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 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.


Cancer not only affects the patient, it affects the people who love and support that patient. Comedians took to the TD Garden stage for the 23rd annual Comics Come Home event in an effort to combat cancer. This is such a great show. I dare you not to laugh. The benefits show was created in 1995 by actor Dennis Leary and former Bruins player Cam Neely, who established the Neely Foundation for cancer research after his parents died from the disease. I love that you guys have the aquarium right between the harbor and a legal sea food. While the show is a night full of laughs, the comics know the heartbreak cancer causes. Jimmy Fallon and Lenny Clark utilized the Neely Foundation's Neely House, where cancer patients can stay during treatment. When my wife went through breast cancer again for the second time, we stayed at the Neely House. Jimmy's dad was there, my wife was there. Over 700 families have been helped by this. Cam Neely saved my dad's life. My dad had prostate cancer and they helped him out and I'll never forget it. Although the comics love performing to raise money for a great cause, they say the best part of it all is coming home to Boston. There's a few cities in America where you really enjoy performing at. Boston is one of those cities. I've done County Hall with Dennis. I've done the Lincoln Center with the God. I trade them all for this. Since its inception in 1995, comics come home as raised over $10 and a half million dollars in proceeds for cancer research. Reporting in TD Garden in Boston, I'm Kristi Matino, BUTV 10 News.


Hello, NJ, root22.com here with another supermarket roundup. But this one's going to be different today because we're going to change things up. For the last six months or so, I've been reporting on various supermarket flyers that we receive and we read them every week and we want to share our insight in terms of what's on sale, what's a good deal. But as each week went by, I found it harder and harder and harder to promote good things because most of the things that supermarkets sell are junk. 90% of it or more is junk.  Most of this stuff was junk and we live a sort of low carb life here pretty much exclusively low carb and mostly meat. And the good deals were hard to find. So we're not going to talk, it was very hard for me.  Not just because of the artificial ingredients and chemical processing but the fact that it raises your blood sugar and it's no good for you, period.  There's one in Flemington and there's one out in Phillipsburg and maybe there's a couple other on the east side of 22 that we haven't been to yet.  You get a whole lot more free money, you can accelerate your trip to the diabetes doctor that much faster and that much cheaper there. So instead of doing the roundup every week where I sit there and flip through the flyer and try to find some humor in it and maybe find a loan deal here and there that might be worth it. I'm going to do something new starting next week, next Friday. And that's going to be the top five deals at the area supermarkets, Shoprite, Kings, what happened. And it's not just going to be food. There's a good deal on paper products for instance because everybody uses paper towels and toilet tissue as they say, toilet paper. I might mention that type of stuff. But it's going to be five deals. The top five deals that includes places that are in supermarkets like Ocean State, Job Lot, they have good deals. So we're going to pick five things every week that we're going to talk about. And I might throw a couple of barbs in at the people who like eating processed foods or bought into the organic thing and want to get ripped off on that type of product as well. I don't care if the chips are organic, garden of eaten chips are still bad for you. They're just as bad for you as Doritos. So don't get tricked into thinking that just because it's some like wholesome corn, it's still bad for you. So those types of things we're going to talk about. And that's about it.  A few people might say, oh yeah, I like the way this guy thinks but nobody has time for that type of stuff anyway. I'm going to make them, my goal is to make them three minutes or under, five deals, three minutes, and that's that. I don't like doing short, really super duper short form video, but I guess in today's day and age you have to do it. So that's it. Signing out and see you next week with our very first issue or episode of the top five New Jersey Route 22.com deals. Have a good weekend.


How's everybody doing welcome back to my channel this week I'm gonna be giving you a different I'm gonna be giving you I'm gonna be doing a different kind of video today and that is giving you tattoo artists red flags giving you tattoo artists red flags red flags of tattoo artists what if you're someone that's looking to get your first tattoo or maybe getting a new tattoo here are some red flags to look for before choosing an artist so without wasting any more time here is my first red flag if your tattoo artist is doing the consultation designing and tattooing all in the same day of the session that's a red flag your tattoo is not going to be unique because the artist is going to go on Pinterest or Google and he's going to start looking for the most famous references that every tattoo artist has used then put a quick design together just so he can start tattooing your tattoo artist is not going to be putting 100% effort into your tattoo because at this point his attention has been divided in three ways the consultation designing and tattooing that is a lot to do in one day just designing can take hours and hours of doing research finding the right references make it flow with the body make it make sense tell the story unless you're doing a portrait then I totally understand that it's all in the same day because you give them that picture and that's what you're getting done by the time he gets to tattooing he's going to be overwhelmed and there's a possibility that he might rush the tattoo and that leads me to my next red flag and that is looking at the client like a transaction rather than a relationship with your client that you're about to tattoo something permanent on their body before I start explaining this red flag I want to say that there is clients that also don't appreciate tattoo artists or they also see the artist as a transaction but for this part I am talking about the artist not the clients maybe I should make a separate video for red flags for clients I just want you to imagine this first the client has to take time off work they have to plan ahead save their money because tattoos are expensive and if they're from out of state they got it by a plane ticket and book a hotel and I don't think it's fair that when the client shows up you just like okay well it's my payday today just give me my money I'll give you whatever you want it doesn't matter here's your tattoo instead make it a little bit more personal like getting to know the client first what do you like what do you like about this piece how do you connect to this piece and then put something unique for that client and make it special for that client because you know it's something permanent that's going on their body forever instead of looking at the client like a transaction look at it as like man I am so honored that I'm gonna give you something that you're gonna carry for the rest of your life if you make the whole experience special for your client the money will follow eventually but do it because you want to another thing is when the artist looks for any inconvenience to raise the price now I don't mean when the client starts adding too many things to their project and something that you guys never talked about that's not the case because that is wrong when you the day of the appointment the client just decides to add way too much and it's just too complicated I'm going to use tattoo gate as an example so um booked a consult with her a couple months ago the consult was non-refundable it was $108 that was listed on our website that's fine I showed her these reference photos as to what I wanted for a half upper arm sleeve she then proceeded to tell me after I had already booked my I paid for the consult she told me that she had these three options for her design fee so the first option was $1,500 plus tax and you get a concept sketch and you can make one minor change and then a final design that you'll review the option number two was $3,500 plus tax where you got two concept sketches and a couple changes and a final design review again and then option three was $6,000 plus tax where you would get multiple sketches and lots of reviews and lots of changes and like a canvas of your concept at first I kind of thought that that price was going to then be taking it out of the final cost of my tattoo but then I find out that it was not so Monday rolls around and she sends me this this is her concept sketch it is nothing like what I sent her it's nothing like what I wanted she said that if I wanted another sketch she was going to charge me the difference between option one and option number two which is $2,260 they were charging so much money to draw a tattoo that didn't even look good and they were charging extra to make any adjustments to the drawing like come on man that is crazy they use any excuse to charge extra and I don't think that's fair either the next red flag is when the artist is laid to their appointment I'm not gonna say much that is just insane to me I understand if something came up in the morning I totally understand communication is key let the client know hey you know what something happened this morning can we push it two hours from now but if you're late because you couldn't get organized with your time or maybe you're sleeping schedule man that's crazy to me for my appointments I like to be at the shop two hours prior to my appointment because I like to get there relax breathe study my image look over my stencil what am I going to use today what kind of cartridges am I going to use today how am I going to record what am I going to do but when the artist is late and the client has been waiting at the shop for a while you have no other choice by to stop rushing your your your station and get ready as fast as possible because you don't want to waste more time so being late to your appointment is very unprofessional and it's a red flag for the next red flag this one is very important pay attention go to their social media and make sure that they post healed work if all they post is fresh tattoos hmm I don't know it's questionable I always say that that tattoo you did is not done until it's fully healed posting healed work is probably one of the best things that you can do as an artist because you get to showcase how good you are at applying how good you are at applying then that's a tongue twister how good you are at applying your tattoos I mean application is key and the way it heals it's going to show the way it's going to hold up over the years if your tattoo artist is only posting fresh tattoos then it's questionable and the last one man I've been seeing a lot of videos that I'm like that's a little cringy my dude and that is how clean is your tattoo artist do they wash their hands do they apply the stencil with no gloves do they shave you with no gloves little things like that can really tell you how clean your tattoo artist is one that I've been seeing a lot lately is when they wrap their phone or tablet or even their Starbucks coffee or their juice and they put it on their tray oh I get I understand it's wrapped but man that phone has been everywhere the juice I don't know my dude I just don't have anything on your tray if you see your artist touching the tray with no gloves on oh just cancel your appointment on the spot just cancel it but that was just a few red flags that I thought about if you have more red flags dropping down on the comments and I'm gonna be reading them and if you have something more to add to my red flags or maybe you disagree with my red flags I totally understand dropping down in the comments and let me know your thoughts and let me know if you like this video maybe on the next video I'll talk about clients red flags thank you so much for coming back and watching this new video I really appreciate you see you guys next week peace


Hello everyone, this is Amit Dan of amitdan.com. Today I'll have another episode of GoPro 10 video And I'll talk about how to start setting up your GoPro So I have done my initial unboxing video yesterday and then today I'm going to show you how you can start setting up your GoPro for the first time. So this is going to be a basic simple and easy beginners guide And I'm just going to touch base on the things you only need to know I'm not going to go through the whole settings and everything to make the video a long video For that reason, this is going to be a short video or I'll try to make it short and simple So let's see have your camera. The first thing you're going to be doing is that you need to update your camera because If you look here, this is the first thing you're going to see that, you know, you cannot Use your camera before updating so your camera has to be updated before you can use it so GoPro did this Because, you know, they want to have the latest software inside the camera So before even you start updating that you're going to have to charge the battery So which I have it right here and it's kind of difficult to open the battery tray And you're going to see that, you know, this is a bit confusing because you're going to see here that if you just, you know Put the GoPro facing towards you, then you're going to see kind of like, you know, two sides one is right here and This kind of looks like a Door tray and there's another one right here But actually this is not anything you can open so you don't try don't try to open this one and this part just to drain the water and You don't have to actually open it or you cannot even open it. But this is where Is the tray or battery tray located and you have to open this part So the way you can remember is that this is where all black There's no like, you know GoPro logo or writing, but this side has GoPro 10 So you are always opening the left side when the camera is facing towards you So the trick is you have to slide it down the door and then you have to slide it up And it took me a while to open it and even I opened it so many times It like, you know, it doesn't happen right away. For example, if you look at the old GoPro This is hero 7 is very easy and simple to open you can just push it and then you can just open it It takes like, you know one second But this one is not like that. So In order to open this what you're going to do push it down and You can see that, you know, it's kind of a little bit Came down. So let's take a look here.  So you're going to like, you know, put your finger right here and press it and To push it down See that, you know, even it's giving me trouble right now, although I just opened it.   So if you're facing the camera, the logo of the SD card is going to be Actually facing the backward or where the battery tray is.  So which is going to be Right there. Now if the SD card is not smooth Then you know that you're doing something wrong. So don't push it too hard to Otherwise you're going to break it. So now we're just going to insert the GoPro camera I already charged the battery. So You have to charge it before you use it.  So now you're going to have to go through the updating So I'm going to just, you know, remove the cover Which is right here Let's put it there. And then I'm going to remove the lens cover which is right here. Just put it there So now you're going to have to update your battery or GoPro and then It tells you actually There's another layer which is right here. So I'm just going to Remove this and then Actually, I don't see any easy way to remove it. So I don't know how why there's nothing in here Okay, so that's kind of giving me hard time. Okay, so it looks like I was able to find it And now I removed it Okay, so it's just giving me French instructions I'm not sure why Okay, so it's not doing anything here.  So I'm just going to restart it Okay, so now, you know, it's turning on for real And you can see that we have the language setting. So I'm going to put it in English Somehow it was in Italian So just put it in here and then you have to agree to the Terms and conditions. So he'd agree and then turn on to your footage with the local speed attitude altitude and more Which is just turning on the GPS, you know, I'm not going to do it now So I'm just going to keep it off. Now you can see that message that your gopro must be updated So what I'm going to do right now There are two ways you can update your gopro one is just by going to the app which you can easily install from the app store or google play store and the app actually Looks like this Which is I have it Right here go pro quick. So that's the best way to, you know, update the phone If you are trying to do it without an app or phone Then it's going to be kind of difficult then you have to go to go pro's website And then you're going to have to like an install the file and you're going to have to go through the process of like, you know Getting the file from your computer, which is, you know, much Difficult and it's not user friendly. So I don't recommend Updating from the website. It's just best to update from the app. Okay. So now We have the gopro here and I'm just going to open the app And see what happens here Okay, so here I'm going to just click on here and then This one is connected to my old gopro. So I'm just going to, you know Try to attach with a new gopro or connect it to a new gopro So here, please turn on the bluetooth to find your camera. So I'm just going to turn on the bluetooth right here And it's searching for the camera. So I'm not sure the bluetooth is on on the gopro But it looks like, you know, it's searching So you have to just select which camera you want. So I'm just going to hit gopro 10 And then it says the swipe down And then tap connection. Okay. So it's kind of telling you what to do on your camera. So swipe down Okay It's not registering any movement here. So I think I'm just going to see here What happens if I hit just Continue and let's see if it can connect automatically Because it's not letting me even swipe down or doing anything Okay, so you can see that it found my camera. Uh, looks like, you know, uh, it Knew what to do. So We found your camera connect. So I have a pairing request on the phone and then pair and connect and pair with gopro 8289 and then I'm just going to say yes and pair Okay, so I'm not sure why it's not connecting right now But let's see what happens now Okay, so Change camera name. So I'm just going to put it gopro 10 and That's going to be the name and save new name And you see that, you know, it's saving right now Okay, so I'm not sure why it's giving problem So leave as is Let's install your update. So continue And It's going to update the camera. So update camera And connect it to your camera's wi-fi Hello everyone. So I got a message that accept and continue It took a while to connect to gopro And I got another message in between that, you know, gopro had to create a temporary wi-fi network Which I clicked yes, then I got this message that accept and continue So I'm just going to hit accept and continue and you can see that started updating and Transferring update to your camera So you can see that, you know, the flashing red and you can see also the updating So just going through the initial updates and this could take sometimes a little time But let's see how long it takes here So I got this message that update complete, but now it's gone. So It could be that it's just restarting. So looks like yes So it has restarted and you have the initial screen right here But however, somehow the phone still says that, you know, it's Keeps updating. So I'll just, you know, leave the phone here So now I'm just going to very quickly show you some of the basics Actually, sorry, we just got the message on the phone that, you know, update has been done So just leave the phone here And then so here you're looking at the main screen So this is your standard Screen or face when you open your GoPro and right now this is set this way You can customize some of the settings later on, but today I'm just going to show you how You can start recording right away. We're not going to go into detail. So here It shows your battery mode, which is 74% and And this is your default video mode. If you want to change it, then you have to just drag it to the left or right And you can see that, you know, it's giving you photo or Video, so we'll just leave it to video right now And here it tells you How much memory you have in your sd card. So you have a four hour 26 Four hour 26 minutes for the current setting and then here it has one x So if you click on here Then you can see that, you know, what is your speed right now? It's normal speed and you can like, you know, go higher or lower So we'll just leave it here and then here if you click on w Then you're going to see that, you know, it's showing wide right now And then if you go up then super view and then if you are looking for Normal video recording just the way let's say your cell phone or your camera records Then you can set it to linear And another mode would be linear horizontal leveling and what is that? That, you know, if you just bent your camera, it just stays horizontal up to a certain degree. So for now, let's put it back to linear and then Here by default is giving you the standard mode, but we're going to just change it to Cinematic and then you can see that there's a small plus sign right here So you can create your own mode different later on, but you cannot delete these three standard The these three modes. So if you click on cinematic So right now we are cinematic mode and if you click on the right So for cinematic mode, these are the standard options you're going to get You can leave it on for for now. The only thing I'd like to change Which is that, you know here, you see the color is natural But if you want to vivid color, then you can change it to Vibrant and or you can just keep it to natural So that's the one thing you can change But everything else you can just leave as is and then later on you can see that, you know, there are options to put the shortcut On the screen and you can do that as well So another thing that from any screen if you'd like to go back to the main You can just press the mode button, which is Right here And it just goes back to your main screen. So that's something To know and then if you want to start recording, then you have to just press it once. So for example, if I press it Starts recording And if you if I just press it again, then you can see that you're gonna stop recording And if you'd like to see your files or you know your recordings, then you can just scroll up And then it's going to show you everything And you have the option to select more than One file and if you scroll Down then it's going to give you, you know, some more settings and here You can change lots of different options And today in today's video, I'm not going to go or discuss About these but definitely I'm going to have more videos in the future So that's how you can update your gopro and start recording for the first time So let's say you want to change to photo instead of video. So just, you know Slide it to the left or right And then you're going to see that you have the photo option And then if you click on here, then it's going to give you some more options Which you can go through and change if you need to But for today's video, I'm just going to stop right here And that's how you basically, you know, update your gopro for the first time and start using it Thank you for watching this video. There'll be more videos coming up. So I'll see you shortly. Thank you


The next one is not every man who wants to be in a relationship with you will want to marry you. And this goes back to like men have issues too, right? And upbringing is really, really important. And you want to learn a lot about his relationship to his parents and his parents' relationship to each other and how he feels about all of that. And just kind of take a look, like especially if you're getting into dating a guy, you're thinking that he might be right, you know, just start asking him about some of this stuff and asking him about his relationship to his parents and their relationship to each other because he has an imprint that he got from when he was a kid in his relationship to his mother, which is a reflection of a lot of his relationships to other women in his life and then how he feels about his relationship to his mother and father and his mother and father's relationship with each other because it will tell you what's normal for him and what you may be experiencing if you end up getting into a real relationship with him because sometimes it's good, sometimes it's bad. Sometimes if it's bad, but he has kind of a negative outlook on that and he wants something different, then, you know, it might be a better situation. So you just want to learn a little bit about what's going on there.


Friends, Kerala's culture has come under immense attack from L.D.F. and U.D.F. Kerala is a land of peace and harmony, but the insensitive communists and congress have brought a culture of political violence The culture of Kerala has come under immense attack from L.D.F. and U.D.F. Kerala is a land of peace and harmony, but the insensitive communists and congress have brought a culture of political violence Several patriotic R.S.S. and B.J.P. Kharkartaj have been attacked and killed for serving the people From Kojikode, I want to ask outdated communists and fake liberals why are you saying that Kerala is a land of peace and harmony, but the insensitive communists and congress have brought a culture of political violence From Kojikode, I want to ask outdated communists and fake liberals why are you silent on the political violence in Kerala What are your compulsions? From Kojikode, I want to ask outdated communists and fake liberals why are you silent on the political violence in Kerala I ask outdated communists and fake liberals why are you silent on the political violence in Kerala I ask outdated communists and fake liberals why are you silent on the political violence in Kerala I assure my fellow B.J.P. Kharkartaj, your struggles will not go in vain Their bullying will not weaken us I ask outdated communists and fake liberals why are you silent on the political violence in Kerala Look at what happened in Tripura Their fear of communists was high They were taking to worse violence They tried to hide their poor governance But the people gave them a strong answer From barely any presence we formed the government with record majority People placed their faith in us The same will happen in Kerala