 It's a lot to the organizers for invitation. So what I would like to do in these lectures is to give you an overview of a number of classical results in holomorphic dynamics, in a number of classical results in one-dimensional holomorphic dynamics, and try to understand what they become in higher dimensions, okay? So I think it's one of the ideas of this conference of how to go from one-dimensions to higher dimension for groups, and I will try to keep this idea in mind of stating classical things in one dimension and see how they evolved in higher dimension. So the one-dimensional part will be very classical, so perhaps many of you have already heard about this with great details, so I apologize for that. I will give a number of proofs, at least for the one-dimensional case. So the overall plan would be something like that. So in the first lecture, so if everything is fine in the first lecture, I want to talk about the Fatou-Juliet dichotomy of dynamical space, so again in one and several dimensions. In lecture two, I want to talk about equidistribution aspects. So I want to talk about equidistribution, so there will be a little bit of ergodic theory, and lecture three should be about parameter spaces and bifurcations. So hopefully I'll be able to maintain the schedule, it's not clear whether this will not contaminate the whole three lectures, we'll see. So again I will try to give you some proofs in one dimension and some statements in higher dimension. Okay, so let's start with lecture one. So lecture one is part R and part A and part B and part A is dimension one. So I want to iterate an algebraic map of one-dimensional algebraic variety. So I take a certain f from x to x, where everything is algebraic and x is one-dimensional. And what I want to do with f is iterate it, so consider forward iterates of this. So I view this as a dynamical system and I want to predict the future of points. I have a point I would like to know what it will become in the future. So the first thing is that if you want to iterate an algebraic map on an algebraic rim and surface, you don't have many, many choices. So first case, so let's start with the easiest case, first case that x... So when you have an algebraic rim and surface, you can always compactify it by adding a few points. Okay? So you can always assume that x is compact. You can always assume that it's a compact rim and surface, okay? And f will extend homomorphically to the compactification. This will be different in higher dimension. So first case is when the genus of x is larger than 2, and in that case, just by looking at the Euler characteristic, you see that f must be an automorphism, and in that case, the automorphism group is finite. So in that case, f must be an automorphism, and this is a finite group. So in that case, of course, for dynamical systems, it's not very interesting. We'll throw that away. Second case, genus 0, genus 1, when x is an elliptic curve, in that case, your map must descend from an affine map on C, and f descends from an affine map. In that case, the dynamics can be complicated, but still it's completely explicit. You can compute the iterates. So you will consider that it's not interesting from the dynamical point of view. So let's say it's easy. Even if, again, the dynamics can be chaotic. And the last case, which will be the one that we are interested in, is the case of rational mappings on the Riemann sphere. So that's the genus 0 case. So genus 0, okay? So in that case, x is the Riemann sphere, or equivalently the one-dimensional complex productive space. So we'll switch between these two ideas of a sphere versus productive space. I mean, I think most people here know about the relationship. And f is a rational map. So the degree of f is the maximum of the degree of the components. Again, the situation is not very interesting when the degree is 1. So I always assume that this is larger than 1. Okay? So this is my dynamical setting. I want to iterate a rational map on the Riemann sphere. So that's the setting for all this one-dimensional part of the lecture. Okay, so that was section 1, setting. So section 2, a number of basic concepts. So one important basic concept here is that of a normal family. So a normal family, polymorphic mappings from a certain domain. So omega is a domain in an open set in the Riemann sphere. So the Riemann sphere is just a relatively compact family of polymorphic mappings. So now this normal stuff is just all terminology. It just means that it is relatively compact in the space of polymorphic mappings. Okay? And it's also very classical. That's, it's the same as being a quick continuous, a quick continuous for the spherical metric. Okay? So there is one important and classical theorem here, which is Montel theorem. So that's one basic tool, Montel's theorem. Any family of polymorphic mappings avoiding three points is normal. So any polymorphic family, no, let me stay like that. Any family of polymorphic mappings from some set omega to the sphere minus three points is normal. Okay? So of course this must be a distinct point, right? And the proof in two lines, the proof is that the universal cover of this is the unit disk. The proof is that the universal cover. So consider the universal cover. So this is covered by the unit disk. And if you take any homomorphic map, if you take any homomorphic map from omega to C minus three points, C hat minus three points, you can lift it to the universal cover. And now a bounded family of homomorphic mapping is, okay, so perhaps you should assume that omega here is not the, is contained in C, right? It's a proper domain. So if it's proper, it avoids one point. And if it avoids one point, you can always assume that this point is infinity. Okay? So now it's a family of homomorphic mappings in a classic, a classical sense from omega to the unit disk. And then it's just, it's equicontinuous. And if you take the homomorphic mapping from omega to the unit disk, it is equicontinuous. Look, equicontinuous I mean on compact subsets, okay? So it's locally equicontinuous. Just because of the Cauchy estimates, okay? And that's the proof. So that's one basic tool. The other important tool in one dimension is the Poincare metric. I think it has been seen a number of times in this conference. So I don't want to say too much about it. So what do we want to say? So a Riemann surface is hyperbolic if its universal cover is the disk. And essentially all Riemann surfaces are hyperbolic except, so X Riemann surface is hyperbolic unless it belongs to a small number of examples. It is, so there's a sphere, a plane, puncture plane, C mod Z. So if X is not one of these examples, then it's a hyperbolic Riemann surface, okay? And then you get a Poincare metric. You get a Poincare metric on X just by pushing the metric from the universal cover. And the important property here is the Schwarz lemma. Schwarz lemma says that a homomorphic mapping between hyperbolic Riemann surface is always weakly contracting. So let's see a homomorphic mapping between a hyperbolic Riemann surface, between hyperbolic Riemann surfaces. Then the distance, it is a weak contraction. So the distance with respect to the matrix of X prime between images is less than the distance at the source. And also you know, even more than that, you know the equality case. And actually it is a strict contraction. So it is a strict contraction unless, unless F is a cover. So of course, you imagine that in dynamical situations it is very useful information to have contractions. Okay. So now let's start to do dynamics. So what's the Fatou-Juliet dichotomy? I think everybody in this room knows what it is. Okay, so let's define the Fatou set. So again, I take F, a rational mapping of degree at least two from the sphere to itself. And I define the Fatou set. So the Fatou set is the local equicontinuity set. It is the set of point Z in the sphere such that there exists a neighborhood of Z such that the family of iterates restricted to this neighborhood is a normal family. So you see that by definition it's an open set and it is invariant under F and also invariant under F inverse. So it is open and totally invariant, meaning that F of the Fatou set is also F inverse of the Fatou set is the Fatou set. And the Julia set, it's just, it's just a complement of the Fatou set. So it is a set of points where the family is not locally equicontinuous. Now I realize that I should have prepared a number of pictures, but I didn't. So you just have to imagine pictures of Julia sets. I think there may be some in the back of the room now. In every mathematical room there should be a picture of a Julia set, but for some reason here there's no. Okay, so let me give you a few examples. No, let me not give you a few examples. So you know that you get these fractal pictures, and the, well, most famous factor is the circle corresponding to F being Z to the power of D. So, example, F of Z is Z square. So what happens to points under iteration of Z square? Well, if the modulus is smaller than one, you go to zero. So it's equicontinuous. If your modulus is bigger than one, you go to infinity, which is also equicontinuous for the spherical metric. And if the modulus is equal to one, you don't know where you want to go. You are separate, well, you are in between the two. So certainly in the neighborhood, the family cannot be normal, right? So the Julia set is the unit circle, okay? And that will be our only example of Julia set. Okay, if yesterday, this morning, we heard the name quasi-circle, okay? So if you perturb a little bit this, and actually, epsilon can be as large as one-four. Then the Julia set is homeomorphic to a circle, is homeomorphic to a circle. So you see a deformation, homeomorphic to the circle. So not only it's homeomorphic, but it deforms, it deforms continuously. So you start with a circle, you start with a circle, and when you start to move epsilon, so I should have used this one, all right? You see this circle moving to a picture which is not smooth, which is a quasi-circle. So this could be a quasi-circle. So a quasi-circle has no inward pointing cusps. So when you draw one, you should make cusps point on the outside. Anyway, so when you deform, when you take small epsilon, you see a picture like that, okay? And when you go, when epsilon is larger than one, real and larger than one-four, you have a counter set. And you get plenty of pictures. Let's proceed without the pictures. And so the Julia set is closed, and also it is totally invariant. So the first thing I will do is to characterize, well, to talk about this closed total invariant set. So let's say that E is an exceptional set, is exceptional if it is finite. So it is a subset of the sphere. If it is finite and total invariant. There's a proposition here, easy proposition. There is a maximal, a maximal exceptional set. Let's call it E naught. And the cardinality of this set is at most two. So what it says that you can have, it is, let's say also that a point is exceptional. The point Z in the sphere is exceptional if it belongs to an exceptional set. It means that equivalently, it has finitely many prime edges. You take all possible prime edges, and you just get a finite set. And what the proposition says is that there are at most two exceptional points, OK? That's the worst you can do. And the proof is? Yes, OK, I don't want to, I don't want, but you don't need to assume that. I mean it's just, yes, yes. Perhaps you need to add it in the definition. I don't think so. Well, the important, perhaps I should have just talked about exceptional point. An exceptional point is a point which has finitely many prime edges. That's the important concept, OK? And the proposition is that there are at most two exceptional points, OK? OK, the proof is easy. The proof is just the remand of its formula. So if you have an exceptional set, you get a mapping from C from the remand sphere minus the exceptional set from itself to itself. So F induces a mapping like that. You apply the remand of its formula. So the Euler characteristic of sphere minus E d times is equal to d times itself minus a certain ramification term which is non-negative, right? So this cannot be a negative number, otherwise it would go to infinity. So it means that this number has to be non-negative. It has to be non-negative. But this number is just 2 minus the cardinality of E. Just 2 minus cardinality of E, OK? So you get the result. So this has consequences. Which consequence? So another proposition. So let me just, OK, just a classification, perhaps a small classification of the possibilities here. It happens in generic cases, the exceptional set is empty. OK, you have no exceptional point. Now if the exceptional set is just one point, you can always declare that this point is infinity. And what is a rational map with infinity total invariant? It's a rational map with no pole. It's a polynomial. So E, one exceptional point corresponds to polynomials. And two exceptional points, you only have the examples of z to the d and z to the minus to conjugacy, OK? So a generic case, no exceptional point. One exceptional point means polynomial. And one last, and there's one last case. OK, proposition. If z is any point which is not exceptional, then if you take the union of prime edges of z and take the closure, then it must contain the Julia set. And the proof is just Montel theorem. The proof is that if I take z not p in the Julia set and a neighborhood of p, then the union of forward images of this neighborhood, it's a certain open subset which misses at most two points. And it is invariant. So it is open. It is forward invariant. And it misses at most two points. So it means that its complement must be an exceptional set. So the complement of this is exceptional. So z is not here. So it means that you take prime edges of z, you must enter n. So when you take prime edges of z, you must get close to p. OK, so that's the end of the proof. And the corollary of that is that if now... So you know, I should have made a remark that the Julia set is never empty. It's just because the degree is at least two. So you know that some complexity should arise under iteration. So the Julia set is not empty. And the corollary of that is that the Julia set is actually infinite and also perfect. And the reason is that if... The proof of that is that if z is a point in the Julia set and we know that it's not empty, then the Julia set is just the union, the closure of union of prime edges of z. And it's... An exceptional point is always in the Fatou set. As I said, one possibility is infinity for a polynomial and the other possibility is zero and infinity for zd. So you are in the Fatou set. So if you start with a point in the Julia set, you know that it's not exceptional. So when you take prime edges, you have this property and then you get equality here. So it says that the Julia set is perfect and infinite, right? So we get one first property of the Julia set. So there are plenty of other... So we saw this property for limit sets this morning, right? Another property which I won't prove is a theorem by Fatou. And Julia, I would say Fatou and Julia. No, excuse me, I will say it later. Sorry, sorry, sorry. First I want to talk about periodic points. So that's section three, four, periodic points. So finding the periodic point of period n for f is just solving an equation of degree d to the n, right? So you know that f has d to the n plus one, periodic points of period n, counting multiplicities of period dividing n. This is just because you solve the equation fn of z equals z. It's just an algebraic equation. So you notice you have plenty of periodic points, right? So now if z naught is periodic, if z naught is periodic of exact... So now when a point is periodic of period n, it's also periodic of period 2n, 4n and so on. So there's a minimal one and I will say that it's the exact period. So if z naught is periodic of exact period n, then you define, you can look at the derivative of fn at this point. This is a number. So you have to consider the spherical metric here, but since you consider it at the beginning and at the end, you just can consider it as a derivative, right? So this number is called the multiplier of z naught, okay? And just by applying the Taylor formula, you know that the multiplier determines the action of f in the small neighborhood, fn in the small neighborhood of z naught. So it determines the... At the first order, it's the main... Excuse me, I missed the word. It's the main character for the local dynamics of fn at this point. So there are a few cases. So let's assume... So you can always assume that n equals 1 just by replacing f by fn. So first case, the multiplier is less than 1 in modulus. Then the point is attracting. It attracts a neighborhood. Z naught is... Let me keep the... Let me keep the... Z naught is attracting. There is not much to say. You have a point z naught here. It has a certain basin of attraction, and all points nearby go to this one. And then it belongs to the fatal set. Then if the multiplier is bigger than 1, and the point is a repelling, so in that case, you cannot describe the whole dynamics of a point in the neighborhood because it will go far away, and then it does something. In this case, you know that the sequence of the derivatives at z naught goes to infinity, so you must be in the Julia set, and you cannot say much more than that. Third case, more interesting one. Multiplier is 1. Then you say it's neutral. Then you can have several subcases. So let me describe a few of them. First there's a lemma. So I would like to know whether a neutral point belongs to the Julia set or the fatal set. So there's a lemma here. So a neutral fixed point, neutral periodic point, fixed point, so again you can always assume it's fixed just by replacing f by fn, belongs to the fatal set if and only if it is linearizable. So linearizable means that you have z naught here, which is excuse me, and you want to know if it is conjugate to the action of the linear part neighborhood of the origin where lambda is just the multiplier of... So I'm going too fast here. You know that the multiplier determines the local dynamics of z naught. So you can always imagine that z naught is zero, of course. And you want to know if it is conjugate or not to the multiplication by lambda. And what I say is that this is true if and only if the periodic point is in the fatal set. So one implication is clear. So the linearizable implies fatal is clear because if the map is conjugate to a multiplication by a complex number from this one you have a quick continuity. For the forward implication, there's a little trick and the trick is this one. You consider hn it's a sequence of mappings and this is this one. 1 over n sum of... So I assume that z naught is equal to zero. So that's a local statement. It has nothing to do with rational mappings. It's a statement for local mappings defined neighborhood of the origin. So I assume that z naught is zero and f is just a germ from c zero to itself with a multiplier of lambda. And I will consider this sequence of mappings. Not good here. So we'll define a sequence of homomorphic functions like that. So it's a cesarean sum of this sequence. So if I name this gk, you remark that gk plus 1 is just lambda gk plus 1 is just gk composed with f. And also you can remark that all these mappings have the relative 1 at the origin. So hn prime at the origin is 1. Now if z naught belongs to the factor set it means that fn is equicontinuous. So it has bounded the relative in the neighborhood. So lambda to the minus k, lambda is a modulus 1, right? So this is gk is equicontinuous. So hn is also a normal family. So you can extract hn is a normal sequence. And now by construction hn plus 1 is the component composed with f minus lambda hn goes to 0 by construction. This is just because when you compose with f it just corresponds to a shift in this sum and you have this 1 over n. It's the same trick as you know constructing a fixed point by taking a cesarean. When you have this when you want to prove this Markov's calculating theorem that when you have a compact convex set it has a fixed point. You just take cesarean and if you take any cluster value of the cesarean it must be a fixed point because the terms just cancel. And here it's exactly the same. When you compute this you get that this is going to 0. So if you take any cluster value h of this sequence satisfies h composed with f is lambda h. And also it has the relative 1 So it is a linearizing map. It's a local diffeomorphism which conjugates f to its derivative. So that's the proof of my lemma. Proving that a neutral point belongs to the Julia set is exactly the same as proving that it's linearizable. So a corollary of that is that so in the neutral case if you have a neutral point you have two cases so you write the multiplier as the exponential of 2 pi theta. So either theta is rational or it's irrational. So if theta is rational you see that it's a parabolic point and if it's not I don't know you say that it's just an irrational neutral point. In that case you see that the linear map has finite order. So it cannot happen for f because for my rational mapping my rational mapping has a degree bigger than 1 so it's not a finite order so a corollary of that is that a parabolic point it is not it cannot be linearizable in my case and actually it belongs to the Julia set. Actually you can say a little bit more about the dynamics near a parabolic point so the typical example is like that so the typical example is f of z is z plus z square so the derivative at the origin is 1 we just have one perturbative term and the dynamics near the origin so on the real axis you know the dynamics on the real axis it is z plus z square something like that it is attracting on one side and repelling on the other side if you start with a point here you do something like that it will go to the origin and on the other side you go to infinity in the complex number it's the same so when you have this so that's 0 0 is attracting on a certain cone on the left on the side so you have an attracting base in here and you have a repelling base in here and actually you can even say a little bit more excuse me the picture is like that you have a certain cardioid shaped picture where the dynamics is attracting like that and it is repelling on the other side so my row here is going to the fixed point so you have a certain base in an attraction and 0 is on the boundary of the base and it's repelling on the other side so certainly this is a point in the Juvia set because you have no neighbourhood with the dynamic system in the irrational case the situation is more complicated so you certainly know that the problem of linearizing holomorphic deformorphism is very complicated so let me just say a few names here so Zigo so in the forties proved that for certain values of theta I don't want to be too precise here certain values of theta every holomorphic mapping with multipliers like that is linearizable F is linearizable and Kramer in the forties I think proved that for certain values of theta F is not linearizable so there are certain values of theta which ensure that F is never linearizable unless it is already linear right? of course you can start with a linear map so actually excuse me so what Kramer proved is that for theta this mapping is not linearizable okay? and actually for quadratic polynomials so we know exactly that's a theorem of Yoko's famous theorem of Yoko's that we know exactly the set of parameters such that the mapping is linearizable there is an open question so for Yoko's we have a complete answer to this problem for quadratic polynomials and also and this is based on the work of Bruno and so there's a sufficient so the situation right now is that there are sufficient conditions for linearizability which is called the Bruno condition it is known that the Bruno condition is optimal for quadratic polynomials but it is not known for in any other setting okay? so that's an open question interesting open question for rational mapping on P1 the question of linearizability depends only on the angle okay? so it's not known if this property of being linearizable or not depends only on theta it is known for quadratic polynomials but not for general rational mapping so that's an important open question in the field okay? I'm going to slow here one theorem by Fatou and Julia which was proved in simultaneously basically the Julia set is the closure of the set of periodic points of repelling points so you look at the set of repelling periodic points and take the closure you get the Julia set so again this is a property which looks like the corresponding one for client groups in one dimension another theorem due to another theorem of a lemma or whatever is that every attracting point every attracting point attracts a critical point and the proof is two lines proof look at the basin so let's look at the the immediate basin of A so you have an attracting point A it has a certain basin so the basin has many components but you look at the component containing A which is called the immediate basin and now F so the proof excuse me okay yes just F is a self-map from the immediate basin to itself and assume that F is a covering so assume that there is no so here there's an implication I'm not completely sure of okay assume there's no critical point then F is a covering so it's true but it requires a few lines of justification so if it's a covering it's an isometry for the Poincaré metric and then you contradict the fact that A is attracting okay so it's not a covering so it must have a critical point okay so it is not a covering so you just conclude by the short slimmer finite so a corollary of that the number of attracting points periodic points is finite just because there's a finite number of critical points okay there's a slightly more complicated theorem that the number of non-repelling points is also finite so actually when you look at periodic points for rational mappings almost all of them are repelling actually the bound is 2d-2 so this bound is due to Shishikura so it's optimal and there's a list of previous bound and I think the first one was due to Fatou it was 6d-6 and a proof looks like that one proof looks like that you take your rational mapping for proof that's not you can make it work in the show take f a rational map okay so you know that it has finite many attracting cycles perhaps it has a certain number of neutral ones now if you perturb f in the space of rational map what you expect is that half of the neutral points will go inside the unit circle and half of them will go outside the unit circle so you know that the number of f is less than 2d-2 number of critical points and the number by perturbation so for nearby f you get that the number of attracting points of f plus half of the neutral points is less than 2d-2 okay so again if you perturb for generic perturbation what you should expect and this is true after some work half of the neutral point becomes repelling and half of them become attracting this is a matter of saying that there are enough parameters and also that the multiplier map so if you want to prove something like that you need to show that the multiplier map is a submersion because the space of rational map is big it's 2d-2 and also if you look at the space okay you write that you need some work at least that's an idea of proof I don't remember if you can make this work or not I don't remember but in any case the idea is that half of them will become neutral and half of them will become repelling I totally agree that you can imagine that some bad situation happens at least from this idea from a graphical analysis from this you get the number of attracting plus neutral point is less than 4d-4 so you still need more work to get this 2d-2 is optimal 2d-2 is optimal and it's done by quasi-conformal techniques okay that's it for periodic points so I want to talk about the fatu set now this item fatu components I should have made a remark I forgot one property of the Julia set so let me add it somewhere so one property of the Julia set is that the interior of the Julia set is empty unless it's the whole sphere okay again this is just mental theorem if not if you take a component of the interior it the iterates must be contained in the interior of the Julia set and must avoid at most two points so it must be written so it's just a montage here so now remember the basic problem of dynamical systems is describe the behavior of orbits or at least describe the behavior of most orbits so since the Julia set is empty interior in first approximation you can try to describe the orbits in the fatu set if you're able to describe the orbits in the fatu set you've done almost all points in topological sense so a remark here so that's the theorem by Buff and Sherita Buff and Sherita it can happen that the the big measure is positive actually in the yes it may happen that the measure of the Julia set is positive but j is not the whole sphere okay so it means that if you want to describe the behavior of almost all points in the Lebesgue sense you need a little bit more but at least in a topological sense you can restrict the fatu components and the good news is that you can describe everything okay so what are the so if you if you want to study the dynamics in the fatu set so the dynamics is locally it's a normal family so in a sense the dynamics is locally constant in fatu components so it's enough to describe the dynamics in the whole fatu components so a fatu component it can be periodic so you have a certain iterate which maps the component to itself let me remove this center here it can be a pre periodic the picture is your component which is mapped after a few iterates to something which becomes periodic and the last possibility is that it is wandering so these are the three possible cases this is so in a few minutes I will say that wandering components do not exist for rational mappings but it is easy to cook up examples of transcendental mappings wandering fatu components so example is something like that if you look at this mapping f of z is z minus the sine of z plus 2 pi so that's an entire mapping from c to c and which satisfies f of z plus 2 pi is equal to f of z so it induces a mapping from c mod 2 pi z to itself so now if you look at 0 so 0 is mapped to 2 pi so on c mod 2 pi z 0 is a fixed point and if you look at the multiplier z minus sine z it's a super attracting fixed point so in the quotient so the quotient is just a cylinder c mod 2 pi z you have 0 here and it has a certain attracting basin now if you look in c what happens you have this attracting basin which comes from the cylinder and you have the same around every multiple of 2 pi but now since I added 2 pi in the formula what is the point here it will be mapped to the next copy of the basin and so good question good question good question I guess this is because you can only have one attracting point because this point is fixed no because this point is mapped to the next one and what and what and yes I agree with that but it's not possible because what the image of 0 is 2 pi so no the argument is not this one yes yes and the same basin okay okay okay okay thanks for the argument I forgot this one okay so the argument is that the picture must be like that because if you look at this mapping you get the same picture in the quotient okay so and in that case each of these points is a fixed point so they must correspond to distinct basins okay and now if you add the 2 pi you see the same picture but now now you have this point okay thanks for the argument so in that case you get a wandering for two components okay and associated to a transcendental map and the theorem is that you can completely describe all possible types of for two components so theorem is like that so it's due to a number of people that the last step and most important one is Solivan something like 95 or 84 85 or 84 or something every so there are no wandering for two components so let f be rational so the first thing is that there are no wandering for two components so every component is pre-periodic so if you want to describe all of them it is enough to describe the periodic ones and every periodic component belongs to fixed list of types periodic factor component is of the following form usually it's written as a five item five item theorem so item one and two are basically the same that's a super attracting or an attracting basin so it means that it's the basin of attraction of some attracting point so we have type three so omega so how do we want to so there exists a certain so let omega be a fixed for two components so another possibility is that there exists a certain point z not at the boundary of omega whose multiplier is one which is fixed with multiplier one and omega is the basin of attraction of this parabolic point and omega and for every z in omega the iterates of z converge to z not so we have a parabolic point and omega is just the attracting basin of this parabolic point and the multiplier must be equal to one if the component is fixed a fourth type so that's the parabolic case and then you have two more types which are called rotation domain omega is bihomophic to a disc so omega associated with dynamics is bihomophic to a disc with irrational rotation right so that's called the Ziegle disc so a picture is like that you have a certain disc with fractal boundary inside you have a linearizable fixed point and the dynamics around is that of a rotation and type five omega f is bihomophic to an annulus with an irrational rotation so same picture so these are called rotation domains ok so this annulus cannot be a punctured disc otherwise you would get an isolated point in the Julia set so it must be a natural annulus so have an annulus like that and it is foliated by smooth circles and dynamics is a rotation on them ok so let me try to prove a theorem ok I will not finish today but let me prove it let me at least give ideas of proof so proof well so I will not give I will not explain the proof of the non wandering factor component so the proof of at least the idea of the proof of the this non wandering theorem is something like that the argument is like if you had a wandering factor component then you would be able to construct by quasi conformal deformation an arbitrary large space of deformations of your rational map ok but now the space of rational maps of degree d is finite dimensional and then you get a contradiction ok perhaps I should add a complement here I should have a complement to this theorem not trivial and a complement there are only finitely many cycles of periodic components ok so we already know that there are only finitely many non-repelling points so it is said that you have finitely many attracting basins, parabolic basins and also z-hole disks but if you want finiteness for the number of hermene... so this is called a hermene ring so the finiteness number of hermene rings is more subtle and you need again some quasi conformal ideas to prove it and so you see that this complement so this morning we heard about the alphos-finiteness theorem ok so it is the exact analog of the alphos-finiteness theorem and actually it is the clue to the proof so Sullivan saw this as the analog of the alphos-finiteness theorem and used the tools to get the theorem here basically the same ideas ok so I don't want to say more about that now let me take so let omega be a fixed fatu component fixed fatu component so there are two cases there are two cases the recurrent case and the transient case so what transient means that for every z in omega the iterates of z go to the boundary point escape to infinity in omega and the recurrent case is just the contraposite there exists some z in omega such that z has infinitely many iterates in a compacted in a compacted so you see that in the transient case all these cases are recurrent except the parabolic one so what we something that we have to prove is that every transient fatu component is a parabolic basin so let's prove that there's one statement so there's a fact statement one transient fatu component invariant fatu component fatu component is a parabolic basin so first argument in the proof so you know that for every z in omega if we iterate go to the boundary now remember that the f is weakly contracting for the poincare metric in omega so now f so of course omega is hyperbolic because it avoids at least two points is weakly contracting for the poincare metric so it means that let me call it z n z n is the nth iterator of z and z n for the poincare metric in omega is less than or equal to the distance from z0 to z1 so it's bounded but now we know that these points go to infinity in omega so it is well known that the poincare metric at the boundary of a poincare metric is proper so when you go to the boundary of a domain the poincare metric diverges with respect to the spherical metric so this implies that z n goes to the boundary implies that the spherical distance between z n plus 1 and z n must go to 0 now if you look at the cluster values of this sequence so every cluster value so the set of cluster value of this sequence set of cluster values is a connected set so that's a classical lemma when you have a sequence which says the set of cluster values is connected it's a connected set of fixed points you have finitely fixed points for your homomorphic map so it is one fixed point so z n converges to some fixed point p in the boundary you know that there must be a fixed point at the boundary and z n converges too and now why the multiplier is equal to 1 so there's something called a snail lemma which is due to Fatou and which does exactly that so it's a local station so let f be a germ of homomorphic map at the origin fixing 0 assume that there exists a certain open set such that f of u is contained in u and for every z in u f n of z converges to 0 and the conclusion is that the derivative is 1 and the proof goes like that so you have your... perhaps it's not a neighborhood of the origin just an open set which is mapped into itself so u is something like that and f of u perhaps is something like that okay it's a domain it's a connected set it's an open connected set so that's my u so z0 is here z1 is here I will consider a certain path joining z0 to z1 and I just take the iterates of this path so let gamma0 be a path joining z0 to z1 okay and let gamma be the union of iterates of this gamma0 okay so this is a path going to the origin because so 0 is somewhere under boundary perhaps so you know that all these points here must converge to 0 under iteration okay so this is a path converging to the origin under iteration okay now excuse me I missed something here the derivative at the origin must be of modulus 1 okay otherwise of course it can be attracting and it's exactly the case here because my fixed point at the boundary cannot be an attracting point and cannot be a repenting point either so it's exactly the situation which I have in my simplification here okay so what does this path look like actually this path must look like a snail because the dominant effect of f so assume so it's approved by contradiction assume that the derivative at 0 is different from 1 so it's a certain rotating complex number so the dominating effect is the rotation so if you look at this path gamma it must converge to the origin like a snail right so gamma must look like that then the picture is that gamma is a certain path going slowly to the origin now you take this snail here so that's the origin take this snail and you cut a piece you close it the domain so the domain is here so I make a little slit here to cut my snail to get the domain and this domain is mapped into itself by the dynamics okay so it should be a contraction for the Poincaré metric okay and again you contradict you contradict the assumption on the multiplier okay so I don't write the details here but you make a certain slit to close the snail and you cut that into itself so that's the proof of the snail lemma okay so what we have done here is prove that any transient component must be the basing of attraction of a certain parabolic point with multiplier one now you have the recurrent case because it's a normal family the distance Poincaré distance is constant well it is decreased under iteration okay another argument is just normal families okay there's a certain limiting mapping and this limit mapping is constant because it values in the boundary okay so the second case is the recurrent case so there exists a certain z in omega such that fn of z accumulates a certain compact set okay again there are a number of sub cases so sub case one f is not a covering okay so in that case if you work a little bit using the contraction of the Poincaré metric the fact that the Poincaré metric must be a strict contraction then you get there's a little bit of work here but it's not very difficult then you get that everybody must convert to a certain point and this point must be attracting so in that case attracting basing okay second case is a covering is an isometry isometry for the Poincaré metric then you have to prove that so it's clear that I will not go to higher dimensions today but at least you have the details of the classification it was the end of the part one dimension so I will do higher dimension tomorrow anyway so f is a isometry for the Poincaré metric so in that case if it's an isometry if you take cluster so it's a factor component so you can always take cluster values of f converging to some mapping g on omega and now since f is a local isometry g cannot be degenerate okay so g is also an isometry it's a local it's a local isometry so g cannot be degenerate g is not degenerate degenerate okay so if you think a little bit this implies that if you look at f and i plus 1 minus n i so composed with f and i of z of course you get f and i plus 1 this is going to g of z this is going to g of z and g is not degenerate so this implies that f and i plus 1 minus n i converges to the identity okay so in your domain there is a certain subsequence of iterates of f and n which converges to the identity and there's a lemma here lemma let me not state the lemma so lemma let omega be a hyperbolic freeman surface and f be a covering of omega okay assume there exists a sequence n i such that f and i goes to identity so it changed the name of my sequence then omega must be a disk or a punctured disk or an annulus or something c star excuse me c is not hyperbolic and f is conjugate to an irrational rotation so the convergence is a strict and f and i is distinct from the identity irrational rotation okay and the proof is not very complicated assume that omega is not simply connected so in our case this does not happen because again the Julia set has no iterated point okay so in our case omega is not simply connected then it must have a hyperbolic geodesic okay and now we have this isometry which is close to the identity it has no other choice of it must send the geodesic to itself so if it must send the geodesic to itself and it's an isometry so i mean you have a holomorphic mapping locally you have a holomorphic mapping mapping a circle to itself which is a certain isometry for a certain metric it has no choice it should be a rotation on that circle so the geodesic is just a question of a geodesic by a luxodromic element right so the mapping must be a question of some translation on the geodesic okay and then locally it should look like an annulus with an irrational rotation and then you propagate using a unique continuation and the domain itself must be an annulus that's a little bit of work here it has no geodesic meaning that it's a disc and then again an automorphism a covering of the disc you don't have many choices and then you get this rotation okay so let me stop here and so tomorrow i will start by comparing that to the higher dimensional case and i will go to this equidistribution business thank you