 Okay, so today I want to talk about dynamics in two complex dimensions and talk about the equidistribution problem that we addressed yesterday. So the setting will be that of holomorphic mapping on P2, meaning that there is no indeterminacy. The mapping is holomorphic everywhere, and the degree is D larger than 2. So yesterday we saw that if you take a point in P1 and you try to look at three images of this point, they tend to equidistribute, right? Now a point in P1 has two analogs in dimension two. Either you can see it as a subset of dimension zero, or you can see it as a subset of co-dimension one, okay? So here we have two equidistribution problems, which are different. So you have equidistribution problem for points. So you take the point Z in P2, and look at three images of Z, and look at three images of Z under Fn. So actually it is set as D to the power two n points. So one simple way of proving this is that Z is, say, the intersection of two lines through Z. So when you take pullbacks of this situation, so that's L, that's L prime, the prime edge of L, so that's the prime edge of L, it is an algebraic curve, which is generically smooth of a degree D to the n, so it's a curve of degree D to the n, and the prime edge of, so that's, and the prime edge of L prime also, okay? So by Bezou's theorem, you get D to the two n points, okay? So by Bezou's theorem, you get D to the power two n points with multiplicity, okay? So if you want to make equidistribution, you take the normalized Dirac measure on this, Fn of Y equals Z, and put a direct mass at each of these points, and the question is, is this equidistributing to something in the sense of probability measures? And now you can look also, so actually it will be a consequence, this equidistribution here, of the equidistribution problem for subvarieties, so for subvarieties. So let's say you start with a line, so as we've just said, the prime edge of this line is a subvariety of degree D to the power n, by the way, a proof of that is simple. The line has a homogeneous equation of the form az plus bw plus ct is equal to zero, okay? And if you write in homogeneous coordinates F as, let's say, PQR, then F to the n is Pn P sub n Q sub n R sub n, where these are homogeneous polynomials of the degree D to the n, and so the equation of the prime edge of L is just apn plus bqn plus cqn equals zero, so it's a curve, you see the equation of a curve of the degree D to the n, Rn, yes, excuse me. So now there's something to understand, so we would like to prove that these kinds of sequences of subvarieties or submanifolds equidistribute to something, but now you can just take, I mean, direct masses, you need to have a formalism to what it means for a sequence of curves to equidistribute, okay? And the right formalism is that of positive closed current, so I will spend some time explaining to you what a positive closed current is, so that's a basic tool here, positive closed current, okay? So what is a current? So we will work in dimension two. So my current will be one-one current, so I will consider, let's work locally for the moment, so we take an open set, so work locally, we take an open set in C2, let's say we have coordinates Z1 and Z2, okay? What is a one-one current on omega? It's just something, an object, which takes a one-one form, which takes a one-one form, and so T takes a one-one form and gives you a number. So it's an element of the dual of one-one forms. So there are two basic examples, so one-one, say, smooth test form, so it's smooth with, say, compact two-part omega. So it takes a smooth form and gives you a number, like, so think about distributions, right, or think about measures in the plane, measure takes a function and gives you a number, so here you act on forms, one-one. So we have two basic examples of currents, so we have two basic examples. The first example is a one-one form itself, so if psi is a one-one form, or even a two-form, well you can do the following, you can take phi and map it to the integral of phi wedge psi. So now this is a form of maximal dimension, two-two form, and you can integrate a two-two form on omega. So this is the current associated, let's say it's t psi, it's the current associated to psi, exactly as when you have a local integrable function, there's a distribution associated to it. You can integrate, so every function gives rise to a distribution by integration and every form gives rise to a current by integration. And the second basic example is if you take a sub-manifold of dimension two, let's say a complex sub-manifold, so here it's a curve, you can integrate, so I have the integration current to a form, associate the integral of the form on this manifold. Again you get a number, so this is the current of integration associated to this sub-manifold. So now there are two other words, positive and closed. So closed is easy. So closed means that the current has no boundary, meaning that if you integrate t against a boundary, then you get zero. So that's the definition of closed. So here by the Stokes theorem it corresponds to the case where v has no boundary. So this is closed when v has no boundary, so this is just the Stokes theorem. So what does positive mean? So this theory of positive currents was invented by LeLon. So in French positive means non-negative, so perhaps the proper translation would be non-negative currents, but I mean now everybody says positive, but writing English words should be non-negative. So what is this positivity here? So there's an important fact that complex manifold carry a certain positive structure. What does it mean? Think about the plane. In one dimension, if you take an object of the form a of z, dz dz bar, let me put an i here by tradition, where a is a real function with positive value and you change coordinates. What happens? If you change coordinates, then z, zeta is f of z is f of zeta. When you change coordinates, what you get is a of f of zeta, modulus square of the modulus of derivative of f times dz, dz bar, and you see that the positivity of this term is invariant on the holomorphic change of coordinates. So this is intrinsic. The positivity of this function here is intrinsic. And the same phenomenon happens in higher dimension, but the definition of positivity is likely more complicated, but I don't want to state the definition. So here it's positivity for one-one forms in one complex variable. And you have the same, there is a corresponding notion of positivity for every by degree. So here in two dimensions, so in C2, I need perhaps more space here, so in C2 a one-one form, if you look at an object of the form sum, so one-one form is a sum of aij, that's a square root of minus one somewhere, dz i dz j bar from ij equals one to two, corresponding to a certain matrix, a one-one, a one-two, a two-one, a two-two. And this is positive if this matrix is Hermitian and non-negative. And this notion, simple computation, is invariant on the holomorphic change of coordinates. So this is an intrinsic notion. Now we say that the current is positive. T is a positive current. If it gives a positive value to every positive form, if whenever phi is a positive one-one form, the value of T against phi is a positive number, non-negative number. So it's the same as a positive measure. So in this respect, of course the integration current associated to a positive form is a positive current. And more importantly, I don't need complex to integrate a two-form. So if any two-dimensional submanifold gives rise to a certain current of dimension two, and now this current is positive if and only if the submanifold is complex. So positive currents, positive close current, are a generalization of complex submanifold without boundary. So this is positive. And the reason why this current is positive is just that positivity is invariant under change of coordinates. So if I take local coordinates, there's a small proof to be done here. OK, what do I want to say about positive currents? So you should think as a positive current as a differential form with measure coefficients. So you put measure here, you put measures here and you have a matrix of measures that would be Hermitian non-negative in a sense. OK, so positive currents, so as we said they generalize complex manifolds. So there's another example which is important from the point of view of complex dynamics, which is this one. Let me just put it here. So currents were introduced by Durand to give a geometric realization of Poincaré duality because you have something which interpolates between homology and cohomology. You have subvarieties, submanifolds, and differential form at the same time. OK, so it's a geometric way of seeing Poincaré duality. And this notion of positivity was introduced by LeLon, Pierre LeLon in the 50s. So just after Durand, I guess. Just one word here. Actually, so one important result by LeLon is that you can integrate against a submanifold, but you can also integrate against a subvariety. So you can take here t, a subvariety, possibility with singularities, and only integrate on the regular part of the subvariety. And this gives you, you can integrate, the volume is locally finite, and the current is closed. If the subvariety has no boundary, these singularities will not create boundary because they are too small. So you can actually integrate with respect to any complex subvariety. So another example here. Assume that you have a certain family, a measurable family of subvarieties with a positive measure on it. A measure on the parameter space. OK, positive measure. Then you can form a current which is the integral of the integration current over your family of varieties. OK, so these family of varieties can be a lamination, or maybe something more complicated. So you can just integrate this integration current, meaning that if you take a test form, you integrate against the alpha, and then you integrate against alpha. This defines the positive closed current, and these kinds of currents are very important in holomorphic dynamics. OK, what is important? So let me now go to P2. So here I was discussing locally. On P2, let me remind that you have this basic fact that you have a Fubini's 2D metric, you have the Fubini's 2D metric, which is associated to the Fubini's 2D carrier form. So here's a certain 2 form which has the remarkable property that if V is a curve, an algebraic curve, so every complex manifold of P2 is algebraic of degree D. Then the area or the volume of V with respect to the Fubini's 2D metric is equal to the integral of this scalar form on the curve. And this is really remarkable, because this is only a homological quantity. This scalar form by definition is closed. So this quantity is just a homological quantity, and actually it's just an integer. It is the degree of the curve. And now if you take a positive closed current on P2, you can define its mass. It is exactly this number, T against the Fubini's 2D form. So it's the analog of the volume for a subvariety. And now the point is that the set, so the important point, so exactly as the set of probability measures on a compact space is compact, the set of currents of bounded mass is compact for the weak topology. The fact, easy fact, the set of currents on P2, actually it's a local fact, but let me state it globally, with mass 1 less than 1 is compact for the weak topology. So now if I have a sequence of varieties Vn of degree Dn, well, I can consider the sequence of normalised integration currents on Dn, and I get the sequence of currents of mass 1. So this leaf is in compact set. It makes sense to wonder whether this converges to something. So it's a relatively compact set. OK, so now I can make sense of the equidissubtion problem for subvarieties. So now the equidissubtion problem for subvarieties. In my dynamical setting is the following. Let V be a curve, curve in P2. And the question is, does the sequence of currents 1 over Dn times the degree of V times integration on the pullback of V converge to something? In the sense of currents. And actually it's not difficult to deal with this kind of question because, as I said in the last talk, a measure in the plane is just the laplacian of a function. So if you want to deal with measure, you just have to deal with function, which is easier. Here is the same. So 1, 1 positive closed fact. A positive closed 1, 1 current on, let's say, it's a local statement. So exactly as on P1, you have no global potential because there are no subharmonic functions. You have the same problem here. So you don't have to define potentials locally. On a ball, say in C2, has a potential which is a function, meaning that this current, T, is the DDC. So DDC is another notation for I over pi Dd bar of a certain function. So if you take a function, so they will do the computer. If you take a function and you take two extra derivatives, you get an object which is of a kind two form. So it's rather a current. So dealing with T is the same as dealing with the potential U, which is a plurisobarmonic function. So the definition of DDCU, just if you want to integrate DDCU against a test function, by definition, so it's exactly as with distribution, this is U against DDC5. So that's a test function. And this is just a classical integral because DDC5 is a function, is a form of maximal degree. So you deal exactly as derivating distributions. Okay, so working with currents is easy. You just have to work with potential. And so actually the equidistribution problem for subvarieties, at least for generic subvarieties, is not very difficult. Because the proof is exactly the same as the one I gave yesterday for measures on P1. So let me state a theorem. So there are many authors. Let me say after that. So if L is a generic line on P2, then a sequence of currents, one over D to the N, a prime edge of L converges to a certain current, which is, of course, independent of the choices, to a certain current T, which is called the green current. Another result here is that the support of the green current is exactly the Julia set. So let me first comment on the first item. So what does generic mean here? So the difficulty of the theorem relies, depends on what you put in the word generic. Okay, so the set of lines on P2 is a P2. So if you want to prove this for almost every line, this is not very difficult. So for almost every line, for instance, so if generic means almost everywhere, this was done by Fornes and Ciboni at the very beginning of higher-dimensional homomorphic dynamics. Now, if you want to prove for every line outside a certain algebraic exceptional set of lines, then it becomes much more difficult. So it's a very delicate problem to understand exactly what is the exceptional set of lines for which this convergent does not hold. And in P2, this problem has been solved completely in higher dimensions. I don't think we have a complete answer to this problem. So in this case, it's theorem by Faber and Johnson, which relies on very elaborate algebraic techniques. They have to do a very delicate algebraic analysis of the singularities that can happen in this delicate theorem. Okay? But if you start with, say, a smooth, positive form and the proof of this theorem is exactly identical to the one I did yesterday. Okay, so we have this interesting fact that the support of this current is exactly the Julia set. So it says something about the Julia set. So it says that the Julia set is an object which has the type of a curve, which has a homology class in a set. So this has some consequences on the Julia set. Consequences on the Julia set. So Julia set is an object which has a, which is a co-dimension one object. So it's a complex co-dimension one object. Complex co-dimension one object. So for instance, it must intersect any algebraic curve. Okay? So you cannot imagine that on P2 you have a, it cannot be contained in a ball in P2, for instance, because it must intersect every algebraic curve. Also, you know from this theorem, so the potential of T, the potential of T is a holder continuous. Again, because it's the limit of a geometrically convergent series. So it says that the householder of dimension of the Julia set is bigger than two. So two is automatic because you have a co-dimension one object. Okay? But it's bigger than two plus alpha for some positive alpha. Okay? Another property is that the Julia set is connected always because the support of a current, I mean, the co-dimension one object in P2 is always connected because two lines must intersect. Okay? So it gives you some information about what the Julia set looks like. Okay, what about equidistribution for points? So actually equidistribution for points is more complicated. So let me come back to the picture. I said before that the point, you can view it as the intersection of two lines. Okay? So the pre-image of a point is the intersection of two algebraic varieties of large degree. And now we know so L, L prime. So we know that the pre-image of N converges in the sense of current to the current T and also pre-image of L prime. And the same for L prime. So one would be tempted to try to intersect these objects. So we are looking at the intersection of these two objects and the good news is that one can intersect currents. So this converges to the self-intersection of the current T. So there's a way of making sense of it. So I should point out that this intersection is a complicated object. Well, it's not obvious to define. And the reason is that you want, so imagine you're dealing with differential forms, you have to multiply the coefficients. Okay? And you don't know how to multiply measures. Oh, this provision, there's no sensible way of multiplying measures. So for this, you have to understand what this object is. And there's a little bit of analysis to do. Perhaps I don't have the time to explain exactly. But this is a... The construction of this wedge product is not obvious, but it's classical in complex dynamics. So it's something which goes back to the 70s and this construction was done by Bedford and Taylor. Bedford and Taylor in the 70s. And it has to do with the potentials. And it works well when the currents have continuous potentials, which is exactly the case here. One difficulty is that this wedge product is not continuous with respect to the weak topology. So actually the implication here is false. You cannot deduce from this convergence, you cannot deduce this from that. Okay? You need to work. It has some analysis to do here. It's not obvious, but at the end of the day, you're able to prove something like that. So let me state a precise result. So actually the way you prove it is not exactly this one. So you cannot just start with two lines and if you try to prove this head-on, you will never manage to do it. Okay? So you first need to start with smooth forms. Use the convergence. So let me write down something precise. Let me write down something precise. So what you do is... So what is true is that if, let's say, phi is a smooth closed one-one form, then as I said, one off-mass one, off-mass one, what about dn? Pullback of phi, as we said, converges to the green current. So this is the smooth version of the equidistration term in the co-dimensional one. And now it is true that you can wedge this. One over dn fn star of phi wedged with itself. So this is exactly the same as the pullback of the positive measure phi squared. So phi squared is a positive measure now. Okay? And the pullback of... The sequence of pullbacks of phi squared converges to t squared. So this is correct. This follows from the definition of the wedge product here because the key fact is uniform convergence of the potential. Uniform convergence of the potential. So we face exactly the same difficulty as yesterday. I told you that the potential of a direct mass is unbounded. Okay? It's not a continuous function. It causes problems. Here the potential is smooth and the sequence of potentials converges uniformly. And so if you look at the definition of qh, which I did not give, actually this is true. So what we get from this is that corollary of that, if you start with a smooth measure, you have convergence. So the corollary of that, if, let's say, nu is a smooth probability measure on P2, then the sequence of pullbacks converges to the self-intersection of t, which by definition is the equilibrium measure. That's the definition. Equilibrium measure. Okay? So this is this argument. And then what you have to prove is an equidistribution theorem. And then you prove an equidistribution theorem. And the equidistribution theorem is that for generic y in P2, if you look at the pullbacks of the corresponding direct masses, this converges to zero. So if you put these two results together, then you get that for a generic corollary, for generic x, the sequence of pullbacks of the two here of the direct mass converged to nu. Okay? This is because a smooth measure is just an integral of the direct mass. Okay? And again, the difficulty of this theorem depends on what you call generic. So again, if you want to prove this for almost every point, this is not very complicated. So if generic means, if this means almost every with respect to the Lebesgue measure, it's not so difficult. You need some complex analysis. I think first by Rusakowski and Schiffman. And so there's a whole bunch of notions of genericity. And the last one is the optimal one is outside the exceptional set. So you can define an exceptional set. The exceptional set is exactly a set of points whose total inverse image is finite. Okay? It is an algebraic set, non-obvious theorem. And actually, this convergence holds exactly outside the exceptional set. So in that case, it was done by, let's say, Brion du Val, and I should add the names of Dean and Siboul. So again, you need to have more complicated techniques here to arrive at this optimal convergence. So yesterday, I explained to you that when you have this equation and you get dynamical consequences of it, so here I have the same dynamical consequences. So theorem. So this is the Brion du Val theorem. So as I said, so first statement, you have convergence outside a certain algebraic exceptional set. So this measure mu that you get is invariant. It's invariant. It is mixing. The measure has positive Diapunov exponents in all directions. So it is a, this measure is repelling. Let me state it like that. For almost every x, for mu, almost every x, for every v in the tangent space at x, if you look at the action of the derivative on v and you look at the norm in logarithmic scale, so dfnx applied to v, look at the norm, take log, take one over n, take lim inf. This is bigger than log d over two. Bigger than or equal to log d over two. So the measure is repelling in all directions. OK, so you have two, basically, you have two complex directions here. Third statement, this measure is the unique measure of maximal entropy. So again, if you know what it means, you're happy otherwise you forget it. So that statement about entropy is maximal. And fourth statement, this measure describes the distribution of repelling periodic orbits. So mu describes the distribution of periodic orbits, of repelling periodic orbits. So let me state something precise here. If you take one over d2n sum of direct masses at repelling points of period n, this converges to the measure mu. Now you know that the total number of periodic points is about d2n2. So actually the consequence of that is that most periodic points are repelling. So as I said yesterday, there are examples where I have infinitely many attracting points. But still, the total number is negligible with respect to d2n2n. OK, so, yes? I will spend the whole remaining of the lecture discussing the support of this measure. So the proof, one word about the proof, the proof is the same as yesterday, basically. So the proof I presented yesterday was, I mean, I presented it, I presented the proof which in a sense extends to two dimensions. One thing that you have to understand, this is the main thing in this theorem, is to understand what the Lubitsch lemma becomes in this case. So you cannot, you don't have exactly the same Lubitsch lemma, so you have to prove one Lubitsch lemma along a certain line that doesn't work to be done. But as soon as you make sense of what the Lubitsch lemma is in this context, then you get the theorem for free, basically. OK, so that's the Breon Duval theorem. OK, so I want to spend the rest of the lecture to the following question, so basic question. Is the support of the measure mu equal to the Julia set? And the answer is truly no. I will give you an example in a moment. And so you see that you have two natural analogs for the Julia set in one dimension. One is the Julia set in two dimensions, so the set of non-normality, and the other analog is this circle. So this is not exactly the closure of repelling orbits, but forget about it. OK, so it may happen that you have sporadic repelling orbits outside the support of the measure, but you forget about them. So this is another analog, this is another natural analog of the Julia set, of the one-dimensional Julia set. OK, this is where the repelling dynamics takes place. OK, so I would like to study, I would like to understand the interplay between these two guys. OK, so let me perhaps first work out an example. So perhaps let me state a definition. So here it's a little stupid, but so maybe the Julia filtration. So here you have the filtration with two terms. So I will define J2, it's the support of the measure mu. J1 is the Julia set, and the Julia filtration is just J2 is contained in J1. OK, it's clear that the support of the measure is contained in the Julia set because you have repelling dynamics here. So perhaps I should add P2 here, OK? So in higher dimension, just a word about higher dimension. If you work in PK, then the green current is a co-dimension one object. So J is a co-dimension one object. So again it is the support of a current of co-dimension one, and you have corresponding Julia sets, corresponding to a co-distribution of sub-varieties of higher and higher co-dimensions. So you have for every Q, Q between one and K, you get the Julia set JQ, which is the support of the current which describes the distribution of co-dimension Q sub-varieties. So the GQ is the support of, actually it's the Qth power of the green current, which describes the distribution of co-dimension Q in co-dimension Q. Pullbacks of co-dimension Q sub-varieties. And then you have this filtration. So JK is the support of the maximum entropy measure, which is contained in JK minus one, up to J1, which is contained in PK. And the right analog of the Julia set in higher dimension is not only the Julia set, but the whole filtration here and the way each Julia set sits in the next one. So this is the last time I will talk about higher dimension. So let me try to explain you at least a picture of what is happening here. So the situation is, in a sense, analogous to the case of Kenyang group fight. You have the small limit set. You have the small limit set and the cool Karny limit set. And so this one is a co-dimension one. So I don't know if it's a limit set on the action of... I don't know what is the mirror in the dictionary, if it is the cool Karny limit set or limit set of the action of lines. So the small limit set should be the closure of repelling points of luxodromic elements. So you have the kind of dictionary between the two. So let me explain you the interplay between these two sets. So let's start with an example. We will work out a little simple example. So let's start with an example. I will talk... So this study is due to Bedford and Johnson. There are three polynomial maps on C2, polynomial maps on C2 of the form Pzwqzw. And remember from yesterday that the term of higher degree... homogenous term of higher degree, highest degree has something to say. And I will assume that it is of the form f is of the form plus lower of the terms Wd plus lower of the terms. So the dominant term is ZdWd. So in that case, so let me draw my C2, so that's C2, and that's the line at infinity here. So we saw yesterday, so I made the computation, in such a case this extends holomorphically to P2. So it extends holomorphically without, in the term, AC points to the line at infinity. And the induced dynamics on the line at infinity is given by this rational function. And f restricted to the... so the line at infinity, of course, is a P1. So you're looking for a rational map on P1. It is just this rational map, ZdWd in homogenous coordinate, which is just the polynomial Zd. So this is a zeta goes to zeta. So we know the Julia set at infinity. So the Julia set at infinity. So in the plane, the Julia set of the polynomial Zd, so that was my only example of Julia set, the Julia set of Zd is the unit circle. So for zeta goes to zeta d on C, the Julia set is the unit circle, and points inside the disk go to zero, and points outside go to infinity. So you have a certain unit circle here on the line at infinity, which is the Julia set at infinity. Now, you also can introduce the set K. This is the set of points with bounded orbits in C2. So if you look at the formula here, ZdWd, you see that if a point is as big norm for this fact, if the norm of the point Zw is large, then the norm of if f of Zw is approximately norm of Zw to the d. So this term is dominating, and the other ones are negligible. So it's not always the case that for a polynomial map, when you take a large point, it goes far away. There can be some complicated cancellations. But in this case, there's no cancellation. OK, so it means that the line at infinity is an attracting set. So it means that the set of points with bounded orbits is compact in C2. So the set K is somewhere here. The set K is somewhere here. So K is compact. Actually, it's closed. It's easy to prove that this is closed. So it's a compact set in C2. OK, so now what if I look at prime edges of points? So it's not difficult to see that the basin of infinity is totally invariant. If you take a point which is attracted by infinity, every prime edge is attracted by infinity. So if I take Z in the basin of infinity, the prime edges of Z, all these points belong to the basin of infinity. Right? And they must converge to a set of points with bounded orbits because you take prime edges of points which are escaping to infinity. So this is set as a set. This is converging to K. And since you come from the outside, this is converging to the boundary of K as a set. Actually, to a subset of the boundary of K. So a consequence of that is the support of the maximal entropy measure is contained in the boundary of points with bounded orbits. So you see that the support of the measure is here, but there's another piece of the Julia set somewhere. So in that case, it is certainly different. It is not equal to the Julia set which must contain the Julia set at infinity. Right? Now there is more. Remember I said that the Julia set is always connected in P2. So there should be something connecting the Julia set at infinity to the boundary of K. Excuse me for that. So what is connecting? Let me do a little bit of hyperbolic dynamics where here. So as I said, the Julia set at infinity is a certain circle contained in the line at infinity. Along the line at infinity, the picture is not here anymore, inside the line at infinity, the circle is repelling. Right? Points in the circle go away from the circle. And actually it is uniformly repelling in a sense. So this is uniformly repelling. Inside the line at infinity. But now the line at infinity is an attractor for the dynamics. It is actually super attracting if you do the change of coordinates. The transverse derivative is equal to zero. So the line at infinity itself is attracting. So it means that this Julia set at infinity is a hyperbolic subset. It is a hyperbolic set. So we don't need to know exactly what hyperbolic set means for the dynamics. It is attracting on one side and repelling on the other side. Saddle set, exactly. It is a hyperbolic saddle set. So now when you have such a set for every, so a consequence of that, for every x in this Julia set at infinity, I don't know why it is doing that. If you look at the set of points, y such that the distance between the orbits iterates of x and y go to zero. Oh, sorry for that. I promise I do nothing to do this noise. If you look at the set of points which come close to the orbit of x, this is actually a piece of manifold. So this is by definition the stable manifold of x and this is a manifold. At least locally it is a manifold. So to every point on the Julia set at infinity is attached a piece, a local piece of stable manifold, which is a complex manifold. And if you look at this stable manifold, let's say the local one, so you have the Julia set at infinity, which is something like that. So if you take a piece, so it's a classical fact from hyperbolic dynamics, and if you take a little piece of this here, this point here converges, at least it aligns with the orbit of x, but on the other side it's exponentially expanding. So this is a sub-stable manifold. This is contained in the Julia set. So it is contained in the Julia set. And the result by Bedford and Johnson is that you get the whole Julia set in the basin of infinity like that. So TRM, the Julia set intersected with the basin of infinity is exactly the union of these stable manifolds for x in the Julia set at infinity. So you know exactly what this Julia set looks like. It's a union of holomorphic submanifolds which connect infinity to the set k. And actually there is more to that. So this family of blue submanifolds is actually a lamination. It's a lamination, and this lamination has a transverse measure. Let me keep the picture here. This lamination has a transverse measure which is given by... So you have a blue lamination by stable manifold and it has a transverse measure which is just the equilibrium measure on this Julia set here at infinity. So you have a lamination with transverse measure. And actually it gives you the current. So you can form a current with that. You can integrate on the blue curves at least this measure mu. And actually you get the growing current like that. So the growing current t, at least in the neighborhood of infinity, is the integral of current of integration on the local stable manifold. So the statement... It's a local statement here at infinity. When you pull back to k, you got to get some topological complications. But still the local picture is always the same. So it's integral of current of integration against the measure on the Julia set, the natural measure on the Julia set at infinity. So you get the structure of the Julia set and the structure of the growing current. So the growing current is foliated and on this foliated part, let me put perhaps a definition. So the consequence of the growing current is foliated. Let me put forward the definition here. So what time is it? Okay. So definition... I say that a holomorphic disc in P2 is a phatodisc if the dynamics along the delta is equicontinuous. If the sequence of iterates along delta is a normal family. Okay, so we see that in this example, the Julia set outside... So you have the support of the maximal entropy measure here and the Julia set outside the support is foliated by phatodiscs. So in the previous example, if you look at the Julia set outside the basin of infinity, then you get this lamination by phatodiscs. It's laminated by phatodiscs. Because of course the dynamics along an unstable manifold converges exponentially, super exponentially fast to the line of infinity. It's certainly contracting. So along the blue disc, it's contracting and transversally it's expanding. Right? So actually I stated the theorem for this special map here, but it's true for every theorem. The same result by Bedford and Johnson. You have basically the same result for every polynomial map on C2, which extends holomorphically at infinity. So there's a little difficulty here, which is that when you consider an arbitrary polynomial at infinity, the Julia set at infinity is not uniformly repelling anymore. Uniform repulsion is by definition hyperbolicity. Not every Julia set is uniformly expanding. It may happen that it's only non-uniformly expanding. So in that case the picture is a little different. You don't have a foliation. So if it's not uniformly expanding, what you can show is that on the line at infinity, to have the Julia set in pink. It's not exactly pink, but whatever. So this Julia set at infinity, something like that, carries a certain measure of mu. So there's a dynamical systems theory for that. Non-uniform hyperbolicity tells you that you have a set of big measures, a measure of more than one-half, where you have a nice foliation with disc of positive size. So you can fill out... There's a set of points of measures, a one-half, where you can find discs like that of positive size. And then if you want to get more measure, you need to reduce the size of the discs. So perhaps you will get more measure if you accept discs of a smaller size, and even more. If you want to get full measure, you must forget about the local triviality of the lamination. So you don't have a lamination with a locally bounded geometry. You get a lamination for every epsilon. You get a lamination in the neighborhood of a set of measure one minus epsilon. So you can make a current with this. You can still integrate on this family of discs, but now you don't have a foliation in a measure theoretic sense. You get a foliation on a set of big measures, and so what you get is a so-called laminar current. So you get something which is foliated, but with no local uniform bound on the geometry. Okay? So what about the general case? What about the general case? So in the general case, so I take again any endomorphism of P2, I have my Julia set support of the maximum entropy measure and Julia set. Okay? Yes? In this case, K2 is... Support of the measure. It is concentrated on the boundary of K. It's the support of the measure by definition. It's not the whole boundary. It's the so-called shield of boundary of K. It's a certain subset. So for instance, in this example, Z2W2, okay? The K is a polydisk, like that, and the support of the measure is the unit torus, not the whole boundary. Okay? So I want to investigate the geometry of J1 minus J. So let's say that on J2, we understand basically the dynamics. The dynamics is repelling. Okay? So dynamics is repelling here. Exponentially repelling. Okay? And with the Yapunov exponent, larger than or equal to log D over 2. Okay? And now I would like to understand what happens on J1 minus J2. So you can imagine a model for the dynamics. So of course, here you have P2, and you could say that this one is J0, and J0 minus J1 is the Fatou set. So in the Fatou set, you have a quick continuity. And so what you should expect as a model for the dynamics is that on J1 minus J2, there should be one direction of a quick continuity and one direction of expansion. So there should be a quick continuity in one direction, one direction, continuity in one direction, and expansion outside, exactly as expansion outside, exactly as in the case of a stable manifold of a saddle point, or saddle set. You have contraction along the blue curve and expansion transverse to the blue curve. Okay? And you can even expect more. You could expect that J1 minus J2 is filled with Fatou disc in a sense. Filled with Fatou discs. So it will never be a lamination, right? Because we know that already for polynomial mappings on C2, you have these non-uniform examples that are only filled with Fatou discs in some measure theoretic sense. Okay? But still you can expect that you have plenty of Fatou discs and the dynamics along this Fatou disc will give you this direction of a quick continuity. Okay, so first, I said several times that there's a measure theoretic sense. So what is the measure on J? So what is the ambient measure on J? So J is a fractal set, right? So you have a picture or something like that. So J is a big set of co-dimension one and somewhere inside you have the measure mu. So mu is certainly not the ambient measure on J. Okay? And J is J1, yes. J1 and this is J2. Okay, so the ambient measure is not this very small one, right? So the ambient measure on J is given by the current. So J, remember that J is the support of a certain current. And as I said, the current is a differential form with measure coefficients. Okay? And this measure is exactly the ambient measure on the Julia set. So this ambient measure, J is called the choice measure. So this is the choice measure of the current T, which is the T wedge, the care of form. So why is this a measure? T wedge, the care of form. If you take a function, T wedge care of form applied to a certain function, phi, it's just integral of T applied to this form phi omega, right? So in the case of my foliated picture with the Julia set at infinity and the blue curves, I said that T is a foliation current associated to this foliation here. The ambient measure is just the associated current. So Lebesgue measure on the leaves times, well, Riemannian measure on the leaves times the transverse measure, right? So that's what the choice measure is in this case. So it's really the ambient measure on the Julia set. Okay, so what you should expect is that there should be this in a measure theoretic sense. Okay? In a measure theoretic sense. And the answer is that this is almost true. Let me erase this. So there are two theorems. So theorem one, which I proved a few years ago. So for almost every point in J1 minus J2 with respect to this measure sigma T, you get the pictorial one. There exists a certain subspace in the tangent space which I call the phi 2 subspace. So we have a measurable field of phi 2 subspaces, such that if V belongs to this phi 2 subspace, then you have no exception. The lim soup of the 1 over n log of the norm of Dfnx of V is non-positive. And if V does not belong to F, sorry for that. Let me just try something. If V does not belong to this phi 2 subspace, then the lim inf of this sequence, 1 over n log of the norm of derivative, is larger than log D over 2. Okay, so you get the expected picture. So it's just a measurable field of subspaces. And so second natural question, can you integrate this field of subspaces to get phi 2 disks? And the answer is no. So theorem two. There are examples where you cannot, so on all examples that you can compute, you can integrate. But there are examples where it's not possible. So there exist examples for which, so it's a little difficult to state precisely, but there is no family of holomorphic disks of phi 2 disks such that these phi 2 subspaces So let me say examples of polynomial maps for which this field of subspaces cannot be integrated, cannot be integrated to a field of phi 2 disks. So how much time do I have? I have two minutes. So it's perhaps not reasonable to explain the proof in two minutes. Perhaps just one word in higher dimension. So what happens in higher dimension? You get a pretty good understanding of what's happening, except that, so there are certain sporadic examples where there are no phi 2 disks. I'm not moving. Don't move. And I don't know any criterion which ensures that you get this field of, except one. So there are these examples of polynomial mappings and there's a theorem by Dottella which says that if f is post-critical finite, meaning that the orbit of the critical set is algebraic, then one can, phi 2 disks exist. So you can integrate. So the current T is a laminar current. So you can integrate this field of phi 2 disks, this field of directions to a field of disks. But I don't know any other criterion. And what about higher dimension? So in higher dimension, there's a conjectural feature which is not a theorem. So you get this filtration of Julia sets. So this JQ here is the support of the current TQ. So the set where prime edges of Q, the Q co-dimensional subset accumulates. And so now it's just a conjecture. It's just a conjecture of JQ minus JQ plus 1 for every Q. Then there exists the Fatou subspace of co-dimension Q, subspace of co-dimension Q, and remaining directions are repelling with the same rate log d over 2. Remaining directions are repelled. So that's a theorem for Q equals 1. So same as this one, actually. So that's a theorem for Q equals 1, but just a conjecture for the other cases. All dimensions. Because again, because working with currents of bi-degree 1, 1 is simpler because they are the DDC of something. So it's always simpler to work with co-dimension 1 curves. And so the proof is just positive currents techniques. And I think I will stop here. And since I'm the last talk of this session, I would like for all of us to thank the organizers for this first beautiful week of lectures. Thank you.