 Okay, before I thank you for the organizing and inviting me to speak here again, I would thank the audience that is here after all these nice talks during the day anyhow, and so I will talk about contracting law as attractors and some statistical properties, and I would say that my interest is to understand the behavior of attractors of three flows. Here I will talk always on three manifolds or three flows, presenting equilibria accumulated by regular trajectories in the sense that Adriana just talked before. And the main example is the geometric Lorentz attractor, but it comes from the Lorentz equations and the first case is the expanding geometric Lorentz attractor, and the second case is the contracting Lorentz attractor. And I want to understand this kind of flows from the topological point of view as well from the statistical point of view. And then let's just recalling once more these attractors because I want to just to fix what is the important thing that I will consider afterwards. And then in the beginning Lorentz in 63 exhibit a three dimensional differential equations whose solutions seem to depend sensitively on the initial point. And the Lorentz equations, the famous one, he was a meteorologist. And then you see that the Lorentz equation is quite simple, two degree polynomial equations and depending on this here that I put in color are fixed parameters and called the classical parameters because that was the one studied by Lorentz and although the equation were so simple, he could not give the explicit solution for these equations and so on. But he, the famous and studying this from the numerical point of view, he conjectured that the flow generated by this vector field should contain a volume zero attractor that is sensitively with respect to initial data. This means that if you take any two points and you iterate by the flow, after some time they separate it. And this is the famous butterfly shape of this attractor and we are interested in just see what is going on with this attractor. And indeed the solution for Lorentz conjecture was given by Tucker at the year 2000 under the advisor of Carlson. And he combines normal form techniques nearby the equilibria together with computer assistance techniques far from the singularity. And meanwhile because there is a big gap between the real, the solution under when this kind of problem appears to the mathematicians and then meanwhile it was introduced a geometric model for this attractor that satisfies all the predictions by Lorentz that we present very briefly in the second. And then that was introduced by Guggenheim at the same time in the West by Guggenheim and Williams and the Est by Aframovic, Bikov and Shilnikov and the model is like this. You have this is a singularity, a hyperbolic singularity and the index of the singularity here is 2 and is hyperbolic and you have a cross section here for the flow. And how you do this? You just put a linear flow in this part with this kind of again values and you complete the pictures like that just composing by a rotation, translation and expand somewhere. But it's possible to realize this picture as a vector field in R3 and the saturation of this flow and the intersection of all of them is what is the geometric Lorentz attractor. And the important thing, the hypothesis, the most important hypothesis to construct such a kind of example was the existence, well first, okay, first this singularity here, why is this is the usual Lorentz attractor because we called expanding Lorentz attractor because the eigenvalues at the singularity satisfy this condition here and this is important and what I will tell you next and the most important hypothesis was the existence of a state of foliation here preserved by the Poncaremap associated to the model, okay. You know that here you have the cross section and then you have a foliation here that it's clear that if you start with a linear flow here, this foliation is preserved because you are just doing a linear flow, but if you do like that, you have to impose the condition that it's preserved, okay, and so what you see is that the map here in this cross section, the Poncaremap is like this, just you have one side this comes in a triangle like this is a caspe triangle and here also the other side and the foliation is preserved and once you have this, you have a one dimensional map associated just projecting this, the quotient map through this foliation, you have a one dimensional map associated and since the eigenvalues here satisfies this condition, you have that the shape of this quotient map is like this, it's just an expanding map with just one singularity and we assume that the derivative everywhere but not at zero is bigger than the square root of two and this map has all the nice properties that we should get from to the Lorentz attract in the sense that once we have this, the properties of F implies the Lorentz conjecture in this case, okay, and okay, how we'll do this, you just notice that as I said F is increasing with the derivative B and here at zero to the left and to the right is infinite and this implies that the maximal invariant set for this one dimensional map is a transitive attract for one dimensional map and again you can push this property to the maximal invariant set to the two dimensional map, the Poincare map and then this is a transitive attract for the Poincare map and finally this implies the conjecture by Lorentz, okay, the moral of this story is that once you have a map, a two dimensional, you have a flow, you have a transversal cross section and a two dimensional map and a foliation that you preserved, you just projecting you from all the properties from the one dimensional map, you recover you get the properties to the flow, okay, and so the Lorentz like attractors, then it was just Adriana I start talking about them, we were interested in this kind of flows and then we generalize this for the first results by Morales, myself and Pujols is that a robust transitive set for three flows are either hyperbolic or singular hyperbolic and the other theorem says that the robust transitive sets for three flows with equilibrium are partially hyperbolic attractors or repellers and singular hyperbolic here means that it's partially hyperbolic with the central bundle expanding area and we call singular hyperbolic attractor is by definition what we denote Lorentz like attractor, okay, and the bundle here is like it's written here, it's just you have that everywhere in the attractor the bundle decomposes like this, you have this one which is uniformly contracting by the derivative of the flow and this one is dominated by the other one but you don't have an expand here but you have a volume area expanding in this sub bundle, okay, and okay, from the statistical point of view now we start studying this kind of flows we what we get, we get that the if you have the Poincaré map associated and me the SBR measure to a geometric Lorentz attractor expanding one then the map has exponential decay of correlations and this implies by some of other results that indeed you have log law for the heating time associated to the geometric not flow but attractor indeed, okay, what is the heating time you know that the heating time is just you fix one point here you take a neighborhood and you consider any other point and for the first time that the orbit is off this point intersects this neighborhood and this is denoted the heating time and when you take the same point it is always when you are considering you take this point you are just waiting for the orbit of this point to coin come back to this neighborhood is the recovery point and in that formula here you have the local dimension of this the measure associated at this point and okay and then you have some meaning for the some statistical property of the flow from this equation there and I will I would like to just to give a brief idea how we get this for expanding Lorentz attractor and how are the steps okay you the idea is to consider first the the first return a map or the Poincare map associated to a cross-section and the first thing is if you take the invariant measure which is a physical measure for this map then this system is exponentially mixing okay this is the first result and then you prove that in fact this measure is exactly in other words that the local dimension exists almost every point and then you you prove that the the law did this theorem here for points in you prove the theorem for the points in the cross-sections and you just iterate this is a easy part because once you have this formula here for the two-dimensional map you just increase one because of the flow direction okay and these steps what is the problem and these steps the problem is indeed to prove well before what is the K of correlation all of all of you already know what is the K of correlation but is this just it means that if you take any the map here and the two observables what you have is that the this formula here that's equivalent to say that in the from the topological point of view is that the what you have is that the the measure is mixing somehow no here it's that's the difference here we will talk in in lip sheets observables yes this for this okay and then the main difficulty in this step is just to prove the K of correlation explanation of the K of correlations for the two-dimensional map associated and to prove this we take the we consider this devising style control which distance which is defined like this you just take two measures and you consider the soup of the difference of the model I of the integral of the two maps of the map considering one and the other measure okay and here we talk just about lip sheets maps on the manifold okay and we try to we study the K of correlation terms of this distance and then we try to what we did was to relate to this kind of a distance if we K and then we have this distance here the decaying in function of the distance that if you have some measure which is absolutely continue with your reference measure here and this one is just the marginal then you this here is just the formula for the decay okay that I'm a simplify this is just this okay with this part here is what I call the that formula that is here CNGF is just the difference between iterates and then you have this kind of a relationship and then you start the distancing function in function of a decay and then you also have something here that the the this is the decay function and you related the with the distance and finally you get this this is the if you're back how the pullback of the measure you relate to with the decay and finally you relate with the disintegration and the approval what I'm trying to tell you you know that you have a two two dimensional map in the cross section and you have a distance and then you try to relate the to obtain and you know that in the beginning that indeed you can disintegrate the measure and then you try to compose the how to get to the the theorem that the F is faxing is explanationly mixing just combining the relation that you get between this distance between the two measures and the decay of correlations okay okay and then let's go to the contracting lawyers attractor and then I would like to repeat once I have the decay of correlations for the two dimensional map associated to the flow I have the log law for the heating time and maybe we can have some water with statistical properties for the for the flow and then I would like to repeat this for the contracting Lawrence attractor and how is this this is much similar to the expanding Lawrence attractor and indeed the difference is that the first one is that you just at the the relation between the eigenvalues now is bigger than zero and not on the contrary okay and we also ask that this is replaced and I have these conditions here by that should be such lambda that 2 1 and 3 are the eigenvalues and I have some conditions between them and this is just the tool this is the appearance of a computer thing of the contracting Lawrence flow attractor and we ask this conditions here because to you have the remember that I told you that the hypothesis there for the expanding Lawrence attractor was the existence of a foliation stable foliation in the cross section here to guarantee the existence of such a foliation and we need the more differentiability to deal with this kind of problem and that's why and the day one dimensional map associate this time is like this here you have a tangents and that the relation that here just about the order of the contact here after this discontinuity and we would like to repeat the what we did before expanding law geometrically lorries to this case and just before we continue let me just commented something in the in this attractor and the first is that this map is not stable because it is is accumulated by maps which have a periodic attracting orbit and no in on this attractor is not a stable or robust in the on opposite to the lorries expanding lorries attractor because expanding lorries attract is robust but this one is not robust in the sense that you perturb and then you don't have the same thing in the neighborhood but you have persistence in a measure theoretical point of view or sense in the sense that if you perturb this family lorries the contracting geometric lorries attractor you show that there is a one parameter family of positive lebegue measure of city close vector fields nearby the first one such that the the starting vector field has a transitive known hyperbolic attractor okay although this result says that if you take a family then you find in the parameter space a full measure set in the parameter corresponding to flows presenting an attractor okay and okay we would like to prove the same result in this case and what is this problem and we go we got this with galatolo misolia and myself and the main point to get it again this property is to prove that the two dimensional map associated to a contracting lorries attractor the two dimensional lorries map has exponential decay of correlations back unfortunately and okay once you have this you have the log law which is simple like that it's the same and how is the strategy in this case in this case we try to follow the same steps but we have to adopt and then the main difficulties again this that I've already told why because the map here the base because of the map now is a cross product and the base map is not expanded as before and then we have to improve the arguments just to repeat the strategy okay and what we do we start the main idea is okay I wanted to recover the properties that the base map has to the flow okay and then now we we take this we recall the definition of convergence to equilibrium but it's simple means that if you have a map and the reference measure move and we say that this map has exponential convergence to equilibrium if if you this hold is here the pullback by T of G you notice that this is almost the the formula of decay of correlations but the difference is here you have one measure and here is you have one under the two okay and then we what is this formula is saying is that it's just the pullback by the other one following one measure converges to to the other one the difference okay okay and then just we give this definition here like the if this is i square and f is some integrable function we denoted this here is just the integral of f in one fiber and we relate this is a theorem telling us what we can get this theorem it's not a it's it's already proved here more or less it was proved in my result with the old paper with galatolo but the point is that if you have a measure a skew product like this such that the this map here that is the quotient map through a foliation preserves a foliation and the this map has exponential decay of correlations with respect to some norm to these norms here T is non-singular with respect to the bag piecewise continuous monotonic in the sense that you have a finite number of intervals such that the restriction of f t t is a homeomorphism onto its image and the map g here is a contraction on the fibers okay once you have this you prove that the the map itself has exponential convergence to equilibrium with respect to the product measure here then you know that if you have a nice skew product such that the base map has nice properties you can prove that the skew product has exponential convergence to equilibrium okay once you have this convergence in the equilibrium is the same as I said before and once you have this you get another result then the next result okay now I have a map that it has okay now the base has this is the skew product okay and the base map has exponential convergence to equilibrium and again the base map is non-singular with respect to the bag measure piecewise continuous and monotonic and you have a finite number of intervals covering the interval such that the t restricted to each interval is a homeomorphism onto its image and the map is a uniform contraction on each vertical leave this means that this map here g is a contraction okay then the map it has exponential decay of correlations okay so we have two theorems here saying what we we have to do with the skew product is such that the base has exponential decay of correlations and another theorem is how to get from exponential convergence to equilibrium to exponential decay of correlation then you have to adapt these two theorems just to apply these results in our case and what is the difficult is to apply these results you see that we have to to work in another panic space which is called the the maps we if you generalize the boundary variation because this fits better in our case then the other one doesn't fit as well and we have to extract the right properties from the base map from the skew product that we are interested which is the case of the contracting Lawrence attractor and this time again the base map is not piecewise increasing anymore then we have to work more on this and the main technical problem is transformed the information we have about the base map in this case in this space of generalized the boundary variation to with respecting I am excuse me I have the main information the technical result is that I have to transform the exponential decay that we have with respect to hold the L infinity observables into information about decay respect to generalize the boundary variation observables and finally we observe that indeed the skew product associated to a contracting Lawrence attractor can be written as this you have exactly this map here preserves the vertical for the asian you see that the map is a contraction in the fibers and the one-dimensional map associated is piecewise and here that enters that case of the relation the extra relation between the again values and okay and you just put that one under less than one are pre-periodic repelling and the one more condition is that this map here again because of the relation that I mentioned in the beginning for this kind of attractor in the singularity the map T has negative Schwarzson derivative and this is important to to deal with this kind of maps and to prove all the nice properties that this map have and the main theorem is that a map satisfying those properties have a exponential decay of correlation risk in this the monarchy space of generalize the boundary the variation observables okay and okay this implies that the F has exponential finally we can transform all these in lip sheets observables and this implies the log law for the hitting time and that's all