 Thank you very much for the invitation. It's special for me to be able to talk about this subject in Montevideo. The C2 manes conjectures for surfaces. Let's see the setting. The name is a closed, without boundary, Riemannian manifold. And a Tornadilla grandian is a C2 function defined on the tangent bundle, such that on the fibers it is convex. It means the second derivative is positive definite and super linear. It grows more than any linear map. The action of a curve, given a C1 curve, say, on the manifold, the action is the integral of the velocity vector of the curve. The curves which locally minimize the action satisfy the Euler Lagrange equation. The Euler Lagrange equation is a second order differential equation, so the initial conditions are given by the velocity vector. The second order differential equation to initial conditions is the position and velocity. The Euler Lagrange equation defines flow on the tangent bundle. You will thank this. Do you hear me better? A little more. A way to look at the flow is to take an initial condition, which is a tangent vector. Then solve the Euler Lagrange equation with that initial condition. At time t, the flow gives the tangent vector of the curve at time t. It is like the geodesic flow. In fact, the geodesic flow is one example of Lagrangian systems. Examples of Lagrangian systems are the geodesic flow for which the Lagrangian is given by the velocity square or the kinetic energy. Another example is the classical mechanics Lagrangian, which is given by kinetic energy minus potential energy, which is this Riemannian Lagrangian minus potential energy, which is a function which only depends on the position. The Euler Lagrange equation in this case is Newton's second law. Force is equal to mass times acceleration. Another example, there is a Mañez example, in which any vector field can be embedded inside a Lagrangian system. A Lagrangian system whose phase space is of the dimension double of the vector field. I wanted to talk about classical mechanics, because we are going to talk about perturbations at La Mañez. Mañez's genericity for Lagrangian system means that we can perturb a Lagrangian but only by adding potentials. A property for Lagrangian systems is Mañez's generic. If any Lagrangian, given any tonality Lagrangian, there is a generic set of functions defined on M, such that the property holds for the Lagrangian when we add this new potential. In the case of classical mechanics, that corresponds to perturbing the potential energy, but not perturbing Newton's law. It is natural in that sense, because a general perturbation of the Lagrangian would involve both S, X and V. This is the same kind of difficulty one has to perturbe just as it flows. Any perturbation cannot be local in the phase space X, V, because a small perturbation in the variable X corresponds to a cube on all the fibers, perturbs all the fibers V. The perturbation is never local. In particular, that's the reason that we don't have a closing lemma for Lagrangian systems, because all the proofs of closing lemmas are made with local perturbation. Besides that, the difficulty of perturbing by a potential or perturbing a Riemannian metric are more or less the same, just technical things. The action of invariant measure, if one has an invariant probability for the L-L-Lagrange flow, which lives in the tangent bundle, the action is the integral of the Lagrangian on the probability. We will look for measures which minimize the action, minimizing measures. The manes conjecture is given, this manes genericity thing, given a tonality Lagrangian, there is a generic set of potentials in CK, if we are talking about CK topology, such that given a potential in this generic set, the Lagrangian perturbed by the potential L plus V has a unique minimizing measure, which is supported on a hyperbolic periodic quality, or a hyperbolic singularity. Manier used to say that conjecture is for free, so he would put CK or C-infinity freely in the conjecture. We can prove it only in the C-topology. C-topology, if you see the L-Lagrange equation, the L-Lagrange equation depends on the derivatives of the Lagrangian, so the vector field for the L-Lagrange flow is written in terms of the derivatives of the Lagrangian. So, C-topology in C2, the Lagrangian corresponds to C1 perturbations of the vector field, so we are going to use arguments resembling some closing lemmas, C1 closing lemmas. So, reductions. The main two reductions here are the uniqueness and the hyperbolicity of the periodic orbit. First, Manier proved that C-infinity generically, Manier's generically, the minimizing measure is unique. And then Manier stated, and then we gave a proof that if the minimizing measure is supported on a periodic orbit, the generically is a hyperbolic periodic orbit. One expects that generically, minimizing things are hyperbolic. Okay, and then the Manier's characterization of minimizing measures, yesterday it was called the revelation theorem, will be a following. An invariant measure is minimizing if and only if it is supported in the over-reset. We will say afterwards what the over-reset is. The name comes from Fatih. There is an inclusion of several invariant sets. One is called the Manier set. This one is the over-reset, a matter set, the support of the minimizing measures. And if you see the inclusions, the names are chosen, such that if you see the inclusions, the word Manier appears. Okay, the set of tonally lagrangians, which has a unique hyperbolic periodic orbit, the orbit is open. The semi-continuity of the minimizing measure and the fact that the hyperbolic things are open. So Manier's conjecture becomes reduced to proving the following. The densely in the over-reset, the over-reset contains periodic orbit or singularity. So we will not deal with the singularity because we just say if there is a singularity, it will be in the over-reset. It will be hyperbolic, but same things. And then that's one of the cases in the conclusion. So we discard that case and look for the periodic orbit. So I have to define the over-reset. We start by talking about the Manier's critical value. So the Manier potential, given two points on M, we want to look at the minimal action of going from X to Y. What may happen is that if we have a closed curve which has an action which is negative, then what we can do is if we want to minimize the action from X to Y, is go from X to a closed curve, go around it infinitely many times, and then come back to Y. And then the action will be many times times this negative number. So the infimum will be minus infinity. So that Manier potential will be minus infinity in that case. So in order to avoid that, we add a constant K to the potential, to the Lagrangian, in such a way that every closed curve has a non-negative action. The minimal K for which that holds, that will imply that the Manier potential is finite. So the minimal K that we add to the Lagrangian so that the Manier potential is finite is called Manier's critical value. Many characterizations, one of them is Manier's ergodic characterization of the critical value which is given by the negative of the minimal action among minimizing measures. So in particular, if mu is an invariant measure, the integral is larger than or equal to zero, and it is zero only if it is minimized. Okay, a curve is semi-static if it is a free-time minimizer. So given two points in the curve, the Manier's potential, the minimal action between those two points is attained by the curve. Okay, a free-time minimizer. And in the pendulum, the semi-static are the solutions which start at the fixed point, at the upper fixed point and go and end in the fixed point in infinite time. That's why I think Manier called them semi-static. And a curve is static if the following is satisfied. First, it is semi-static, a free-time minimizer. And second, the Manier's potential among points in the curve satisfies the following, going from one point to the other. That's the action, the minimal action of the curve. Plus, coming back gives zero action. So we have the curve is a free-time minimizer. And the action of going from here to here along the curve or the minimal action plus, the minimal action of coming back is zero. Perhaps this coming back is not realized by any curve. It could be something like, usually something like this. Say here we have another, one semi-static going to the right and another semi-static coming back. All free-time minimizers. If the total action is zero, then they are in the opposite, we say. And the opposite contains in this case the red ones and the blue ones and green ones. So that is our, when that was this, no? So the sets, the Manier's set is called, is the set of tangent vectors to semi-static orbits. Tangent vectors to free-time minimizers. And the obrisset is the tangent sector to static orbits. The ones which are free-time minimizers. Also, the action of coming back gives zero. Well, and I said this obrisset satisfies the fact that if an invariant measure is, an invariant measure is minimizing if and only if it is supported on the obrisset. That also happens with the Manier's set, which is larger because everything which is recurrent on the Manier's set is in the obrisset. Because the potential of going from one point to itself is zero. So if you have something which is recurrent and realizes the potential, realize this zero. So our theorem will be the following. We have a closed manifold and a tonerian Lagrangian. And the obrisset is hyperbolic. And suppose that the obrisset is hyperbolic and without singularities. Then we can perturbe with a la Manier in the situ topology, such that the oblique set contains a periodic orbit. If the obrisset contains a periodic orbit, then there is a minimizing measure supported on the periodic orbit. So that is what we have to prove. The corollary is that if we talk about the set of Lagrangians, or potential for which the obrisset is hyperbolic, that's H and we define P as a set of potential for which the obrisset is a hyperbolic periodic orbit. Then to be a hyperbolic periodic orbit is open and dense in the set of Lagrangians which are hyperbolic. So this is a Manier's conjecture restricted to hyperbolic obrissets. But luckily we have another theorem which says that if M is a surface, this is why we restrict the conjecture to surfaces. If M is a surface, then to have a hyperbolic obrisset is open and dense in the situ topology made by Fidali-Riford and Contreras. This was done by using the theory developed by Fidali and Riford about perturbations of obrissets and a theory of Hamiltonian points without conjugate points developed by several people and inspired by the people in grey. So as a corollary we don't have to restrict ourselves to hyperbolic obrissets because they are already generic in the case of surfaces. We obtain the Manier's conjecture in situ topology on surfaces. The set of potential for which the obrisset is a hyperbolic periodic orbit is open and dense in the situ topology surfaces. Remarks, the proof uses our ideas of a... You can write the same problem for functions on the shift, symbolic dynamics. You want to minimize, maximize the action, the integral of a function on the integral by invariant measures of a function on a shift. And that subject is called ergodic optimization which started as a baby version of Manier's problem. And that could be solved. I mean we could prove that generically on the function which you are integrating in Leibschild's topology, the minimizing measure is supported on a periodic orbit. And this work is that those arguments can be translated to the Lagrangian setting. The proof in the case of ergodic optimization, and here as well because here is a free writing of that proof for Lagrangians relies on work of Serra and people which approach the conjection and prove approximations to the conjection. We use all those approximations and give the final remark. They are John, Hunt, Bressot, Quas, Sifken and Morvis. And then symbolic dynamics corresponds to hyperbolic obrises. That's why we use this theorem with the Fiegelian riffle. So, steps of the proof. There are two main steps of the proof. First, we prove that generically that's in the CK topology. The obrises has zero entropy, zero topological entropy. And the second step is knowing that the obrises has zero topological entropy. How to obtain this closed periodic orbit in the obrises? So the first step to have zero topological entropy is done in the following way. First, one obtains a closed orbit nearby the obrises with a period which is not too large. Those are the estimates. The action can be compared with the distance to the obrises and that decreases super exponentially on M. M is comparable to the logarithm of time. The way it is done is something like this. You have the obrises set and take something like a T delta generating set or something like 2T generating set on the obrises. So that the dynamic balls of time 2T cover all the obrises. And then make a subset of finite type in the following way. We can follow from one point we can follow to another point if they are shadowed for a time T. So that makes that these pieces of orbits are all in the very set and are very near to each other because they are shadowed by time T and this thing is hyperbolic. Then we have a projection from the shift defined like this to a neighborhood of the obrises set by shadowing. We have a sequence of points and then this hyperbolic set and then we obtain a curve which shadows the sequence of points. That curve will not be in the reset but will be very nearby and how nearby this is because this shadow. And then use the arguments for the shift. In the shift one takes a closed orbit with a minimal period and that will satisfy these estimates. So that was made for shifts originally by Bressaud and Quas and Anthony Quas. They were trying to approach the conjecture so they couldn't obtain a periodic orbit generically but they said how a minimizing measure is approached by periodic orbits. By measures on periodic orbits and this was. And then we use the arguments of Morris made also in the Ergodic Optimization case to prove that the entropy is generically zero and it is done in the following way. You have this closed orbit with a small period. It is very near the obrises set so it inherits the obrises set. It is a graph. It is a leapshift graph over the manifold. Since this periodic orbit is very near to the obrises set it inherits part of the graph property. So suppose it is a graph. And then put a potential which is like a half a channel around this periodic orbit. That makes that for nearby potentials the minimizing measures will be in this tunnel half pipe. So we have this closed curve and we put a potential which is like this small. And then since this was the minimum. And then we make estimates in such a way that the new obrises set or the new minimizing measures have to be in the neighborhood of this periodic orbit. And then in the neighborhood a small neighborhood and those are the estimates of a periodic orbit. Something cannot have large entropy because I mean you have a periodic orbit and then in the neighborhood you can have a horseshoe. But the horseshoe appears after the period. So it has a small entropy. You calculate how much of the measure has to be in the neighborhood with a small entropy and how much of the measure is outside. That is the argument. The second part is knowing that the obrises has zero entropy how to close the obrises, how to obtain a periodic orbit. That again has two parts. One which was known for shifts I will tell and the new part is this second using the zero entropy. So we will perturb a return. If we have a special return inside the obrises we can perturb it to obtain a new obrises with a periodic orbit. The special returns we are going to use. So it's a situation similar but not equal to a closing limit. If we have the following situation. We have an orbit in the obrises, a piece of orbit which returns at a distance delta and which the minimal distance among pieces of the orbit, well the minimal, say here, it's called gamma. And suppose that we have this situation in which so many of these ones in which this quotient is tending to zero. So the distance here with respect to the minimal distance in the middle tends to zero. If we have this situation then we can obtain a periodic orbit which is a periodic orbit. In fact we will do something a little different from this. Instead of one return we will allow to have a, instead of an orbit a pseudo orbit. So we can jump and we will do it for two jumps. So instead of an orbit a pseudo orbit with at most two jumps. So we need to obtain this situation, returns with arbitrarily small distance to return. But in the middle the orbit cannot approach too much. So the sketch of the proof that this is enough to obtain a periodic orbit. Wow, not much time. I will jump to the other part which is how to obtain this situation. This is the sketch of the proof of why this situation is enough. So suppose that we have already orbits. No, we have to prove why we always have orbits which have this behavior. Situation where we come perturbed. And we do an argument which is inspired in part of, it's not the same but it is inspired in part of Manier's conjecture. I'm sorry, in part of the proof of the stability conjectured by Manier. Which was Manier was trying to find a homoclinic connection. And then he would say something like this. If you have a hyperbolic periodic orbit and you have invariant measures such that in the limit the measure of this point is positive. So invariant measures which accumulate here. Then you can perturbed to obtain a homoclinic trajectory. And the method was by contradiction. Suppose I cannot perturbed. Then if I cannot perturbed then the measure should give a lot of weight outside. And then the limiting weight of this cannot be positive. So in our case we want to obtain this situation. And by contradiction suppose that we never obtain this situation. Then the proposition is the following. If we take a minimizing measure, an ergodic minimizing measure and a generic point for the ergodic minimizing measure and a transversal section and we look at the return times to the transversal section. Then they last this amount of time between one and the following. And then we calculate the entropy nearby the point using the Brinkatox theorem. The measure of a small ball around this point is given by how many returns I have to the transversal section. And if the return lasts a lot of time then the measure of the transversal section has to be small because it is an invariant measure. The measure has to be spread over the orbit. And then the measure will be shrinking exponentially with time and hence by Brinkatox theorem the entropy would be positive. And it was zero by hypothesis. So how do we prove this proposition that the return times lasts a lot? The argument is as follows. I will write the pseudo-orbit as a circle. So I start here and then come back as a pseudo-orbit. And so suppose that I cannot have this situation. So in the middle there should be a return. Okay. I should have a... Let's see if I... I need to discretize. So I have a return and it does not hold the conditions. So say gamma one would be larger than something like... Yes, smaller than... So since it does not satisfy the condition there is a point in the middle, a return which is small. And then we split this into pseudo-orbits. So the return is small so I can do the following. I start here and then jump here. And that will be a new pseudo-orbit with two jumps. And the red one will be another pseudo-orbit with two jumps. That corresponds to going here, jump, and then come back. And then this is a new pseudo-orbit. But then these two pseudo-orbits are in the same... They could be used to obtain the theorem. By contradiction we are saying that we cannot. Then there must be two returns here of the next order. So this one was a smaller return and then there will be another order next return. And then these ones are pseudo-orbits with two jumps and then the argument repeats. There should be a return here and probably here. So if we see the quantity of returns is growing exponentially. So that means that the quantity of points in the pseudo-orbit is exponentially large. So the original return lasted an exponentially large time. There is one thing that the exponential growth can only be proven for the second iterate. And then we have to look at all the possibilities. There are not many. That's why I talked only about two jumps. Because for example this one, here we have one jump, another jump, and a third jump. So this one I will not count. It will stop there. Because the combinatorics is too large. So I will continue only here. But I am sure that after two iterations I get the doubling of the quantity of returns. And that is the reason that the exponential is not 2 to be something but the square root of 2. Thank you very much. Yes, the hyperboleicity in the second step is actually more important. So you have this situation. And then the first thing is you close the orbit using the shadow in lemma. Then you have to prove that this orbit can be made, one can make the obrisset to be this orbit. And it is done as follows first. It also inherits some graph properties. So you put a potential which is a half pipe around it. And then I have to prove that for this new Lagrangian, the obrisset is a periodic orbit. It is done as follows. We will prove that any semi-static orbit, which is defined in the past, will have a semi-static. Then the alpha limit is in the closed orbit. That will imply because alpha limits of semi-statics are in the obrisset. That will imply that gamma is in the obrisset. And that is what we want. How do we do that? So suppose that the alpha limit of a semi-static is not inside gamma. Then by the expansivity, it will, perhaps we will approach gamma, but we will have to go out of gamma. It cannot be near by gamma forever because it would be gamma. And we have to go out of gamma infinitely many times. So we have to be a little while outside of gamma infinitely many times. So each time it will be adding a positive cost in the Lagrangian. So the total action will have to be infinity. But the action of a semi-static is given by the potential, the manier potential. The manier potential is lipstick and then it is bounded. So that situation cannot be so after a while it is not outside of a neighborhood of gamma. And then the alpha limit is gamma because it is also hyperbolic.