Thanks, so the last Last lecture was supposed to talk about fat two sets in one and higher dimensions So I could not go to higher dimensions. So I will start with it. So the plan for today is first I will finish the previous lecture so talk about fat two sets in higher dimension and then I will attack the Julia set so we'll attack the ergo dictionary and the Equal distribution properties of the constructing the length measure. So the second part will be in dimension one only and Should do the the last part. So the ergo dictionary in higher dimension tomorrow Okay, so that's lecture one part B first. So the fat two Julia the composition in higher dimension and actually I wish Mainly talk about the fat two set today and the Julia set in higher dimension is a little more Delicate and we will encounter the kind of difficulties you have with different limit sets So we will see that in higher dimension. There are different Julia sets with Big Julia sets small Julia sets. This should look similar to the situation of Clangian groups in higher dimension Okay, so Again, so what is the setting? So what are the kind of mappings? That we want to iterate so in her. So let's start with the same setting So I take a certain algebraic map on Uncertain algebraic manifold and I want to iterate it. I Want to iterate and as before to describe the future of points So in that case so I explained that in dimension one you you don't have many choices either you can have Compact roman surfaces of higher genus in which case there's no dynamics You have to write and then you have p1 in higher dimension. It's very different. We have plenty of possible algebraic varieties and You have to make choices. So let us start with X Equals C2 and basically we will only be talking about C2 in the talk if you start so you take a certain mapping from C2 to C2 and You would like for Many reasons to compactify you it would be much easier So as we were working in one dimension, we compactified the plane Complex plane with projective plane here. You'd like to compactify it, right? So there's a There are many choices are many compactifications of C2 many possible compactifications of C2 for instance. So one natural one is the Complex projective space, but why not a product of p1's and Why not blowing up all these guys at points at infinity have many choices and One important part of higher dimensional dynamics at least is to find the good if you start with the same point of mapping from C2 to C2 to Find the right compactification in which you you want to work So I don't I won't enter to the details here, but perhaps let me do a few computations Okay, a few computations related to CP2 so if I take coordinates the W on C2 you can view these are You can homogenize these coordinates. So I will consider corresponding coordinates in CP2 Big Z big W and T and Small Z will be Z over T and small W is W over T. Okay, let me take a certain polynomial map if you want to understand so P2 so this is C2 with the Z axis and the W axis and If you can compactify it at P2 you add a certain line at infinity Which is actually the T equals 0 Axis so C2 here is T equals 1 the chart T equals 1 and the line infinity is just T equals 0 Right, so if you want to analyze what this mapping does to the line at infinity you homogenize these Equations, so let's D be the degree so the maximum of the degree of P and degree of Q and so in homogeneous coordinates ZWT goes to So you homogenize TD P of Z over T W over T TD Q of Z over T W over T and TD Okay, so first example first example assume that so the there will be different behaviors according to the Homogeneous part of higher degree of these polynomials P and Q so first first case for instance assume that P of ZW is something like ZD plus terms of lower degree and Also Q of ZW is WD plus terms of Lower degree then when you homogenize You see something like that What you get is ZD Capital Z now plus terms of lower degree of ZW and T WD plus terms of lower degree and TD Now when you restrict the line at infinity When you restrict to the line at infinity what you see is ZW so it's a P1 So now you have two coordinates maps to ZD WD So you get a well-defined mapping at infinity So in that case the mapping extends holomorphically to infinity and you get a holomorphic map on the whole projective space So that was a special case actually the right Assumption is that You should have no common zeros between the terms of higher degree the homogenous term of higher degree of these two polynomials P and Q So let's look at the second example. So in that case the conclusion is that F extends To a holomorphic map On P2 Okay, so we say that so sometimes we say that the mappings Polynomial mappings on C2 which extend holomorphically at infinity are called regular mappings. Okay, that's one terminology Now second case Let's assume that you start with this famous mapping So let's start with a famous automorphism of C2 FzW is Z square plus C plus W AW So this is a degree two mapping from the plane to itself And if you look at it a little bit, you will see that it's an automorphism It's a point of automorphism if you if you try to solve F of ZW equals Z prime W prime then you you get It's easy here Then you get Z from the second coordinate and then you can Plug it into the first one. So that's an automorphism and it's it's called a non-automorphism It has been studied a lot by people in real dynamics. If you do the same thing What you get is Z square plus C t square plus W t A Z t t square Now when you do t equals zero What remains is Z square Zero on the line at infinity. So what happens here? So this is C2. This is the line at infinity When Z is different from zero Everything goes so when Z Well, when Z is different from zero, this is exactly the same as 1 0, right? It's homogeneous coordinates So it means that the mapping sends everybody on the line at infinity to this point, which is 1 0 here So this is 1 0 0 But when Z is equal to zero, the mapping is undefined. So here The mapping is indeterminate So we are in a situation where you have a holomorphic map on C2, a very simple one But when you try to extend to P2 you get an indeterminate C point, okay? So you can say okay, perhaps the choice of the compactification was not good. I chose P2 Perhaps it was not the good choice and the answer is that there is no good choice in this case So there's a theorem due to a dealer and far Which states that there exists no, so there is no complex surface. There is no compactification of C2 such that F extends to a holomorphic self-map on X So you can make F holomorphic, you can resolve the singularities But then you will fall in some other manifold. So you cannot find the self-map which is holomorphic Okay, so it means that in this special simple case that you have to deal with rational maps which are not defined everywhere that's an important fact and actually this So you have plenty of classes of mappings with different behavior So perhaps you cannot find you cannot find the compactification when the maps become holomorphic But perhaps you can find the compactification which is well adapted into the dynamics in a sense So I don't want to be too precise here but there is a notion of good compactification and So let me just say that good good compactification still exists So in that case the good compactification is simply P2 after all there's a point of indeterminacy, but it's not so bad In for every so that there's a another theorem by dealer and fab which is very difficult By Faber and Johnson, I don't need to state it. So this theorem by Faber and Johnson says that good compactifications always exist for point of maps of C2 So this problem of compactifying is well understood in two dimensions and it's completely open in a higher dimension So there's a lot of difficulty here. We have to live with that Okay, so So there's a whole zoo of different behaviors different mappings So in two dimensions in two dimensions in C2 Two good representatives of the possible of the diversity of behaviors are these two guys. So There are two classes of interesting mappings. So the holomorphic mapping You have these mappings which extend holomorphically to P2 no No, excuse me.  So in two dimensions in two dimensions basically you have two possible behaviors one is that of automorphisms of C2 and more generally of Perrational transformations and the other one is holomorphic mappings So if you understand these two classes of mappings, you basically understand everything that can happen in two dimensions And in higher dimension have more choices. So in the talk, I will only Focus on these these classes of examples and mainly on this one I will just state a few results that are specific to C2 automorphisms Yes Okay, so I mean that I add something I add a certain divisor to C2 and I ask that if extends to holomorphic map So that's the extension of it. So that's So f okay Okay, okay f. Okay is for f the Heno automorphism. Is that a question? Yeah And and it is true for every automorphism of C2 which has non-trivial dynamics Each time you have an automorphism of C2 with non-trivial dynamics you have this result Okay Okay, so now let me talk about the Fattu Giulia the composition So here the definition exactly the same as in one dimension So you have a notion of normal family in this context. So the Fattu set Again it is so here I take f from x to x where x is C2 or P2 and F is one of these two examples. Okay an automorphism of C2 or mapping on P2 So the Fattu set is a set of points x such that fn is normal In the neighborhood of x and the Giulia sets the complement. So we will see in the next lecture that This definition of the I mean this is a nice definition for the Giulia set But perhaps there are ways of refining the definition as you can refine the definition for higher dimensional cloning groups of just the limit set Okay Um So there's a basic difficulty here There are several basic difficulties But one is that there is no Montel theorem. So the Montels Montels theorem is false In C2 so you can have Non-normal families of holomorphic mappings in two dimensions which avoid pretty big sets in C2 So there's no reasonable analog of Montel theorem and the reason is this one Something very well known which is called the Fatoubi-Berber phenomenon So this is due to the Fatoubi-Berber phenomenon There exist proper open subsets of C2 which are by holomorphic to C2. So there exists Proper subsets of C2 and so open And perhaps a set which avoid a ball or avoid something something big open avoiding avoiding such that The closure of omega is not equal to C2 And such that omega is by holomorphic to C2 So you can map C2 into itself in a strictly smaller subset. Okay, so uh from these kind of examples Of course, it's easy to find normal families With values in such a non-normal families in values in such a set omega, right? And the construction just in one word So it's a dynamical construction the easiest construction of the Fatoubi-Berber example is dynamical is just You take f an automorphism of C2 Which which admits an attracting point with p an attracting point Okay, then So we saw uh in one dimension that each time you have an attracting point It attracts a critical point, okay And if you look at the argument the argument is something like if it did not attract a critical point Then the basin would be c But here you have an automorphism So it cannot attract a critical point and the answer is that the basin is by holomorphic to C2 then The basin of p is by holomorphic To c2 just a simple exercise Okay, but now the automorphism can have several attracting points So each time you get a basin which is c2. So this basin most often it's a Fatoubi-Berber domain, right? So that's one difficulty and Another difficulty that you don't have the Poincare metric. So you have You have the Kobayashi metric or the Bergman metric So the Bergman metric or Kobayashi metric these are efficient tools, but they don't work all the time For instance in the Fatoubi-Berber domain the Kobayashi metric is just zero So you don't have the Poincare metric in every domain in c you have the Poincare Besides except c star you have the Poincare metric and you can work with it here in many cases It's just ineffective Does not is not always Effective, okay, so these are difficulties we have to deal with Yes, because if you try to prove the same argument Since it doesn't work it see it's implied that there's a critical point in the basin Yes, yes, yes Okay, um, so let me talk about the classification of Fatoubi-Berber, yes It means that it's a nice metric, but sometimes it is he's he's he's uh equal to zero So you cannot use it Yes, but in many cases just vanishes, okay, so you cannot use it to to to find fixed points or whatever Okay, so let me talk about Fatou components. So there will not be many proofs here Just a collection of results about Fatou components So what do we know about Fatou components? so there's a Starting classification of Fatou components So again, so of course you can construct many examples of Fatou components just by taking products, okay, so Uh, just by taking products of one dimensional maps you can you can get various behaviors for instance If you take I don't know if you take a mapping of this form p of z q of w where p and q are polynomials in one variable Well, you can construct examples with for instance a component, which is a z-gold disk times a parabolic basin So in the Fatou component everybody will go to the boundary by turning in the z-gold disk So you can you can construct a few examples like that and the question of classification of Fatou components is Is there something else and people tend to think that most often you you get all possible Behaviors just by taking products, okay, it's not a theorem right now. There are some indications Okay, so that's the first theorem. So let me recall this definition the Fatou component Is recurrent if there exists an orbit which does not go to infinity If there exists p an invariant excuse me, so an invariant Fatou component Is recurrent if there exists p such that fn of p does not converge to the boundary of omega Okay, and uh, actually this So we would like to classify invariant Fatou components invariant Fatou components They fall into two types recurrent and transient. So it is transient otherwise And it turns out that recurrent Fatou components can be classified So that's the result and it's essentially the same as in dimension one. So that's a theorem So it is due to Bedford and Smiley For automorphisms and to Farnes and Ciboni for holomorphic maps on p2 Okay, so the statement is let f be as before so automorphism a holomorphic map of p2 And omega an invariant Fatou component A recurrent Fatou component Then omega can be of the following types. The first case is that omega is an attracting basin So first possibility omega is an attracting basin or super attracting perhaps basin Second case there exists a certain closed Submanifold In omega closed Uh, so here we are in dimension two. Okay, so a closed submanifold Which is biholographic To a disc A punctured disc or an annulus And on which f acts as an irrational rotation Okay, and every point in the Fatou component is attracted by sigma And for every p in omega the orbit of p converges to sigma. So you have a a z-gold disc somewhere in the component So orbits are turning there and everybody's converging to that z-gold disc. And last case Omega is a z-gold domain Meaning that there exists a certain sequence of iterates converging to identity not right Down So this means that there exists a certain sequence f n i converging to identity So like in a z-gold disc and in that case actually the closure of of f n as a subgroup of The automorphism of omega is actually a torus of dimension one or two so is Is a tk Times f where f is a finite group and k is one or two So in dimension one when you're in a z-gold disc the closure of the of the f n is just a circle right in the in the subgroup In the in the group of automorphism of omega. So here we can have a circle of a torus Okay in the torus case you can just get from a linearizable fixed points with two different irrational eigenvalues Okay, so that's all the recurrent case Yes, this is the definition. Yes. Yes. Yes. Yes. This is the definition Uh It depends on the cases So for automorphism of c2 most often basing So if for instance for automorphisms of c2, you know that omega is by homomorphic to a disc cross c Or uh annulus cross c or by disc Or you have a list of possibilities Okay, and nobody knows if uh, there are airman rings. So if annulus cross c is possible That's one open question. So it's an open question whether there exists a certain complex and on map With a fat two component, which is of by homomorphic to an annulus cross c That's an open question. So I think this the last possibility last thing that is not known about fat two components of a non-map So now in the transient case Almost nothing is known There's only one uh nice theorem Which is due to lubic And peters which is quite recent And which says that if f is a dissipative a non-map Is a hand on map, uh a polynomial automorphism Of c2 with a certain assumption on the Jacobian So an automorphism of c2 always has constant Jacobian because the Jacobian is a Poinomial polynomial, which does not vanish. Okay, and you assume that the Jacobian is less than one over the degree of f Square the reason for this assumption is very delicate. I don't want to explain where it comes from, but it's called moderately dissipative Then if Omega is an invariant transient fat two components transient Fat two components Then it is the basin of a parabolic cycle Then there exists A certain p in the boundary of omega with eigenvalues one and uh The other one smaller than one because we are in the dissipative case And everybody's converging to p such that for every q in omega for every z in omega the iterate of z goes to So it's a version of the snail lemma, which we saw in the last lecture Okay, you know that on the boundary there is a fixed point So the fact that there's a fixed point at the boundary is absolutely not obvious because you don't have the Kobayashi hyperbolicity argument in that case omega is biomorphic to c2 Okay, actually it's consequence of the theorem that omega is biomorphic to c2 So you cannot start the argument just by saying you have the Kobayashi metric and then everybody goes to a single point That's the difficulty Okay, so that's for invariant components Yes Excuse me the writing Polynomial automorphism Polynomial automorphism So, uh, that's for the classification of periodic components So it's the only general class of mappings where we know something about transient components. Otherwise, it's completely open So now there's another natural question, which is the question of wondering for two components the possibility of wondering for two components And uh, there is an answer now So the theorem is like that So that's a theorem that we proved with a mature store Xavier Buff Myself and peters and jasmine racy So there are examples of polynomial mappings of c2 with wondering for two components So there exists Polynomial mappings of c2 Wondering for two components arbitrary close to the to the region for two components So the Sylvan theorem does not work So it was known that the method of the Sylvan theorem does not work But the theorem Is false either so the the theorem actually gives explicit example. So let me give you one explicit example So this is a mapping Z w goes to z So let me check z plus z square. So it's a kind of funny minus So point 21 36 z4 Plus pi square over 4 w w minus w square So that's one example of a mapping with a wondering for two components and let me show it to you if The guy accepts to work So I have pictures Here we go so Actually in that I chose this mapping this particular map is a real map, right? So it induces a mapping from r2 to r2 Okay, so what you see here is a window in r2 and what you see is a window I think it is minus three three times zero one Okay, and all the sets So there's this grayish picture It is the set of points with bounded orbits in r2 every point in white is escaping under the dynamics Okay, and every point outside this window is also escaping So the non escaping locus is contained here and it's this gray region And now I will do a zoom of a very very tiny Piece here of the bottom of the of the of the image. So you will see a picture a zoom About One so minus three three times one over one million Okay So the zoom is here So that's the magnified picture on the left Okay, so that's a this picture is a zoom of the very thin slides Of the of the previous one and you start with the Red dot on the upper left and follow its orbit So the red dot is contained in a fat two component Which is so there's a zoom of the fat two component here in green So the red dot on the upper left What you see the all the black points are the iterates of this red dot. Okay, so here What you see is it working now Yes, it is working. So you have this red dot here What you see is 2001 iterates Of this red dot and you arrive here after 2001 iterates you are you arrive here Okay, so you travel to the to the right and by slowly decaying This is because the second coordinate has a parabolic point at the origin So you do 2001 iterates of these points. So again Of these red you arrive here And then you continue so this one this this point is in the southern wandering factor component in the fat two component in green And now you make 2003 iterates your idea and so on So you have an orbit which does something like it goes to the right and come back goes to the right and come back But it takes longer and longer time to come back So it is not in a periodic component the component must be wandering and so we have these specific examples like that Okay So the fat two uh the sylvan theorem does not hold in the in that setting So let me uh Oh, that's a long story It's uh, so you see You see in both coordinates are parabolic points Okay, and it's based on uh parabolic dynamics in one dimension and the phenomenon of So-called parabolic implosion So but it's a very long story. Okay, very technical Okay, another theorem Another counter example, which is classical Is that also the analog of the alphastereiniteness is not true So theorem So there exists So automorphisms of c2 Or holomorphic maps on p2 With infinitely many things So this is a classical phenomenon in dynamical systems. So that's found due to new house So even if it was not formulated in this holomorphic context, it works exactly the same Actually, you can just take real maps of surfaces And so it means that the finiteness of non-repelling orbits is not true in that case either. Okay Okay, so that's for the uh fatou set in higher dimension What time is it? Okay, so now I would like to uh go back to one dimension and do some ergodic theory So now lecture two starts So you will never see lecture three So I want to talk about equidistribution and ergodic theory Okay, and again, I will start with one dimension. So part a Is one dimension and part b is for tomorrow Okay, so uh, let me remind the setting. So the setting is just you take a rational map on p1 Of degree Larger than two and the degree is d Okay and I want to study the following problem. So the problem is this one I take a point a On the sphere And I look at the solution at the pre-images of a so I look at the solutions Of this equation So this equation has d to the power n solutions with multiplicity Counting multiplicity, of course, you have d to the power n solutions with equation And I would like to know how they are distributed on the sphere And what we will see is that actually they always accept very few exceptions They always the equidistribute to the same thing and this has very strong consequences on the dynamics Okay, uh, so what what do I mean by equidistribute? I just consider, uh, what I will call the pullback So I consider the normalized pullback of the direct mass at a which is just the sum normalized sum of direct masses At the solutions of equi of the equation, of course, I I add multiplicities, okay I count the solutions as many times are there solutions Okay, and the theorem is this one So the theorem is proved by by brolin in the 60s for polynomial for polynomials And then in the general case independently by u beach on one side and Lopez and manier at the beginning of the 80s in the general case And the theorem is this one For every so remember that there's a certain there's an exceptional set which contains at most two points So remember the exceptional set It is the maximal finite set which is totally invariant and its cardinality Is at least two so remember that there are basically one two possibilities either the mapping is a polynomial and the exceptional set is one point Or the mapping is z d or z to the minus d and you have two points and, uh The statement here is if a is not An exceptional point then the sequence of measure converges this sequence of probability measure converges to A certain measure mu which is independent of the choices Which is independent of a of course and And that's it And that's it which is called the equilibrium measure of f by definition Okay, so a consequence of this Just a restatement If you start with a measure which now is not a direct mass So if nu is any probability measure on the sphere on the sphere Such that nu gives no mass To the critical to the exceptional locus then the prime edges of nu converge to The equilibrium measure Okay, it's just the proof of the corollary. It's just that every measure is an integral of direct masses Okay, so how Does one prove such a theorem? So the proof is based on potential theory so that several proofs, but the easiest one so the proof Is based on potential theory So actually we don't need much Of potential theory. We just need basic properties of subharmonic functions and I maybe the only thing we need from The only idea we need from potential theory is that So the idea Let's work on seed. So the potential theory on p1 is not very Pleasant because there are no subharmonic functions on p1 because of the maximum principle Okay, every subharmonic function on p1 is constant. So of course the theory of subharmonic function on p1 is a little poor So it's easier to think about c for the moment. Okay, so the on on the plane On the plane every or maybe on a certain open subset of the plane every positive measure positive measure Is the laplacian of a certain subharmonic function Okay, where where u Is subharmonic And then working with measure comes down to working with functions and it's much easier to work with functions You can multiply you can you can do plenty of things that you can do cannot do with measures. Okay, so working with so we say that U is the potential so it's not unique it's defined up to a harmonic function, but it's not a problem here So u is the potential of the measure or one potential of the measure Okay, and for instance an operation like Taking the pullback of mu by f is just composing the potential with f Because I mean it's just taking the pullback of the function but the pullback of a function is just Just the composition So of course it's easier to understand this kind of thing This kind of thing Yes, yes, yes, yes, yes.  So we will start Start with the corollary So we start so it's more complicated to work with direct masses than To work with smooth measures So we will first prove the theorem for smooth measures so proof So start With a smooth let's say smooth measure Measure new probability probability measure New on p1 So as I said You cannot find the potential of a probability measure on p1 because every sub harmonic function is constant But what you can do is find the potential of a difference of probability measures So think about it if you look at the difference Of direct masses Then this Has a potential on p1 because You have in c In c you have this formula which is well known That the direct mass on a is the laplacian of the logarithm of this function Maybe just Canonical solution to the laplacian, okay Now if you want to think about it in homogeneous terms So z in homogeneous coordinates is z1 z2 a is a1 a2 So if you homogenize this this Modulus here what you get is z1 a2 minus z2 a1 Okay, so that's the Homogenization of this So of course, this is not a well defined function on p1 because you there's an homogenizing factor But now if you take the difference of these two direct masses You can write it as 1 over 2 pi Laplacian Of the logarithm of the quotient of these guys z1 a2 minus z2 a1 over z1 b2 minus z2 b1 And now you have a well defined function on projective space If you multiply z z1 and z2 by a constant, of course this cancels So now this is a well defined function on projective space and this potential is well defined on projective space It's just locally the difference of two sub harmonic functions So this is well defined on p1 This is well defined of p1 and locally it's just the difference of two sub harmonic functions Okay So it has basically it has the same property it shares So the space of differences of sub harmonic functions with just the vector space Generated by sub harmonic function. It shares a number of properties with sub harmonic functions regularity and so on so now What I did with two Direct masses I can do now with two probability measures just by integration So now so the consequence of this is If nu1 and nu2 Are probability measure on the sphere and the difference Is the laplacian of a certain So a certain function u certain potential u which is a difference of some of sub harmonic function Okay, so now let me start so back to the dynamics. I start with a smooth measure nu on p1 So I will consider the following difference of Probability measure. So that's a difference of probability measures, right? Let me write it the other way around Plus minus So that's the laplacian of someone Which I called u0 Okay, and now I can eat so you not so if nu is smooth u0 is smooth too so that the Regularizing effect of the laplacian. So you notice move Okay, and I can iterate that I take one pull back And I have this formula and you can sort of course get this Telescopic behavior And what you get at the end if you sum k terms like this What you get at the end is just A converging series You get something like nu is equal to a minus one of the dk fk star Of nu is the laplacian of the sum of u0 composed with fk I'm missing something here fj Over dj from j of zero to k minus one Okay, that's just the telescopic version Now, uh, this is a converging series Yes Yes, I need to divide so I apply here I apply One over df star to both sides of the equations Okay, is it right now? Okay, so I apply this and I apply again just telescope. I use the telescopic sum Okay, and I get something like that So now this is a uniform This is a uniformly convergent sequence of functions So the limit will not be smooth. It will only be continuous But so this is a converging sequence So by just taking the laplacian in the sense of distribution So this is fixed you get that this sequence is converging to Okay, so the consequence of that is that mu is uh Nu minus the laplacian of this infinite series Okay, so you may wonder if the measure that you get depends on nu But it does not because if you start with another one if you look at the difference between If you start with a need another A probability measure And you increase to you both of them. It is just the laplacian So nu one minus nu two is the laplacian of a certain g So it's just the laplacian of g composed with fn over dn So it converges to zero Okay, so you get yes Yes, exactly. So I I apply This again I apply one over df star to this equation and then I sum in in columns And then everything can the terms can solve two by two Okay, uh, so if I start with two different probability measures, then I take the difference and make this operation I get zero Okay, so you get we get this equation for probability measures So now for general measures the difficulty if you start with a direct mass for instance, this function is not Which is not bounded Okay, it's a logarithm. So you cannot apply just this This uniform convergence here Okay, so if you want to work with points It's more complicated and you have to understand a little bit the geometry of the problem And this is how it works So now the consequence of that is we get equidistribution for all smooth measures So what do we do for points? Well for points, we uh, let me introduce a notation here So that's the critical set Which plays an important role. So it's a set of points where the derivative vanishes And let me define the post-critical set of order l. It's just the union of the l's first images of the critical set So that's c Enough fj of c from j equals zero to l Okay, and the pc infinite Is just the limit the union of these forward images Okay, so there's this famous lemma due to Lubitsch Which says the following So what is the problem the problem is that One one something that you have to understand is when you take inverse branches by f Of course the invent inverse branches are not always well defined Because of the critical set. Okay, you cannot solve the equation holomorphically And the point is you start with a certain point a And you want to understand the tree of frame edges of a some can be multiple some are simple If you have a simple pray if this one is a simple solution of the equation Then you're in good shape because you can pull back a little disc from here holomorphically You have a certain inverse branch and if here if this point belongs to the critical set If this point is critical you cannot pull back a disc holomorphically So here you you have a certain difficulty Okay, and the language lemma says that if you go sufficiently far away in the tree Then you will mostly see non-critical branches So the language lemma says the following for every epsilon positive there exists a certain l such that If z does not belong to the post-critical set of order l Then for every n large enough Large enough n Let me put the quantifiers in the right way There exists a certain disc about u The topological disc such that for every large enough n f n admits D to the n times 1 minus epsilon holomorphic in the inverse branches inverse branches On u Okay, so we have a certain obstruction which is a critical set So this is a finite set, okay So outside the certain finite if you give yourself a certain epsilon Outside the certain finite set then you can pull back univalently f on small discs And u should be disjoint so this neighborhood there exists So of course it will be disjoint from this finite set and furthermore the diameter The diameter of the prime edge. So these inverse branches. I will denote them by f i Minus n so the i here is just to remind that there are several of them Okay, so i is supposed to range between all value all possible values And the diameter of f i minus n of u is less is an o Of d to the power of minus n over 2 Okay, so that's an elementary lemma as very difficult to prove So first before proving the lemma, let me explain why it implies the the theorem So why Does it imply The equidistribution, okay So if so start with a point z which does not belong to The infinite critical set so that's a countable set, okay, if you start with two points z z prime Which do not belong to this Countable set then it means that for large l z z prime do not belong to No for every l Excuse me, so you start with two points outside this countable set Okay, you give yourself an epsilon and now So uh, you which lemma gives you a value of l so you look so that z And that's z prime. I'm missing something here Perhaps I should have said excuse me that's for every topological disc Disjoint from pcl this joint from pcl and containing u So the post-critical set of order l is just The finite number of points so you choose a certain topological disc joining the two and avoiding the Avoiding the post-critical set And then by the Lubitsch lemma, so you give yourself your epsilon here by the Lubitsch lemma You know that you can pull back univalently this thin disc And you get the overwhelming majority of prime edges You will get so when you pull back By all possible branches of f n in the in most cases what you will see is a very very tiny disc Concerning two Prime edges of z and z prime which are very close Okay, and the small number of x of bad discs which you can forget because the they count for epsilon in the measure Okay, so that's for uh points not belong so you get equal distribution for all points outside the certain countable set and of course you must equal distribute to the same measure as before just by a smoothing and When you are in the in this set, there's a certain combinatorial argument saying that Unless you are an exceptional point if you are an exceptional point You will never escape this infinite post-critical set But if you are not an exceptional point after a few iterates You will get an overwhelming majority of prime edges outside this post-critical set and then you apply the previous argument Okay, so you get equal distribution from this From this lemma so how Do you prove the Lubitsch lemma? Check the time Okay so I start with an open set which is this joint From a post-critical set of order l Okay, so it has This is an empty set So it implies that u has d to the power l Univalent prime edge not f excuse me f l as d to the power l univalent Prime edges on u. Okay, so let me draw the tree of prime edges Yes, that's a disc topological disc Okay, so let me start With this disc excuse me this joint from the post-critical set of order l and I put it back l times Okay So you get d to the l good prime edges now Maybe you are in the you you meet the post-critical set of order l plus one Okay, so among these prime edges a certain number of them may intersect the critical set but the critical set as admits as most To the d minus one or to the minus one 2d minus So the cardinality so the derivative is p prime q minus q prime p blah blah blah So the cardinality is 2d minus one two 2d minus one that's something I'm shocked by this formula 2d minus one does not sound right wait Yes Yes, because of consolation of 2d minus two is better. So the cardinality of the critical set Is 2d minus two so it means that among these so l is already very large So among these d to the l prime edges at most 2d minus two meet the critical set so you just You just discard them and continue So you discard this so that's a critical set you discard this guy these ones You discard these components and keep going Okay, and at each step you simply discard the component You simply discard the component containing a critical point and you count Okay, and so you see that at step l plus one How many good prime edges you have?    So i'm counting good prime edges Okay, so at step l plus one you discard possibly not not d 2d minus two excuse me That's 2d minus two, okay and You do the same so that's the cardinality of critical set. Let me say it's degree Of c so at the next step you have this You you you take this you remove degree of c and you start again and so the formula should be something like Sum from j equals zero to k minus one of degree of c d j I guess that's the formula that you the formula that you obtained Is it okay?  So perhaps there may be a small mistake here, but it's not the idea here Okay, so that's for the lubech lemma and then you have the Equal distribution So now let me explain the consequences of this So now we know that Primitives of generic points Equal distribute always the same measure So now this measure has nice properties. So that's a again a theorem of lubech And uh frère Lopez and manier So the measure mu So the equilibrium measure Has a number of nice properties So first it is invariant So first the support of the measure is exactly the Julia set So the measure is supported exactly on the Julia set Second the measure mu is in is an invariant measure. So it It has dynamical meaning And even it is mixing So in particular, it's an ergodic measure third property Mu is as positive yapunov as a positive yapunov exponent. So mu Has a yapunov exponent Bigger than log d over two This means that for almost every for mu almost every x If you look at the norm of the derivative Of f at x and look at this on logarithmic scale This is bigger The lim inf of this is bigger than log d over two So this means that on the Julia set you really have Um chaotic dynamics, okay, you have Exponential sensitivity to initial conditions. So on the fact to set Everything is a quick continuous and and you have this gap on the Julia set. You have Sensitivity to initial conditions with a certain rate log d over two, which is optimal. This is sharp This is sharp fourth property mu Describes also the asymptotic distribution of periodic orbits describes The distribution of saddle point of repelling points distribution of periodic points Meaning that if you put A direct mass at each periodic point of period n normalized by the number and let n go to infinity you converge to mu And the last property which I will not comment is that mu is a measure of maximal entropy. So the entropy Of mu is equal to log d For those who know what entropy means and it's the topological entropy of the map. I don't want to Explain what it means. It's not Not the point Okay, so how where does this come from? Let me give you some proofs Ha ha good question So there are plenty of measures that you can construct on the limit set, right? Uh, so I don't think this one is the Paterson-Sullivan measure. So Paterson-Sullivan measure would more correspond to um And it's looking for the world for to geometric Hossdorff measures on the not Hossdorff measures, but conformal measures on the Julia set So you can in certain situations in the geometrically finite situation You can construct conformal measures on the Julia set with basically the same procedure in the as the as in the As in the Kleinian group setting and these are the analogous of So I would say that so we have a work with Bertrand on random works on Kleinian groups And if you give yourself a certain if you give yourself a certain Finitely generative group and look like a surface group you can fix you can work at random on the group If you do this random work on the group Then you get a measure on p1 which is the unique stationary measure under the under the random work Which satisfies an equation probability which looks more like this one meaning that you don't need to put Modular modulate of the derivatives against the direct masses you just have plain equation for these measures So I would say that the analog in the Sullivan dictionary are more stationary measures for random works Okay Yes, see you you were you were talking about uh how's that I mentioned in Here so just the fact that you have this uh the definition of mu You have this geometric convergence series. So the potential of mu is held are continuous So just looking at the sum the potential of mu Is always under continuous Is say is holder continuous So this implies Continuous this implies that the household dimension of the Julia set is always Always positive Okay, because the laplacian of a hold of alpha holder continuous function Uh is supported by a set with the household dimension is at least alpha Okay, so you can you have household dimensions estimate for free Okay, just a few ideas of the proof and I think I will be done with the time. Yes Okay, so some ideas of the proof So I will not prove this equality actually it's not completely obvious So it is obvious that mu is supported on the Julia set At least it is obvious from this property because at almost every point you have exponential expansion of the derivative So you cannot be in the fact to set what what is less obvious is that when you Uh on the Julia set you are on the support of you there's something to prove which is not Not completely obvious. So let me skip this part The fact that mu is invariant is obvious. Okay mu by construction The measure mu satisfies this functional equation. Okay, if you push If you push it by f You get that mu is invariant in the usual sense Okay, so why is it mixing? Why is it mixing so mixing you have to prove that for every continuous if you take a pair of continuous function if you Consider this sequence of integral This must converge to integral of phi times integral of Psi that's mixing. Okay, so why is this true? Well, you just look at this equation integral of phi composed with fn times psi Mu So phi composed with fn is the pullback of psi by f. So it's f and star Of psi of phi Times psi integrated against mu. But now mu is the same as f and star Of mu over d to the n Okay, so what you see here So let me arise all this Is integral of f and star Over dn Of phi times mu Against psi But now this is a certain phi times. So Now we can apply the distribution theorem to this sequence of measures Okay, this also converges to mu. So not exactly mu because this is not a probability measure So it converges to a certain normalizing constant times mu. And what is the normalizing constant? It's just the integral of this function. So this converges to integral of phi Against mu times mu Okay, but now when you put the psi here, you get exactly what you want, which is mixing, right? So the measure is mixing The bound on the Lyapunov exponent The bound on the Lyapunov exponent just is a obvious consequence of the Lyubitsch lemma. So what what does the Lyubitsch lemma say? The Lyubitsch lemma says that for typical branches if you take The point x the tipi for in the small ball about x if you take a typical branch Of fn it will contract this ball by a factor Oh, I forgot to explain the factor d to the n over 2 And let me I will come come back to this point just after So what the Lyubitsch lemma says that if you take for for a generic situation If you take a generic point and a small ball about it for a generic frame edge You will contract this ball by a factor exponential to the D to the n over 2, okay? So this is exactly I mean by definition this means that the Lyapunov exponent of Of mu is at least log d over 2. So let me come back to this So why let me come back to the Lyubitsch lemma why the inverse branches Have diameter d to the minus n over 2 So it's just a very easy argument. So so remember you have we have constructed the number of good inverse branches Univalent ones and there are d to the n of them So on the sphere of p1 we have constructed plenty of Of this joint open subsets about d to the n this joint open subsets Plus perhaps some crap which we The bad the bad inverse branches you forget about them So if you look at all these inverse branches most of them have area so most inverse branches Have that the area of the prime edges of u is less than D constant d to the minus n over 2 Okay, just by counting And then you apply cuba distortion. So by the cuba distortion lemma if you have this Round disc mapping to a certain component to a certain small set of area D to the minus n over 2 if you reduce The disc just a little bit since the mapping is univalent what you get here is diameter D to the n over 2 so you just conclude by cuba distortion Okay, and I think I have just one minute to explain the The equidistribution of periodic points So how do you come so? How do you prove that? Repelling points are equidistributed Well, you look at this picture again So you take your generic point x And a small disc so this disc is supposed to be small about x And you look for prime edges Which come back? Which come from this disc so that's f to the n So you know that for most prime edges most branches of f so most are well defined. Okay, most Branches are univalent and well defined and most Give right to a small disc like that and you look at those small discs which are contained in the big one Well, it's just the set of points. So the measure of the set of Points which do this is just the measure Of set of points which come back to D under under n iterations By mixing this is approximately mu of D Square Okay Now, uh, this is also so each time you have a point coming back So we have plenty of nice Small inverse branch like that. So that's branches of fn So this is also uh, the sum This is also the sum. So here i'm cheating a little bit because Okay, uh, let me do it like that. This is the sum of all inverse branches This is approximately equal to the sum of all inverse branches of fn on D Okay Of The measure of this small ball here. So actually this set is the union of small balls like that plus a little bit of crap okay, so This is the sum of the measures of the prime edges of the inverse branch of D over the set of inverse branches Right now since the measure satisfies this functional equation You know that the measure of something like that is exactly one over dn times the measure of the disc So it's the sum One over dn times the sum over the set of inverse branches of mu of D Okay, now each time you have a small disc mapping into a big one You have a repelling point, right? Just uh, you have a this You don't know a Brewer fixed point theorem for the inverse And the point must be repelling because of the Poincare metric Okay, so to each of these inverse branches is associated a certain repelling point So this is just one over dn mu of D Time the number of repelling point in D At least so if you don't know you can construct a set of repelling point which which has exactly this cardinality So now you can sell out this mu of D And you see that the average number of repelling points in the disc is essentially mu of D Okay, so this is exactly a equistribution Okay, uh, so I think I will stop here and tomorrow I will explain what becomes of all this in two dimensions


All right folks welcome on back here this Tuesday July 6th shortly after 2 30 p.m. Eastern time We'll come on back of course for our afternoon meeting it hosted the morning meeting Fausto was out earlier and he'll issue I think he is out tomorrow. Also. He was taking a small trip out with family He'll be back on Thursday But great to see all of us live inside the cybergroup room and of course for all of us in social media Facebook live and YouTube Let's hop right into it. So the BSQR trade was looking great going into like two o'clock It was looking great going into more even like 215 It was pushing up to you know break a new high the day. It was testing and popping over 750 The false pop here at first was to be expected. In fact, actually there was a huge iceberg order I forgot the size of it for my level four, but you know, let's just say like anywhere between 50 to 100k Oh, I see is like dark red line going across my level four program I don't have the chance to show it right now for all of us on the stream But you know, that was a big level and in fact going into the afternoon this stock I was doing pretty well on but then I just gave back some profits So it ended up kind of filling me on a trade right off of 730 here at first Then stop me out. All right. Well, you know, no problem. I don't mind, you know jumping in once getting stopped out It's a big level. So it ended up pushing right back up You know, I didn't look elsewhere and jump right back in on the same trade right from the same level I was expecting the breakout. Oh, I said to you folks live inside the trading room Said looking for BSQR to break HOD. What do you think that means?  I'm in this trade right from the level We are calling out from 730 ended up Unfortunately failing to break that HOD it held right under the next big resistance off of 750 That is and I ended up wasting that profit. I ended up taking a two cent hit I call it a break even trade or scratch trade more so but um, you know Hey started to give some back there and you know for what it's worth now I guess pushing back up towards that same 730 level. I'll check back in with this stock at the end of this meeting I wouldn't mind adding it on our list. I'm not fully convinced yet though You know very volatile. So if it ends up moving all the way back up during this meeting I'm sure that you folks will let me know otherwise, but uh, you know, we'll come back to it at the end All right, so a couple chats though just beforehand from rich you was saying the XEL a trade Let's take a look on that one Yeah, geez. Yeah, I was wasting my time looks like they're rich, right damn this XEL a trade Who this thing ended up running right over to for a quick 40 cent move. No big deal Geez ended up running up very swiftly. You would expect for it to pull back down now In fact, let me get you the volume bars here. We're talking about this in our, you know Workshop earlier. I was gonna say webinar our workshop earlier trader stock Well, you know for the meantime a lot of volume traded off the top here I'm not looking at that as much, you know, that's resistance here in the meantime, but As low as 208, I would say more more so like to 19 to 20 on a pullback That could be a really big, you know dip point heading into the afternoon. So they easily enough I mean, we would have ran into this trade anyway as we went from the gainers to losers list in just a bit But the XEL a trade definitely familiar name to be traded the stock like a week ago You know less than a week ago, perhaps even pretty well on it All right, so otherwise though, and I'm gonna show Bitcoin at the end Don't worry for any of the crypto people out there on the social media streams. I you know that tend to you know Follow me on Bitcoin. I said I was gonna show that in the afternoon. We'll get to that the end You know, but we ended up doing pretty well across the board though You know really from the morning watch list that we had we did pretty decent And what I mean by that is even on some of the stocks that didn't make the big move that we are hoping for them to Make we recognized it. We knew what was going to happen So we didn't get like, you know really misled on some stocks. Actually, I'll begin with one of those We talked about it in our traders talk workshop the OCG entry This was one of my better trades across the more afternoon took a long time But you know stock ended up providing me an opportunity at first off of eight I ended up wasting that trade great entry. It allowed for a great, you know Profit for about 12 to 15 cents. If not, maybe a penny or two more But you know, I unfortunately I wanted more my greed got in the way and dropped whatever now Nonetheless though, there was never really a good buying opportunity on that stock until 750 until $7.50 and You know, I'd call that out in the trading room. It broke below 750 broke above 750 It was like geez like the slowest slingshot of all time But ended up beginning to make its move, you know a bit later on I think this looks fast, but I ended up jumping in initially here So it took me nearly like, you know 40 45 minutes back to, you know come to fruition But yeah, I mean even this, you know, it wasn't the best play on the morning But just goes to show it gave us some good opportunity, right now to those that did well at least the BLIN stock and this doesn't look like it did too hot But you know, we talked about this also in our traders talk workshop So based on the level that I was calling out from pre market 12 10 12 17 I copied and pasted my chart in traders talk for all of us at least live there It's in the chat board should be in there feeling a scroll all the way up if you're there You'll see the picture of the chart. I mean, I it's practically these levels But you know, I drew this during the workshop and it went to show that you know Hey off the first attempt of this testing as support held perfect and providing a really good bounce opportunity Not the previous trade but based on the technique based on following the order flow and and certainly more from pre market anticipating Such a big you know break above a level Where is it gonna bounce from not when but where so, you know, you focus on whole numbers But otherwise from where the big volume is being traded very very similar to what we were talking about on OCG And this for this one here, you know Hey once it ended up kind of snapping back down towards 12 10 12 17 led to a great trade And then even here heading into the early afternoon. I got tricked on this I ended up jumping in and out here off at 12.  So by the time I came back. It was already at like 12 50 I felt as if I was gonna chase it reason I bring it up is because one of our students Felix ended up getting like 68 70 cents on the trade I forgot what he had posted. He was in from right around 12 30. So ended up paying off really nicely for him I just spiked up way more than I thought it was gonna move up to when I first got in my trade there But you know for right now came back down the danger on stocks like this we talked about it You know the stocks pulling back down pretty quickly at this rate heading into the afternoon wouldn't shock me to see See break lower and fill the gap lower in the after hours. That's all There we go. All right, so for right now BLIN a decent trade from earlier this morning And even here going into the you know mid-afternoon off the breakthrough this level but at this point now I'm not really too convinced of another run up. All right So listen we covered BSQR. That's already number three on the gainers list. What we'll do is go from top to bottom We have a you know, I guess a warrant here at the top I'm kind of gonna skip over that to begin a rarely trade war and so unless it looks stellar you folks Let me know otherwise. I'm just gonna be pressed for time, but move forward So the oh out OSAT and I hopefully not too pressed for time bang my elbow OSAT ended up making a nice push more so leading into the open pre-market And then like as soon as the market opened up off of that pop ended up making a slingshot from eight Right and then ended up breaking higher suddenly. I wasn't really there for that But you know if you were there to catch it off the break from nine or maybe what appears to be like nine ten ish You know that paid off pretty nicely, but for right now looks pretty choppy. I mean, I'll come back to I'll come back to this trade at the end Just like kind of like the the other one the BSQR and I guess number three will just check in with it right now Man this thing pulled back over seven ended up slipping back down I mean it bounced off a seven pretty nicely here, but I don't know It's got a mega nice convincing move back up so far holding over seven is good But lower highs comes lower lows. We know that so, you know, sooner is better than later here on the street All right, we keep it going PBTS number four this one I know rich called out earlier more so heading into the early afternoon breaking out over to 50 Breaking out over three on the on the rebake here later on and you know really nice run I mean one of us said that they were in the trade from a much lower price and You know they were holding on for a little bit I mean what they had said made sense at the time They said there wasn't really any sign In as many words they said, you know, there wasn't really any reason to sell, you know It was just all buying right and I agreed like yeah, definitely But what I told them was focused on 50 cent levels and whole numbers so I think I was like even over here when it ended up already like popping and dropping moving around moving around 350 and Unfortunately, this thing ended up coming right back down over 50 cents lower Broke below, I mean almost a dollar lower look at that down to 260 and bounced from there. Jeez So definitely volatile definitely good stock to keep in mind across this week, you know good sideless stock perhaps I don't know. You know come back to it at the end So far the only one I really like is the XCLA and we didn't really even get to that on the gainer side You know, we kind of jumped ahead to that but uh, This PBTS we'll check back in at the end. We'll kind of do like a cherry pick kind of BLIN this one we know we're not really gonna follow as much this one. I need to put on the side XCLA we skip over we'll kind of skim through right now see some other you know Stacks that are on the gainer side here so far familiar familiar names, perhaps I don't think I'm gonna add it just gonna check in with it the ember and trade Just so choppy order book is so thin All right, remember I said sooner is better than later on BSQR. Let's keep it going. Let's keep it going Right back over 730.  We'll put this on our main list BSQR See if you can hold over 730 and get going over 750 there Let's see geez talk about traders dyslexia here. You have BSQR than SQBG This one thankfully we're not gonna watch but Geez tongue twister day as fed this one has been creeping up across the more like early afternoon and since It's almost like too slow order books not too filled in at least you have a big order on the bid 87,000 shares roughly on the bid at 150 Yeah, I don't know I'm gonna pass for right now Some other familiar name CCXI from pre-market. I didn't really like put this on our main list at the time You know, I said it had a good range The MMAT trade formally torch torch light TRCH that stock over the last month You know, you can see based on the pattern here practically the same stock ended up running up came all the way back down Over the last week. They all but did a merger with another company meta materials So I you know, we're following this now It's up 7% ended up making the pop more so after the market opened up, you know, good trade good runner Keeps as a good sideless stock That's all All right jinko solar at the ends. I just see it at the bottom. I'm just gonna check in expensive stock I knew that going into it, you know more of a swing trade Perhaps Fausto tends to follow this every now and then, you know, it's nice. It's nice to see at the highs Just very thin volume on the order book and very light volume traded per minute so far across the afternoon All right got a chat here from YouTube and I you know, I learned my lesson this morning I went to YouTube a little late.  I am pretty familiar with it as are a lot of us inside cyber group Call this out a lot off of news. Let me just jump to Benzinga pro really quick Like do that quick board a break myself All right, so apparently the FDA clears SR and ease phase 2 trial of non-opioid product candidate Blah blah blah for treatment of knee pain and osteoarthritis patients Interesting It's 30 deleted his comment. I see everything that's all right. So with that in mind No, I'm just I'm just kidding. But I you know for the meantime s already I mean hey popped off of news. It is shortable. So if it ends up completely dumping off I reckon there's an opportunity there, but we'll follow it more on the way up And we'll put this on our watch list for right now. I don't mind not really at a Excessive stocks on our list and then also Norman from the trading room just checking You know the chat there he had called out the pop just about five minutes ago So I'll give him credit there right away All right, cool. All right, let's hop over to the loser side Then I'll just check back in I'll go over my Bitcoin chart here at the end Not a whole lot really happened up and down it went inside the same channel But I'll talk about what could be ahead I'm thinking there could be a bit of a drop on Bitcoin throughout this week and perhaps even after or you know After the week is over. We'll see a drop. We'll see I'm not gonna time it But I think there could be a bit of a drop coming on Bitcoin. I'll explain why DD how flat is this? I mean we're gonna stay away from the stock certainly for the afternoon Keep it in mind for pre-market, but if it ends up acting as choppy as this has been I mean it's pretty similar to wish just you know really high float You know, we don't want to trade a stock that's too low float But you don't want to trade a stock that's always, you know Really high float because it's gonna take a lot to move the stock and although the stock doesn't have like too filled in of an Order, but I mean it is but we've seen stocks like you know, for example, you know a little time here I always love to show this in class. I mean you look at Siri look at the order book on Siri 94,000 on the ask 48,000 after 27 and so on and so forth. All right Well for DD it doesn't look like that but first for a stock that has such a high float And you think you know bad news is on it. Don't buy it It will look what happened off at 12 made for a great day trade opportunity once it really broke above 12 here You know good for at least 50 cents keep the good times rolling here at TIGR. I didn't really put on our main list as much Did I? Said at the end like that was the last one. I was looking at maybe maybe not and I chose not dropped off after the Market opened up made the bounce after you know thin order books So, you know perhaps I'll keep it in mind for tomorrow for as much as it's been dropping not only from Seemingly this morning in pre-market, but let alone the last couple of days at the end of last week You know that that probably is due for a bounce, you know 20 would be a big level of watch going forward for this week Food for thought so I mean just like globe at 10. It's not like a swing trader investment Opportunity, but if you're looking for a quick play 10s major support for as much as close been dropping This was the easiest nine cents my life not the biggest trade of the world for me But you know it was just on a thousand shares, but just to get in and out from 10 just relatively simple That was just my thought process on that so for for Tiger. I'm referring to you know, perhaps later on this week It's a ways away from 20, you know, it's at 21 50 right now But you know that that could be due for a bounce at some point keep it in mind ASTR about goes up comes right back down. I don't know. I don't really see many other Familiar names here or anything.  All right, we'll check out the chat board Then we'll jump to a Bitcoin then we'll wrap it up. So I got nothing else on YouTube So I'm good to go there at least for the meantime From Norman Fay. He's asking about that a whole lot really happening there Don't need to put it on our list Check back in on stocks. I kind of put on the side for the meantime like the OSAT trade The BSQR I guess proved itself a little bit there squeezing up the time. So that's on our main list PBTS Could still be in play How much more could it move up is the question? I mean three is gonna be a big level Looks like three 15 ish just from the chart looks to be pretty big So given the range this stock has had today up and then down and it's already been bouncing since 230 So, you know, hey, if you liked it as I went through during the meeting and caught it Nice trade you're doing better than some of us right now looking at it not in the trades But I don't know where's it gonna top off at that's the question. So I'll add it to her list. You know, we only have three stocks on there so far Doesn't hurt. All right, let me pull up the Binance chart really quickly Bitcoin that is and then we'll wrap it up That BSQR is pushing up nicely up towards 740 745 iceberg than 750 All right, so for Bitcoin, this is where we left off now This is what I'm gonna do like when I catch back up the folks on Mondays So this is practically where we left off on Bitcoin. So from the afternoon meeting back from Friday I ended up making a bit of a move up now I like to show a few charts here looking at the hourly the six hour chart Maybe I'll jump to but you know, otherwise the daily and then weekly but um and weekly if need be but you know Really we've just been in a channel the last like two weeks here, right up and down up and down roughly So it seems like it's making lower highs and with lower highs we would expect a Lower low. So, you know, that's always in play naturally and that is true I mean, even if you're not following, you know Bitcoin or familiar with it I mean those patterns are extremely You know true or at least, you know, you see him all the time on crypto as frequent as you would an equity A stock but it's more than just that though. So What I show on the hourly chart I have the 500 EMA in yellow or I in orange. Sorry So it's been having a real tough time trying to break over that 500 EMA again I mean most recently and or first try doing it here. I mean it popped it, but You know for as much as it dropped off from it just like any other, you know Horizontal level you'd expect that to hold as resistance at first So, you know seeing this pull back that was expected I called that out right away when we saw that at the time But the thing is is that we're seeing it begin to kind of peek its head again peek its head again What's gonna happen here coming up one of two things.  I'll look back at the daily chart in just a moment or Otherwise we continue to see it slump off drop back down and perhaps retest 30 30,000 So with that it's just to say you would expect ride and mara and cannon all those bitcoin mining affiliated stacks to go with it But just on the crypto side, let me just jump to the daily chart here quick And I'm gonna delete this line here really fast So again, we've been in this range really for less like two weeks let alone if not more So at this point if this ends up breaking the 500 EMA on the hourly chart I didn't even get the price on that but you know the orange line I was teasing just before on the last example or last chart I would expect this to make a further run up perhaps up towards the 200 EMA and yellow or the or the 100 and green So that could be up a up as much as around 41 42 We'll see what happens across this week It's not gonna be an overnight story unless of news comes out That is for of course always an Elon tweet away from this dropping off again Any sort of China restriction type news that comes out on Bitcoin. I mean hey, we saw it happen on DD from Friday afternoon, right? Literally China government ended up blocking that from being listed on app stores So, you know, they've of course been playing around with Bitcoin and you know coming out with news to block mining and banks from allowing Bitcoin is custody. So, you know, we'll see and anything like that can of course create that drop-off any sort of good news with companies I Incorporating Bitcoin as treasury for the company itself Then you're gonna expect to see the stock pop like it did, you know a few months ago in Elon did with Tesla I think that we're kind of overdue for that too So I'm gonna lean more towards the latter on that but alas here. We are now at least, you know We're gonna finish up the afternoon being folks. So and all the time. I was flat my gums on crypto I'll re ask if there's anything up down left right more than 5% you would want me to look at or check out Check back in one last time on that OS a teacher. It actually before even went into Bitcoin there. So Felix make a chat on an iceberg. So He was called calling out an order at 989. I Don't know if I got filled Maybe 990 probably iceberg It hasn't tested it yet. It seems like Felix if you happen to see a level it seems to be holding well under that Let alone 10 You know, I'm gonna keep this on the side for right now I'm not really as interested yet seems to be like coiling up. So, you know, if it ends up popping Be honest, I'm you'll probably be the first to call it out But you know if not you then I will be so one of us will catch it and we'll work together as a team and see if We'll be able to capitalize but if it's not this then otherwise I like the list to begin with the XCLA the BSQR the SR&E even if that dumps off that could be a good short And then the PBTS, you know Finally from there. All right. All right, so that about wraps up folks for all of us in social media I appreciate you sticking with me each and every afternoon if you're tuning with me tuning in tuning into me earlier in the morning I appreciate you sticking with me from there as well. Again, Faust will be back in the mornings I believe on Thursday. You might be back tomorrow. I just don't want to hold hold don't hold me to that But otherwise either way, I look forward to speaking with you folks at least tomorrow Make sure to like subscribe notify right on our YouTube channel that being youtube comm slash cyber trading you Otherwise if you've yet to join our live trading room If you're interested in listening to more of our commentary heading into this afternoon afternoon or going forward You can go right down to the link on the banner below just at the stream just to get yourself started All right for all of us there right now. Give me a quick couple of minutes. I'm going to copy and tweet the pics I'm gonna refold my water bottle. It's been a lot of talking like a Expecting it to be at least from this morning, but nonetheless, I'll be back on the mic folks right around 3 3 30 I think rich will jump on pretty shortly from now. I'll talk to you soon


Live from San Francisco, extracting the signal from the noise, it's theCUBE covering Oracle Open World 2015 from Studio C, brought to you by Cisco. Now your host, Stu Miniman. Welcome back to Oracle Open World 2015. I'm Stu Miniman with wikibond.com and this is theCUBE. We go out to all the shows, help extract the signal from the noise. Happy to have back on the program, a regular of our program, Eric Herzog, who's the Vice President of Product Marketing at IBM. Eric, welcome back to the program. Stu, thank you. We love being on theCUBE and love everything you guys do. Really appreciate having the opportunity to talk to you at Oracle World. All right, yeah. Eric, I mean, you've seen us at many shows. Matter of fact, some of my guys might be at an IBM show right now, I think, but we're going to talk about Oracle today. What's new in your world since the last time we caught up? Well, a lot of things. We've done a lot of product announcements starting at VMworld on both our VersaStack offering, which you do jointly with Cisco. That's why Apropos being here in the Cisco booth. And added two new configurations. One is an all flash configuration, ideal for Oracle workloads or for things like SAP or other apps that sit on top of an Oracle database, as well as a low-end product. We've introduced some new high-end products, our greater than 69, available at DS8000. We actually announced a new version two weeks ago and we've also got some other announcements coming next month. So been a very exciting time starting in August and running through the end of the year at IBM Storage. Great, so let's start with the VersaStack piece. We're familiar with the announcement when it came out. Covered some of the stuff at VMworld, but Cisco's got a lot of partnerships. How does IBM gain general awareness in this product line and is there any kind of momentum, customer stories you can share with us when it comes to VersaStack? Well, we've got a number of public references posted either on the Cisco YouTube site as well as our YouTube site. Everything running from traditional IT configurations to a number of cloud service providers, the VersaStack, these integrated solutions combining compute, hypervisor, networking, and most importantly, of course, storage, since I'm a storage guy, are very critical to the build-out of virtualized and cloud workloads. Whether that be a private cloud, a hybrid cloud, or again, a number of our public reference are actually public cloud providers that are using VersaStack to make their expansion, to make their growth, to be able to easily add new capacity and new compute to their clouds built around VersaStack and a number of public references built around that. So very strong in a multitude of workloads, cloud, virtualization, Oracle, SAP, all ideal workloads for VersaStack solutions. Yeah, so Eric, you mentioned cloud a couple of times here. Brian Gracely from Wikibon joked that, we should really call this Oracle Cloud World, from all the cloud discussions. Can you talk a little bit about how IBM Storage interacts with the cloud group? And I guess one of the things that I thought was really interesting, the clever, safe acquisition, which sounds like it's going to sit in the cloud group, but tightly tied, obviously, to the storage group also. Well, as you've probably announced on the clever, safe acquisition, we're going to be offering standalone object store, but also be integrating clever, safe into our software offerings as well. So from our perspective, we see that in today's environment, there are a number of different ways that people want to consume storage. One is traditional arrays. Obviously today, all flash arrays and hybrid arrays that are part flash, part hard drive. Second method is software defined storage, where you go all software. And the third way is cloud. So what we'll do is be able to offer clever, safe in both an array sort of configuration as a software defined configuration, which they already have today, and then adding software as well. So whichever method the company wants to consume their storage, cloud-based, software-defined storage-based, or traditional array-based, with clever, safe will have an object store in all three of those categories, and the customer can choose whichever is right for them. Yeah, so Eric, it was interesting. We were talking off camera. We've got the Dell booth next to us here. Just bought EMC for 67 billion. We've got the Sandus booth right behind us. Just got acquired for 19 billion. I mean, it's almost $100 billion of storage-related acquisitions in the last couple of weeks. I mean, I know you can't comment from an IBM standpoint, but you and I both worked at EMC. What's your thoughts on just, I mean, massive change in the storage industry? Well, I think there's no doubt that this is going to cause change throughout the industry in all segments. The OEM segment, what's the supply base going to do now as people continue to consolidate? Are people going to try to move up the stack, AKA the traditional OEM suppliers that supply people like IBM or our competitors? B, how does this impact cloud environments? Is Dell going to offer some sort of cloud configuration? As you know, they're not too strong in that area. How is that going to impact things? And what's going to happen with things like the channel? Dell is not known as a channel-friendly company. How are channel partners going to deal with and see about how are they going to do business and how are they going to continue to survive in a market that's going under incredible change in all areas, whether it be all the new technologies such as all flash and software defined. As you know, there's a raft of startups. We of course just recently announced our intention to acquire Cleversafe. We bought TMS, which now are flash systems. So we ourselves have been acquisitive with the startups. So a lot of change going through and these two acquisitions are going to add even more fuel to that fire. The good thing about storage is it's always changing and it's never boring.  And it's critically important to work on it. As I look at things, that hybrid story that we've been dancing around for a bunch of years, I feel like the stacks and what sits on either side of hybrid is kind of being a little bit more clear now. So you mentioned Dell actually. They made a big announcement with Microsoft that Microsoft inside their public cloud Azure is using the Dell hardware. So the Dell says, okay, you want Dell and eventually Dell EMC stuff, we're going to put it both places on the cloud. IBM, you've got SoftLayer, you've got obviously a large portfolio that you can put inside data centers. How do you look at that hybrid cloud story and where storage sits in kind of your environment, service providers, public cloud, what's IBM's role? So for us, cloud is a great opportunity for us. Whether that be a public cloud provider, we have a number of public cloud providers that use our storage, whether it be XIV. For example, Netflix is a public reference. So if you use Netflix, which is sort of a, we will video server as a service kind of public cloud environment that comes off IBM storage. We have a number of cloud service providers who use our technology, whether that be our all flash. We mentioned already earlier our versus stack configurations. So that's an ideal offering as well. The other thing we do is make sure that our storage can easily tier to the cloud. So for example, as we talked about last time on theCUBE at VMworld, we announced our new version of Spectrum Protect, and Spectrum Protect can see a cloud as just a target device. So Spectrum Protect can back up to a hard drive, can back up to a tape drive, can back up to an optical drive, or it now can back up to the cloud and treats cloud as if it was just another tape drive or hard drive, but it's really a cloud. So at IBM, hybrid clouds, private cloud and public clouds are huge opportunities. And the key thing is when you're in the cloud, you have all sorts of other vicissitudes. What's my network connect? How's the traffic working? The last thing you want is that storage not to be a highly available, highly reliable, and also provide high performance because you're subject to the traffic on the network. So you want that storage to be rock solid, and IBM is proving with all of our hybrid cloud customers and our public cloud customers that we can deliver on that promise. You know, when we talk to practitioners as our community, what's really clear on cloud is it's not one size fits all. Most customers have, you know, they're using public cloud, they've got their own data centers, they're using SaaS. And one of the big challenges they have is how do I manage all this? How do I, you know, orchestrate for how the piece is? You know, what has APIs, what plugs in?  Maybe if you could talk, you know, IBM's view on kind of that management orchestration layer, you know, what you tie in with Cisco and other partners and how much IBM does themselves. Sure, well, IBM is very active with all our partner base. In fact, not only are we here working with Cisco with VersaStack, but if you go over to the VMware booth, they're demoing our flash systems, running a virtualized workload of Oracle. So whether that be Cisco, VMware, Microsoft, you know, Oracle of themselves is a huge partner of IBM. So we work with all of the vendors. From our perspective, it's very critical to support. Again, as we already talked about, we have on-premise offerings that are array-based, if you will, physical entity. Software-defined storage, which is also on-premise, but you can use anybody's hardware. And one of the good things about our software-defined storage and two of the industry analysts have named us number one in software-defined storage now for two years in a row, is it works with anybody's hardware. Not just IBM's, but all of our competitors, or just generic white box storage, and we can manage that. And then lastly, we have, of course, cloud offerings. Spectrum Accelerate is available as a cloud offering. Spectrum Scale is available as a cloud offering. And Spectrum Control, which is our orchestration layer, our control layer, which, by the way, works not only on IBM storage, but on all our competitors' storage, is also available through software. So however you choose to consume, and I do think one of the key things you hit on, is most big companies don't consume one way. They're not all in the cloud, they're not all on-premise. If they have a cloud, it's probably part hybrid, but off in the corner over there, the marketing guys, or one of the engineering teams, is using a public cloud provider, whether IT knows it or not. So from an IBM perspective, the key thing is being able to span the gamut from on-premise to software-only on-premise to cloud integration, and make sure that our control plane, our Spectrum Control, which is available as a cloud, or on-premise as a software defined, not only works with IBM's gear, but all our competitors' gear as well. And that's how we see the world, and it's mostly a heterogeneous world out there as well, Stu. So you've got to make sure your gear works with everyone's gear. Absolutely, Eric. I would definitely give IBM kudos when it comes to working in a heterogeneous environment. I've hard to find another company that does a better job at that. The other thing that's been thrown around this week is, you know, how open is a company?  But I mean, IBM's got a very strong heritage when it comes to open-source. I'm curious how that plays into what goes on with storage. I mean, I know IBM has a strong presence out at the OpenStack Tokyo event this week. The cloud group's doing a lot there, but when it comes to open-source, what you're looking at. Well, when you look at IBM across the board, right, from the things we're doing in the cloud team, what we've done with now Linux and VMware on our system Z, what we have with our power open initiative, right, which is an opening up, not only running power on Linux, but literally offering our power chips to essentially people who would be competitors to create their own servers that can compete with our own power servers and let them use the chip in our infrastructure. On the storage side, we do things again, like tier to the cloud, like our Spectrum Protects, so again, supporting that. We're very active. Our Spectrum Scale product is not only scale out NAS, but supports the OpenStandard Swift for ObjectStore. So we've got support for OpenStandards. We have Hadoop connectors, for example, on a number of the products, again, trying to be very open and supporting all the new issues. And as you know, IBM as a global company is one of the largest contributor to the entire Linux space across all the divisions. Probably the biggest contributor to Linux is IBM Corporation and the software groups and the systems groups. So we have a strong tension to support OpenStandards and OpenAPIs across all of our products from storage to the rest of the systems group with Z and power and into the software groups and into, of course, our software infrastructure. Yeah, so Eric, you talked a little bit about analytics and we know IBM has a strong strength there. They've got a huge portfolio, as a matter of fact, that's the IBM Insight Show that's going on right now. Another item that's been talking here that has huge impact on kind of the infrastructure and architectures is IoT. So, you know, the IBM storage portfolio, how's that fitting to IoT? How's that getting ready for some of these, you know, potentially disruptive new architectures? So one of the key things is IBM Flash Systems and our spectrum scale products are heavily used in big data analytic environments. So if someone's going to do IoT and have all thousands and thousands, tens of thousands, hundreds of thousands of data points that they need to analyze to control their network better, to control their security environment better, the best thing to do is run that on IBM Flash Systems. Low latency, high IOPS, incredible performance. We're across our spectrum scale infrastructure which can scale to exabytes. As we've already publicly talked about, we have one of the global enterprise that has an exabyte with spectrum scale in production. Can you think of any other storage company that's got a customer with an exabyte in production? Probably not. And they're running all kinds of workloads including heavy data analytics workloads. So, you know, big data, IoT, just generic analytics, even the older form of analytics, the old data warehouses, the Oracle data warehouse, SAP data warehouse, SAS data warehouse, we support all of it. We've been with analytics before it was called big data and we're in analytics today and with Flash Systems and spectrum scale, we offer ideal solutions for big data, for analytics workloads and for the IoT infrastructure. Yeah, so, Eric, the storage industry so often doesn't make big moves fast. Talk about customers take a while to adopt. Talk about market share. It's highly fragmented. See, Dell by an EMC, huge shift in market. The other piece that I saw, I mean, AWS, can be an eight billion dollar company this year, probably at least two billion in that that's storage by the end of this year.  Why are customers coming to IBM for storage and how do you position yourself? Well, there's a couple of things. First of all, IBM is betting on those waves of change. We are the number one provider of all flash array capacity and all flash array units and number two in revenue. By the way, it's interesting we sell more units. I guess we charge a fair price for our all flash arrays. Some of the other guys may overcharge for theirs. We're also number one in software defined storage, the use of our software with anyone's hardware, whether it be our own or someone else's. Integrated infrastructure, which we do of course with Cisco versus DAC, but we also have some versions with our own power technology, so integrated infrastructure. So when you look at how storage is consumed and then obviously all of those solutions, whether it be integrated infrastructure, software defined and all flash arrays are heavily used by cloud providers and hybrid cloud configurations. So the key thing and why people come to IBM is we're riding the wave. We're not fighting wave. We're riding the wave of all flash. We're riding the wave of integrated infrastructure with both Cisco versus DAC and pure power. We're riding the wave of software defined, being number one in software defined storage according to the industry and it's now for the last two years in a row. So they come to IBM because we're on the cusp. We're not conservative. We keep moving forward with technology. We invented the hard drive. We're still one of the biggest guys in the tape space. We were one of the first storage subsystems in the market space that was not embedded in a mainframe back in the day, back in the 90s. So we're always trying to widen the wave and that's why people come to IBM because we have a reputation, not only for rock-solid company and with highly-proppled rock-solid support and service but people view us as you know. IBM does more patents every year than the entire rest of high-tech industry can find. That's innovation. All right. I think we'll close it on that, Eric. Eric Herzog with IBM. We really appreciate you coming back, sharing with us all the new things that are going on in the IBM Storage Portfolio. We'll be right back with lots more coverage here from Oracle Open World 2015. This is theCUBE. Thanks for watching.


Namaste. My name is Malvika Ayur. I was 13 years old when I lost both my hands in a bomb blast accident in Bikaner Rajasthan. Along with my hands, both my legs were severely injured with multiple fractures and loss of sensation. Luckily, I survived that night and I was able to understand the big picture as to what lies in front of me. The most important thing that helped me overcome my disability is, I believe, education. I gave my 10th grade even though I was a private candidate. I studied in a coaching centre and I wrote my exams with the help of a writer. I got 97% in my final exams with just three months of preparation and I think there's never been a time where I looked back as to what happened or I ever felt bad or I ever pitied myself for what has happened to me or the loss that happened to me. The day I got my state rank score, I got the opportunity to meet Dr. Kalam who was the president and that moment I promised to myself that I'm never going to look back in life. From being a young girl with disability to receiving the Nari Shakti Kuraskar from the president of India, my journey was challenging at every step. As a disability rights activist, I'm trying to create awareness about the discriminatory attitude and the stigma that surrounds people with disability. Also, the inaccessible public spaces. I am a product of inclusion. I have been included in my school, in my workplace and by my society and I truly believe that only if a person is treated as an equal if the person is included in the society and only then can the person achieve whatever they want in this life. I dream of an inclusive India where we are truly accepting of each other despite our differences. Acceptance is truly the greatest reward that we can give ourselves. I want to thank our Prime Minister, Shri Narendra Modiji for creating this wonderful campaign. She inspires us to share success stories of women and inspire each one of us. Happy women's day to each and every extraordinary woman out there.  Keep inspiring today and every day. Be unstoppable.


Article 1. India is indestructible union of destructible states. It means the formation of India is not the result of the agreement between the states like the USA and the Indian states do not hype autonomy to secede from the union of India.