 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. I'm saying something stupid here. 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 here f is my f f is my rational function Okay, so Let's prove the theorem. 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 you have dl at step l At step l plus one you have d times dl minus d. So you have to remove d components and you multiply by d. So that's Prime edge number l plus one And so on so and so on means that you get d to the on l plus k d to the k times d l So so at step l you have dl good prime edges. 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? I guess so I guess In any case when you're normalized by d to the power l plus k what you get is d to the l plus k times one minus something which is small Depending on l Okay, so if you choose l large enough at the beginning, this is smaller than epsilon and then you are done, 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