then automatically you have, that there exists u, continuous here, such that uf minus u is equal to 5. And this means that if this holds for one point, automatically holds for any other point, not exactly with n, but you will have that the birch of sum in any other point is also uniformly bounded. And this is just a consequence of this, it's a corollary of this theorem. It's said that if you have a certain direction, when you have bounded deviation with respect to the rotation, then you automatically have an invariant pseudo-foliation. It is not, as I said before, it's not necessarily a foliation, so using this theorem here, you will see that if you want to study the existence of invariant pseudo-foliation, what you have to do is to analyze the problem of rotational deviations. And so that means that we will consider the case, for instance, I will consider in this case where the rotation set is a vertical segment, and we will try to analyze what happened, we will try to show that such a system exhibit bounded deviation in the horizontal direction. Bounded deviation with respect to this rotation. Yeah, alpha. And this is the way that you construct the invariant foliage. So how you can study the problem of rotational deviations? Let's start with this case, which has nothing to do with the minimal case, because in certain cases it can be, but suppose that you have this, you have the rotation set, it's contained in the y-axis, okay? So if you have such a situation by a classical work of Birkoff, you can show that you can define this set here, you can show that it's not empty. What is the idea of this set here? You are saying this is the plane, you are considering this is R, the vertical, and you are considering this semi-plane, okay? And after that, you take the maximal invariant set of this semi-plane, and this is the connected components, the unbounded connected components of this invariant set, okay? And Birkoff showed that these sets are not empty whenever you have this property here. If the rotation set is contained, the first coordinate of every rotation vector is zero, then you have that the maximal invariant set here has unbounded components. And that means that this set is not empty, okay? And what's the topology of this set? If your f here exhibits bounded deviation, uniformly bounded deviation, then these sets here are very thick in the sense that you have that, you have all the semi-plane containing this set. And this is essentially if the point is here and exhibits bounded deviation, it cannot cross this with some sort of iterate. And when you have unbounded deviations, this cannot happen, or I'm cheating a little bit, but you have something like a hertz, okay? It's not filled. They are just hertz. And this is very extremely useful to study the problem of rotational deviations, because when you have an invariant set which have something like a hertz structure, this imposes certain restriction on the possible dynamics on the vertical direction. And for instance, this set were used by Nancy Gellman, Andres Koropecki, and Fabio Tal to show that if you have an area preserving an area preserving homeomorphism, and the rotation set is a vertical segment, then you automatically have uniformly bounded rotational deviations. There is no chance of having such a hairy invariant set. Okay? Okay, so what happened for minimal homeomorphisms? For minimal homeomorphisms, you know that if f is minimal, then you have this property due to Frank's result, and if it's a vertical segment, you are supposing that it's not a pseudo-rotation, then automatically you have that alpha that does not belong to Q. So in particular, it's different from zero. So if you try to imitate the situation in the previous slide, and you say, okay, I will try to define exactly the same set, you will see that this set is empty, because everybody, suppose that alpha is positive, everybody is going in this direction, so the inverse is going in this direction, and every point escapes from this set. So it is not a good idea, it is empty. So you have to change the definition. Okay. So in fact, what we want to study here is the following. We are trying to estimate this expression here. This is n alpha, we are trying to estimate the deviation with respect to the horizontal, in the horizontal direction, and so you want to estimate this quantity here. So one thing that you can do to imitate the situation in the, where the rotation set was on the y-axis, sorry, is the following, you could define this set, for instance, and you will have exactly the same thing that you have before. Okay, that means that when f exhibits, if x exhibits bounded deviation, then you have a thick set, like this, and if it exhibits unbounded deviation, then you have a hairy set, but it is not empty. So it's this definition, this attempt of definition is much better than this one. But there is a problem. This set is not dynamically defined. It is not invariant, I mean, by anything. So it's not very useful to do dynamics. It's not an f-invariant set. It is not an f-invariant set. So you have the set, you have something very similar to a previous case, it seems to be rather hard to use it to get any result. So what can we do is the following. First of all, they will work in the area-preserving case. Minimum homeomorphisms, you can suppose that they are area-preserving because you can conjugate to preserve the back. This is SOC-STOV-ULAM theory. I will consider the space of lifts of area-preserving homeomorphisms, which are isotopic to the identity. And you have, in this case, you have the rotation vector of Lebeck. This is exactly the same definition I used before. But in this case, you have a group homeomorphism. Okay? So you have a group homeomorphism and the kernel of this group homeomorphisms are the so-called Hamiltonian homeomorphisms, homeomorphism, sorry. I mean, I just consider this as a definition. Hamiltonian homeomorphism is a homeomorphism such that its area-preserving and the rotation vector of Lebeck is zero. And when you have such a situation here, you have a simple group theoretical remark that this short exact sequence split. This essentially means that this projection here has an inverse. Forget it, it's not important what is interesting here is that the symplectic homeomorphism or area-preserving homeomorphisms can be written as a direct product, semi-direct product of R2 and Hamiltonians. And when in algebra you have a semi-direct product in dynamics you have a skew product. So if you consider this decomposition here, you can construct a skew product which is very useful for dynamics in the following way. You consider an area-preserving homeomorphism and a lift. And let us write rho for the rotation vector of Lebeck. In this case, then you can define the following map which is an skew product over T2 times R2. This is essentially motivated by the previous formula, the previous remark I said, that the group of symplectic homeomorphisms can be written as a semi-direct product of R2 and Hamiltonian homeomorphisms. So you have this structure here and notice that the interesting point here is this. This is a Hamiltonian homeomorphism. And why this is interesting for us? Well, you can write the lift in this way. You can easily show that you have this expression here. And that means that when you take the n iterate, you have this property. Forget about the first coordinate. Consider the second property. The second coordinate is exactly what you want. You want to iterate n times. You are iterating here, and you are pushing back to the origin in a certain way. So this expression here is always Hamiltonian. So that's the reason why I define this as a Hamiltonian skew product. You have on the base you have a rotation and on the fiber you have Hamiltonian homeomorphisms. So that means that you can now apply exactly, not exactly, but you can define the birch of stable sets using this skew product. You know because suppose that the rotation set is always alpha AB, you consider once again the vertical semi-plane and now you define for each T you will have a fiber invariant set for each T in T2. You consider exactly the same set, but now for the skew product. And you intersect this invariant set with the fiber of T and you consider the unbounded connected components on this set. It seems to be a rather cumbersome in a certain way, but on the other hand you have this. This is really an F invariant set. This is a dynamically defined set. So taking the union on each fiber, you have a dynamically defined set. And now you can try to get the same result you had before. You can try to get here. The first remark is the following. Of course we are trying to prove through MA that we are trying to show that we have uniformly bounded deviation with respect to alpha in the horizontal direction. So let's suppose that this is not the case. Suppose that you have unbounded deviation. And due to the Gotchal-Hettlund theorem you can suppose if you want that you have one point with unbounded. You have one point with unbounded deviation. In this case, in fact to prove that the set is not empty, it is not necessary to have unbounded deviation. But in this case you have that this set is not empty. The second case is also rather easy. If you are considering the maximal invariant set with respect to this semi-plane, of course this contains the maximal invariant set starting here to the right, this is clear. But the two fundamental properties are the following. If you take R very, very negative in this way and you take the union, you have something dense. And on the other hand due to Gotchal-Hettlund theorem if you have a maximal invariant set by the right by the left they cannot intersect. Because if you have one intersection you have a point with uniformly bounded deviation and if you have a point with uniformly bounded deviation you have every point with uniformly bounded deviation. So you have two hairy invariant sets such that if you translate in this direction you get a dense set. If you translate in this direction you have a dense set but they don't intersect. They don't intersect. So you start to see that it's very particular. The idea of the proof is that always assuming this unbounded deviation the first thing is that you can show that the homomorphism, the original homomorphism is spreading. What is spreading is something stronger than topologically mixing. In fact, what you can show is that you take any invariant set, sorry, if you take any small set U and you take a ball of rises R, very large and you take epsilon very small then you have after some time N for every small N you have that this set the N iterate of this set is epsilon dense in a ball of rises R. So this is in the universal cup. So you have something which is much stronger than topologically mixing. For instance, in the Anosov case you always have mixing but always along the direction of the unstable, of the unstable direction. Here you are getting a mixing in every homological direction. How you can prove this essentially it's rather complicated. This was a result of a student of mine in Brazil. The idea is that you have this head structure here. This is this set of minus infinity and you have some head structure here from plus infinity and you have inside the set U you have point that goes up and point that goes down and you have to avoid this set here if you started initially you have chosen U the joint of this set you have to avoid this and so that means that the image of U have to oscillate a lot. This is essentially the idea of the proof. Secondly, you have this the horizontal set and you can also define the vertical set. It's essentially the same thing here you are considering not this semi-plane here but this semi-plane for instance the upper one. You can show that this set is completely empty but the argument is completely different to the Birkhoff argument. This is a combination of Lefchitz-Fixhorn theorem a result of Patrice Le Calvea and Jean-Claire Stofiot-Cos for the index of iterates of isolated fixed point and another theorem of Réderique Le Roux about also the index of not isolated of fixed point but combining these three results you can show that this set is non-empty and due to the spreading condition you have two Hailey sets in this direction you have two Hailey sets in this direction and all these sets are disjoint they cannot intersect I mean this third argument is a very crafted argument it's not easy at all but you get a contradiction from the existence of this fourth set that cannot intersect and well I have three minutes and for the case of the dent twist the argument is similar at first we thought that the dent twist should be harder but in fact it's easier this is less involved so you have when you have a homeomorphism which is isotopic to the identity to the dent twist of this form here any lift have the following form here in this case you cannot consider the rotation vector because this function here is not a co-cycle for the homeomorphism but this one is a co-cycle so you can imitate the rotation theory you can define in this case the vertical rotation set essentially in the same way okay when I say when this is a co-cycle I mean that if you take the n-iter here what you get is the Birkhoff sum and in this case it is not true here you don't have the Birkhoff sum okay or not the same or what the problem is this m here so you have the vertical rotation set and there is a recent result of Salvador Adasanata García Fabiotal who showed that if you have a minimal homeomorphism which is homotopic to the dent twist then the rotation set is just a point and this point is Russian this point is Russian so what happened here well you can construct a very similar skew product the problem is that the skew product now it doesn't exist on T2 times R2 but it works in T times the open annulus essentially the formula is the same you are also using this idea of the semi-direct product here you will see that here if your this area preserving is area preserving yes and this is a vertical rotation vector you will have here something which has zero rotation number with respect to the vector so you are pushing all the time things down and so you can define exactly the same set but in this case you define the vertical set well you have unbounded vertical deviations you show that the set are non-empty closed and unbounded and using the argument once again of minimality you can show that this the set from below and from above are disjoint so they have non-empty interior and again you have the same property that the translates are dense where essentially this is the whole idea because when you have a twist map there is no invariant set like this like that one if you have an unbounded invariant set for a twist map with zero a complete invariant set automatically it is the neighborhood of infinity it is completely filled but in this case it is not necessarily true that it is a twist map but when you regard a homeomorphism which is isotopic to the identity if you look them from very far away it seems to be a twist map so you can repeat some arguments here and to show that this is not the case thank you very much