 Okay well good morning and good afternoon everyone thank you very much for coming to the this week's ICTP mathematics associate seminar which I believe is the last one before the summer break is that correct or is there no there's one there is another one sorry so there will be another one next week next Thursday as well. So this week I'm very pleased to introduce Luna Lomonico. Luna Lomonico is Italian mathematician however she has been working and living in Brazil for a while and she recently just took up a ten-attack position at IMPA in Brazil and we are very very pleased that she accepted our invitation to give this math associate seminar and Luna is going to talk about making quadratic maps with the modular group. It's a complex dynamics seminar. Thank you very much Stefano. It's a great pleasure to be here well even online but to see Trieste from some cameras. Before starting the talk I will do one second of advertisement of another talk I will give tomorrow because there is this nice event at the University of Rio de Janeiro federal University of Rio de Janeiro that started two three days ago and will end tomorrow in between half past nine and half past six in the afternoon Brazilian time so you need to add the five hours for you in Italy and this is the event and here and let's see if I can manage to no I don't manage to put it in the big screen but then I will just I think I think Luna you should be able to choose which window you want to show when you share your screen on zoom yeah you should be able to choose the specific window not just to choose share generically everything that's on your screen I chose the desktops very sharing let's start the sharing again ectp talk wow you're a genius thank you very much okay and let me start saying that everything I'm saying is joined to work with shambulet well actually this is a lie because the first half of the talk is actually work of shambulet and Chris Penrose and I was in primary school when they did it actually I enter after 15 years today we're working in the subject and I'm going to tell you about making quadratic maps with the model group and of course during the talk I will tell you what I'm eating is but I will start by explaining a little bit the dynamics of quadratic maps on the Riemann sphere and the modular group and how I will speak about another object that can actually be both a quadratic map and a modular group and we will see how this object can actually be a quadratic map and the model group at the same time this is and being two different things at the same time is being a making we call it making okay this is the plan of the talk I try to start as gentle as I can so we are interested about the dynamics of rational maps on the Riemann sphere so the Riemann sphere is the compactification of the complex plane adding infinity we have a rational map which is a fraction polynomial over polynomial going from the Riemann sphere to itself we ask the degree of my rational map to be at least two meaning that every point on the Riemann sphere has two pre-images well has deeper images and at least has to be two for technical reason because if we have a degree one rational map on acting on the Riemann sphere is a movies transformation and the dynamics of one movies transformation is kind of boring okay we have this rational map and we iterated on the Riemann sphere we define the orbit of a point on the Riemann sphere as the sequence of iterate under this rational map so we have zeta r of zeta r of r of zeta r of r of r of zeta and blah blah blah blah r to the n of zeta for every n when I speak about r to the n in this case is composition and we want to know what the orbit do in particular we are interested in in know if I have two points that are nearby I can yes if I have two points zeta and w that they start to nearby will they stay nearby forever and ever after this under these iterates or will grow apart what will they do at the limit actually the dynamics of a rational map of the Riemann sphere and partitioning partitioned what divides the Riemann sphere into two sets one is called the fact to set and it's very roughly speaking the set of points that like each other under this rational map so the set of points zeta that if they start nearby they will stay nearby forever and ever the definition is of the fact to set is that is the set of point on the Riemann sphere where the family of iterates it's a nitty continuous family or a normal family in every sequence has a convergent subsequence uniform and the complement of the fact to set is called the Julia set so we have the Riemann sphere we have this rational map iterating and this partitioning in the fact to set and the Julia set our Riemann sphere the fact to set is the set of points roughly that like each other's the Julia set are the set of points which very roughly hate each other's if one goes somewhere the other one goes somewhere else a as a very basic example let's consider a polynomial on degree two actually let's consider the map PZ of zeta zeta square and we see that in under zeta square zero is a fixed point zero squared is zero and every point in the open unit desk if we start iterating and pick the square we'll eventually get closer and closer and closer and closer to zero so the open unit desk is part of the fact to set the points here like each other they all converge to zero if we take points outside the field Julia set sorry outside the set and the point will have modules bigger than one and the iterating under zeta square will the modules will become bigger and bigger and bigger and bigger so the points eventually will go to infinity and if I have a point in nearby also eventually we go to infinity so the outside of the unit desk at the close unit is is also part of the set on the other hand if we take a point on the unit circle the point is we can write it as e to pi i zeta and well if zeta is irrational this point it will turn here around forever and ever and ever and ever and ever if it is a rational eventually we'll come back here if I take a point nearby this point but inside the unit desk eventually we'll go to zero if I take a point outside the unit desk close to this point eventually we'll go to infinity so if we take a neighborhood of this point you will see that the points in this neighborhood they really hate each other they try to go as far away as possible one from the other one so the unit circle it's the Julia set for zeta square sorry okay let's do something like that and then share again so we are particularly interested in polynomials in particular quadratic polynomials after a change of coordinates if you have a quadratic polynomial you can write it as zeta square plus c and for a quadratic polynomial for any polynomial on the Riemann sphere infinity is fixed and super attracting infinity to the end is infinity and all the points nearby infinity will get mapped closer and closer and closer to infinity so we this set of points that will be mapped to infinity after iteration is called the basic of attraction of infinity and we write it a of infinity in this case we can define the field Julia set of the polynomial which is the complement of the basic of attraction of infinity so the set of points with bounded orbit and in this case the Julia set is the common boundary between the basic of attraction of infinity and the field Julia set if we look the example of the previous slide this is the field Julia set the closed unit this case the field the Julia set of the map zeta goes to zeta square outside there is the basic of attraction of infinity on the boundaries there is the Julia set and inside is the interior of the field in Julia set which is the basic of attraction of zero and the here I draw the field Julia set of different polynomial different quadratic polynomial for different value of c here c is equal to zero here c is equal to one quarter here is minus one here is one quarter plus epsilon and the first thing that you look is something like well I mean this is connected that this is connected this is not connected and the set of parameters c for which the field Julia set k of c is connected is called the Mandelbrot set and it's also called the connectedness locus for the family because it's the locus is waiting for place and connectedness is place of connectedness for the family this is the Mandelbrot set so it's a set of parameters for each c inside I mean in painted black the field Julia set is connected zero is here for this c we have this guy for this c we have this guy for a c here nearby but outside we have this guy this guy lives here this guy lives here and something that it will be important later it's a direct c if we are in a neighborhood of infinity and we start iterate a quadratic polynomial zeta square plus c the c doesn't really play a big role right I mean if the modulus of this guy is very good if the modulus of zeta is very very very big this is much more important than this guy and there exists a conjugacy there is a map that a uniformize the dynamics of your polynomial in a neighborhood of infinity to a the dynamics of a the map given by the leading term so zeta square this will be important later and if the field Julia set is connected that this map extends to the whole basic of attraction of infinity and if the field Julia set is locally connected it extends to the boundary semi conjugating the action of your polynomial to the action of the doubly map okay if there is any question please stop me I apologize with the people who are in dynamics and a foregoing us was look but please see if you have question stop me okay so now we change word we have this word of iteration of rational maps in particular polynomials and we change word and we look a PSL to see PSL to see is the group of matrices two times two with complex entrances under this equivalence of relations so the elements of a PSL to see are may use transformation from the Riemann sphere to the Riemann sphere so rational maps of degree one as I told you before iterating on the Riemann sphere a rational map of degree one is slightly boring so we started iterating bench of maybe transformation a clean and group is a district subgroup of PSL to see so you take a bench of a maybe transformation acting on the Riemann sphere you start to compose them in all possible ways and you see what happens you make this on every point of the Riemann sphere and you want to know where your points are not it happens that in the action of a clean and group also gives on the Riemann sphere also gives a partition of the Riemann sphere into two sets both in bite invariant means that the set is mapped to itself under the the action of the group one set is the ordinary set which is defined as the set of points where the elements form a normal family on a neighborhood and the other set is its complement the limit set a so we have the ordinary set which is an open set where very roughly the points like each other and the limit set which is the complement a demodular group it's arguably one of the most studied clenion groups is the clenion group always has S generators tau one which is zeta plus one and tau two which is zeta over one over zeta and here in this picture there is a fascination of the upper half plane under the module group the ordinary set for the module group is the upper and the lower half plane and the limit set is the real line but since it on the upper half plane and on the lower half plane the same thing happened we usually are represented just on the upper half plane and this is the tasselation okay so these words of iterating rational maps on the Riemann sphere and the action of clenion groups on the Riemann sphere are not this different as one may think and for sight we saw that the action of the operational map partition the Riemann sphere on two sets the one set which is the photo set where the family which is the set of points around which the family of iterate it's a normal family and the complement similarly in the world of clenion groups the action of a clenion group on the Riemann sphere give a partition of the Riemann sphere in two sets the set of points around which the clenion group a the elements of the in clenion groups forms a normal family and the limit set and Salivan started realizing it in the early 90s and started writing a dictionary between these two words during a lot of parallels that will end up in the Salivan no wandering theory which is an amazing result which takes very much in inspiration on the half force finiteness theorem in the world of clenion groups but we are not interested on that in this talk we are just interested about the first similarity a both object iterating on the Riemann sphere partition the Riemann sphere in a set open where the dynamics is tamed and and the complement where the dynamic is a mess and these words are really not that different and there is even an object that can be both these objects is called a onomorphic correspondence our more of the correspondence F acting on the Riemann sphere is a multi-valued map F that goes from Zeta to W the degree of this multi-valued map is usually N to M meaning that every point here as N pretty much and every point here as M images and they are defined by a polynomial relation so in this case my polynomial will be degree M on Z time degree N on not degree N on Z time M on W so we take our favorite polynomial in Zeta W of degree N on Zeta and M in W and let's put N and M equal to and to let's say that we have two two correspondence so we have a polynomial of the Greek two in Zeta the Greek two in Zeta and the Greek two in W and we take the Riemann surface that is given by the zeros of this polynomial and this is what called this thing S is equal the surface given by pH Zeta W equal zero so this is S and the correspondence is the implicit map defined by this polynomial if we consider the projection on the Riemann sphere where Zeta lives and the projection on the Riemann sphere where W lives for each Zeta we have 2W killing the polynomial right so that's Zeta we have W1 and W2 associated and the projection sends it here W1 and W2 so my correspondence sends Zeta to W1 and W2 but on the other hand for every W we have two Zeta that kill this polynomial right so each W has a super images every map has two images and every point has two images and every point has two pretty much and why am I telling you about the this complicated objects because actually we can write a rational map as a correspondence if I have my rational map of Zeta equal pH Z over Q of Z of degree n this becomes an n to 1 correspondence you just state as polynomial QW minus P of Z for every W I'm going to have mz calling this killing this polynomial so my correspondence will be n to 1 also planning group can be written as correspondences if your planning group has generator gamma jota equal your maybe transformation your polynomial it's written like the product of the polynomial given by killing each of these guys please read the expression not to make me pronunciate this okay so a correspondence can be a rational map and can be a planning group now the question is I know I have to wait again so the question is can a correspondence behave like a planning group and a rational map at the same time and in the early 90s the Sean and Sean bullet and Chris Penrose started in the city in these questions and they started plotting the limit set of a particular family of correspondences and they saw this and they start convincing themselves that this has to be true because this picture looks like two copies of this field really a set with around the tasselation of the module group this looks exactly like you take one and another and then you clue here outside the module group so they start wondering can we meet a planning group and a rational map in a correspondence what's a meeting a meeting between two objects a and b is a third object see which behaves like a on an invariant subset of its domain and sb on the complement so we have this object and this object simply has two different behaviors like an object a and an object b this is why we call it a meeting between a and b meetings exist in the polynomial I told you before that if we have a polynomial with connected the future a set let's say I don't know this guy which because it's the easiest the future a set for zeta square the dynamics outside is given by a zeta square and on the well let's put the zeta square plus c because in other cases trivial outside we have the dynamics of zeta square outside the field Julia set and on the boundary of the Julia set if the Julia set is locally connected we have the dynamics to zeta now if we take another field Julia set of quadratic polynomial outside we have the dynamics of zeta square and if this guy is connected and locally connected on the boundary we have the dynamics of two zeta which is the same dynamics so we can glue this set to this set using the map to zeta because it's the same dynamics on one side and on the other so we glue together by this map and we obtain a topological sphere and then you have some work to do but if you're lucky you have a rational maps that behave like this a two polynomials on one side and the other polynomial on the other side a shelter in his web page has very very nice video about making polynomial and how the the meeting construction is done and the meetings also exist in the world of planning groups so I don't understand why when I start to writing I have to start the sharing and then share again but yeah so what would be a meeting between a rational map a quadratic polynomial and the modular group well it will be a correspondence because we need an object that can be a rational a quadratic polynomial and the modular group a rational map cannot behave like the modular group and the modular group do not behave like a quadratic polynomial a correspondence can behave like both so like a true to correspondence in such that we have a completely invariant open disk where the dynamics is the dynamics of the modular group and in the complement we have a the one point a connected set made by two copies of field Julia set of a quadratic polynomial and my correspondence behaved like the quadratic polynomial on one side and the inverse of this quadratic polynomial on this other side the inverse is in order to fit the the degree okay and now that we have defined what I'm making between a quadratic polynomial and the modular group is the question is do they exist and when we ask if something exists this question really has two sides and on one side we are asking can this a object exist and on the other side we ask a do we find them so on the first side a the existence problem can we make quadratic maps with the modular group the question actually a also have two sides and the first question is do quadratic polynomial and the modular group fit together because if I have two polynomials of the same degree I know that on the boundary of their field Julia set they have the same dynamics the same fit together but here I have the modular group and here I have the doubling map and here if I have the inverse of the polynomial I have the halting map so do the generator of the modular group on the real line behave like a doubling map here and halting map here because if they don't we don't even have at this to start the second question is does it exist the sample respondent that actually could realize this that actually can behave like this if we answer positively these questions then we can ask okay but can we find that these makings in the sense that there exists a family of correspondents such that every member it's a meeting between quadratic map and the modular group and then you can ask even further if there is a family such that every mate the meeting does this family organize internally like the family of rational maps they are meeting with the modular group in shitty terms yes the connectedness locus of this family of correspondences all neomorphic to the mandible set or to the connectedness locus of the family of rational maps we take as model and okay shown increase answered positively the first two questions in particular there exists a map the minkowski map which is a homeomorphism from the extending line to zero one it's only ascending a number real represented in continuous fraction expansion so we have x zero x one x two which means that is x zero plus one over x one plus one over x two plus blah blah blah blah to the binary number given by zero dot x not copies of one followed by x one copies of not followed by x two copies of one and it happens that this map conjugates the action of the let me take the action of the generators of the modular group so let's put them this is the apparel plane with the modular group here is zero here is minus infinity here is plus infinity and here we have acting taiwan and tau two and the minkowski map conjugates to conjugate the means that it brings one dynamics to the other and it conjugates the action of the modular group on the negative realign with the doubling map which is exactly the dynamics of our polynomial on the neighborhood of the field in julia set here i have to consider and on the positive realign with the dynamics of the halting map which is exactly the dynamics that i have on the boundary of the field julia set if instead of p i i consider the inverse of my polynomial so thanks to the minkowski map which i write m we can it's possible to glue the upper half plane around two copies of the field julia set actually we need to attach them up here we glue zero here and here plus and minus infinity identify so here basically we have the negative real axis and here the positive real axis and here you have the upper half plane is a topological construction that becomes possible because the two things the two dynamics fit together okay so now reassured by the existence of this conjugacy we start wondering okay but there exists a family of correspondents that actually can behave like modular group on some open space set and polynomials quadratic maps so there's a question from the chat asking is a question just asking whether does any kind of intuitive realistic explanation for why continued fractions coming to play in this situation a not that i mean that i i i could give a different talk on the minkowski map um for the i i i i cannot give you what intuitive explanation in this moment maybe shown can you fast no it's not answering but um you'll have to repeat the question for me yes do you have any intuitive explanation why continue fraction expansion comes into play only because it can be thought of as the two operations can be thought of as the doubling map on the circle and so it has to topologically exactly the same thing as as the binary shift yeah the thing is that this map morally is kind of a miracle because it's a conjugated to pretty different dynamics and we could speak a lot about this map but then i won't give you the results and would actually give you um i think you can go on it but okay i agree that it's very fascinating because it at least for me it's unusual to see why you know the number these entries of the continued fractions associated to the numbers of zero than one you know somehow that that does not not something i've seen before i i know but if you are asking me a picture for an explanation in immediate i don't have it in my mind this moment i'm sorry i apologize an interesting question thank you okay so now that we are sure about the fact that these things can these meetings can exist the question is do these meetings exist is there a family of correspondences that can behave like in the modular group and quadratic maps the answer is yes the result is by shown increase and they prove that the meetings between quadratic polynomials and the modular group lie in a family of onomorphic two two correspondences given by this relation and here there are some limits set for different parameters of a could explain later how this family has been cooked up if you're interested but i won't explain it now because it will take a bit so but i will show you some this is the connectedness locus of this family actually is the connectedness locus intersection the disk of center four and radius three and here are some limit sets for these correspondences this guy lives here this guy lives here this guy lives here and so on so when you see this picture the natural conjecture is this guy that we're going to call m gamma is on the morphic to the motherboard set and everybody living here it's a meeting between a quadratic polynomial and the modular group i mean look the picture it's pretty reasonable to guess this right and the conjecture is from the 94 and since then there has been a lot of partial results and the state of art so since 2007 was that there exists a meeting between the modular group and quadratic polynomials q of c for a large class of body c but they never managed to say that the whole family is a is a meeting why because well let's see how is the dynamics of our family a of correspondences it happens and i'm going to state it as fax it comes from the construction of this family that we have a disk that in this picture is the left house plane we're in the correspondence just another question just to make sure i'm calling so this is the map for every point of two images and two pre-images is that correct yeah so the orbit is the full set of all images of images and pre-images of pre-images that's correct yeah but the thing here is that for how this family of correspondences has been constructed there is a disk which is the left hot plane which is invariant after the images and the complement which is invariant under the pre-image what am i saying i'm saying that this set is sense it's sent one to two inside itself this line is mount here and here one to two it has two images and everybody here has two images here inside then we started writing again this guy is sent here and here and well and this guy is sense here and here and this so this thing is sense one to two here inside the complement of this disk uh so in the right half no the left half plane sorry i have a problem with the right and left on this part this disk is invariant by the pre-image so the pre-image of this line is this guy and this guy or say in a different way the correspondence sends this line here on the imaginary axis we are on the Riemann sphere there is also infinity yeah and they are all circles and in this line this is the pre-image of this point is sense here and here and the pre-image of this guy is this so we have that this thing is sent actually two to one to this guy to to everything i mean this heart bounded the part this one well let me annotate maybe it becomes clearer this part is sense two to one to everything on the left part and the pre-image is i'm going to make the pre-image of this red it's this part bounding by the red curve so we have a the correspondence behaving like a map inside this guy like something one to two like the inverse of a map on this part and then we have a branch that sends this guy to the other one which is actually an involution in this case is zeta goes to minus zeta so everybody here has one image here and one image there and it has two pre-images here and everybody here has two images here and has one pre-image here and one there okay if this guy this point is fixed and by construction is a parabolic fixed point a parabolic fixed point is a point which has an attraction direction and a repelling direction at least it can have many but let's give it a simple and by construction here there is an attracting direction because this disk is mapped inside and here there is a repelling direction because this disk is mapped outside and we can define that the forward limit set to be the complement of this to be the intersection of this disk by iterating so it's this intersecting to the image with the image with the image with the image and the forward and the backward sorry limit set to be the intersection of the pre-images and the rest of this disk let's remember that our correspondence behaved like a map basically everywhere here but a neighborhood of this point because let's remember that this is sense to everything and this is also sense to everything so a neighborhood of this point is mapped one to two to a neighborhood of this point for the rest here we have basically a two to one one okay this is for every eight so this family has what is called the persistent parabolic fixed point and the way the theorem has been proved the previous theorem i showed you was by breaking this parabolic fixed point every time polynomials do not have always a parabolic fixed point in the in particular the family of quadratic polynomials so how did they prove it they made for every a the first perturbed this correspondence in order to break the fixed point then they used a tool that is called polynomial like mapping in order to prove that here inside actually you have a polynomial and then they made another surgery for a pinching back in order to come back to the family of origin and of course you can use it you can do it but the meaning is not particularly efficient if you have a family with a persistent parabolic fixed point and you are planning to prove something perturbing healing the parabolic fixed point for everybody is not the most efficient way and it's not just this it also suggests that maybe these guys are not meetings between polynomials and the module group but we are meeting between a family of a family of quadratic rational maps with a parabolic fixed point per system and the module group right i mean there is this parabolic fixed point why do we want to kill it all the time and maybe we should start working with him you start wondering okay do we have available a family of quadratic rational maps with a persistent parabolic fixed point yes we do it's called the family per one one and yes the maps rational maps pa of zeta is zeta plus one over zeta plus a with a in the complex plane for every a infinity is a parabolic fixed point and for example this is a parabolic fixed point with basic of attraction call it lambda a which is the set of points that are set are mapped to the parabolic fixed point under iteration and in this case we can define a field reasset as in the case of polynomials like exactly parallel my field reasset is the complement of the basic of attraction of infinity just for polynomials the basic of attraction of infinity was the basic of an attracting point here the basic of attraction of infinity is the basic of parabolic point but who cares and here we see the field Julia set of the map p0 no and yes zeta no p1 sorry zeta plus one over zeta plus one is this guy here we see infinity this is the preimage of infinity so here outside colored is the basic of attraction of this guy everybody here is going to this guy and in black there is the field reasset the set of points that are not mapped to infinity eventually this is the connectedness locals for the family per one this guy lives here this guy lives here and this guy lives here and actually this connectedness locals which is called the parabolic manderbrot set is only more fit to the manderbrot set by a result of by Pascal Horsch and Carson Peterson so a one instead of mating polynomials with the module group but we don't we made the disguise with the module so we define we say that f a is a mating between some rational map of the form zeta plus one over zeta plus a and the module group if well a f a on the backward the limit set behave like a member of PA and outside the limit set the correspondence behave like the module group and the result that we obtained is that yes finally for every parameter in the connectedness locals the correspondence is a mating between a rational map PA what happens sorry okay let's do this again okay can you see this or am i or did i stop well okay sorry so for every a in the connectedness locals your the correspondence is a mating between some rational map and the module group by the way actually for every a for which this thing is a is a reasonable correspondence in the limit set you have a PA but if the limit set is disconnected it's difficult to prove that you have the the module group outside because we use the Rima map and if you have the your disconnected set and then the the complement is not a disk and then is complicated but yeah this is the result and this is a illustration of the result this is the parabolic mender board set and this is m gamma our connectedness locals he lives here and here outside we have the connect the module group and here we have exactly the action of PA here p1 this guy lives here outside we have the module group and here we have exactly the dynamics of PA of this guy in a neighborhood of the digital gss this guy lives here and here we have the dynamics of the PA which lives here which is this guy just an illustration and how how much time do i have because i started something like five minutes later or something like that yeah i think another five minutes is fine don't know okay yeah i don't want to say basically anything about the proof is a bit technical the only thing that i will say is that what we need to do it's making a surgery in order to kill this guy we want a map we want this thing to behave like a map in a neighborhood of this guy and if we take a neighborhood of this guy and we take the pre-image we have a map everywhere but here because here oh okay because here a f the correspondence sends this neighborhood one to two here and this is annoying because maps do not have two images you just have one so what do we do we have to kill one image by quasi-conformal surgery i won't say anything else just that if you want to do things one by one then find a you kill the image one by one but if you want to do it in some uniform way in order to try to prove that the connectedness law which is homomorphic to the parabolic manoeuvre set you need a technical condition which is angles which is a disk containing the field reassert that the backward linear set with angles less than pi for everybody in the family and this technical condition well this was the surgery this technical condition is ensured if we proved for technical stuff that this connectedness law is inside what is called a lune a lune is the the topological disk bounded by two arcs of conferences and it happens that if this guy is inside a lune then everybody then the backward linear set of everybody living here inside is inside a lune and we can do our surgery with some a regularity with the parameter for trying to speak about families but in order to get here and also we needed to well in order to get here we needed a to prove that this guy is shrink shrink shrink shrink shrink for saying that they don't go out so basically we needed what is called a yakuza inequality and in order to get a yakuza inequality you need an insane amount of things so a basically we ended up again I lost sorry I lost I just wanted to show this table that I just remember that I forgot to get problem compiling we ended up developing a full complete dynamical theory for this family of correspondences which parallels completely they do a de Hubbard theory for polynomials and to me this is amazing in the sense that I mean correspondences are mess I mean the only nice thing about iterating a function is that the points just have one where orbits go here every point have two images at the third iterate you want to cry because it's a huge mess but magically these a family of correspondences be paid very very very very very very very very very similarly as polynomials not just rational maps not just on an able of the limit set but much more globally and okay and now I can share the last thing which was this okay and we are as a working progress proving that the connectedness locus is actually homeomorphic to the parabolic mathematical side while working progress close to one end actually and this is everything I want you to check so thank you for your attention but thank you very much Luna on everyone's behalf um that was a very very nice talk I think it was really perfectly pitched to for us all to to learn a little bit about correspondences so let me ask anyone who has some question to your welcome to unmute and ask the question or or something in the chat does anyone have any questions so so let me ask you Luna um if to see if I understood correctly the so what you've done is you've defined a specific family of correspondences is a right that you are saying is somehow kind of canonical or standard similar to what the quadratic family of our show matches well um honestly it was shown bullet and three spenders who actually constructed these family of correspondences they constructed in such a way um they considered how the correspondence modular group behaves and basically they took a correspondence that behaves very much like that one above different enough for having a limit set where you could have a different behavior I can give you the construction if you want but so my question is so you you uh Sean and Chris or you know you you you you define to construct the correspondence with a specific motivation of of constructing this mating but what you are saying at the end now is that this interest in this particular family of correspondences is kind of independent of the original motivation for the mating basically yes I entered the picture that the family was already there was already cooked cooked up because they figured out that this family was special and there could be a lot of meetings there inside but they couldn't really prove that everybody was a meeting because they were tried to kill the parabolic exponent every time and blah blah blah and I entered the picture with the family already there and what they proved which shown is that actually everybody there inside is a meeting and they have a wonderful dynamical theory and blah blah blah blah but this is my question the dynamical theory of this correspondence and the fact that they are meetings are they kind of independent of each other or they connected no they are connected most of the things okay when the do you see this slide right sure okay we want to prove that outside in saying that a meeting you saying that inside you have your rational map but outside you have the module right and for proving that outside you have the module group in the case your limit set is connected you have your Riemann map from the complement to a disk so let's say the upper half plane and you prove that actually your Riemann map conjugates a the dynamics of the branch of the correspondence to the dynamics of the generator of the module after you have this this map basically becomes your picture map is the uniformization of the outside to your model then we wanted the Yoko's inequality so you want it external rates right and you don't really have infinity here but you have periodic judicics which you can pull back by your esomorphism and the let me show you the table that is not on this guy it's on the other guy of the university no not this one sorry yeah we have our butcher map which is this this one and the external judicics that they play exactly the role as external rays and then you start proving that as periodic race land the periodic judicic lands this is not the difficult and then you want to prove that every repellent fixed point is the landing point of a periodic judicic and this is extremely difficult then you want to prove a Yoko's inequality and and we actually have a better Yoko's inequality than the original one and the difference means that you a Yoko's inequality tells you morally how these things shrink going to the root and the fact that our Yoko's inequality is better we have a square here says that our limbs here shrinks faster than these ones and you can see it because I mean the ratio between these guys and this one is much bigger than the ratio between these guys and this guy it all comes from the mating structure there is a lot of work to do in the sense that I mean you need to I mean you can replace what is the what the how about the dynamics of polynomial like mappings but you don't have polynomials you don't have infinity so basically you have to use invent new tools so it's really something that it comes in the in the union of the two two words because the fact that geodesic behave like external rays I mean to me this is it this is one of the pieces of math I'm the most proud of of my work because to me this is a a lamp in the night you know I mean it's a family of correspondences it's a mess and somehow these families perfectly describe can I just add a couple of points step on that you're you the dynamics forces you into this particular family of correspondences and you can think of these correspondences as you've got the modular group on one side you've got the squaring map if you like on the circle on the other side and in this particular in this particular family we're degenerating the squaring side so that we get the whole modular group we sorry we get the whole Mandel Broad set we get all the quadratic maps you can also degenerate the modular group side and get all sorts of interesting things other than the sort of pure modular group on the outside so there's a great richness to the way that the group theory fits together with the rational map theory and there's a lot more to explore yeah and actually shown with the Bill Harvey really proved that you can make polynomials with a lot of planning groups I mean if you have the you have to have the right time I mean set to match them yeah but I mean very roughly speaking let me be a rough shown please if this is the discreteness locus of the faithful discrete representation of order two times order three groups the modular groups lives here but actually you can make polynomial with a bench of guys living here inside so one could actually investigate the c2 complex parameter which is basically this guy times Mandel Broad set and parabolic Mandel Broad set it it's a new word somehow right right I had a very general question this this is an amazing talk thank you very much yeah it's I mean I was always been fascinated by Sullivan's dictionary ever since I heard about it from my teachers and yeah this this is completely mind-blowing that you can do this way of unifying both sides of the dictionary so I had a very very naive question when you go back to the Sullivan dictionary slide wait because I need to stop sharing and share again because I'm not technological enough to sorry no okay let's go to the Sullivan dictionary so this is something that that sort of haunted me for a while and I don't understand it quite so on the top line you have these rational maps and you have these finitely generated Kleinian groups right and then there's the the Fatou set and the ordinary set and then the limit sets so my question is on the right hand side of your column on the Kleinian side right it's here thank you on the Kleinian side you have this notion of a discrete subgroup okay so there's greatness I'd like to know your thoughts and Sean's thoughts on what's the analog of discreteness on the rational map side do you understand my question I'm saying on there's entries on the right hand side there's an entry discrete right because Kleinian group yeah I'd like to know what the entry is on the left side and I mean really this is a question that is haunted I mean I don't know the answer it's a genuine I think in some sense they're all discrete on the left hand side kind of thinking the same it's just that giving you the picture of why is something I need to think I mean the nice questions they used to be the deepest ones so I need to think a little bit but I okay no no that's that's great I mean I had another question about what maybe you're you hinted at about this so the the this particular meeting was very special because you were able to glue the dynamics um when the limit set was this sort of you know circle like object right homeomorphic to a circle dynamics on the circle um so so is there a way to to sort of do this when when the when the the dynamics on the circle is like more like say dynamics on a canter set you know something that's disconnected um see I mean the problem it's what does it mean to glue in the sense that I mean as I was saying before I mean in order to prove that outside you have the modular group we start by the RIMOMA because we need to uniform step one is uniformizing this thing to the upper half plane right then you try to say that the dynamics here is the same dynamics here but I mean if you don't have the upper half plane for starting with gets complicated of course you could try to get the biggest set get into the bigger part a these things we have been thinking about but I mean things becomes extremely technical and complicated because your starting point breaks basically and the and you get a lot of difficulties yeah so it's something that it's uniformizing sort of say like a complex plane or the RIMON sphere no the the the upper it's if this guy's connected right it means that this is a disc topologically speaking right maybe maybe the challenge is that it's harder to talk about uniformization when you're looking at you don't have the starting point yes I mean let me see if I have the picture of a disconnected one the Europe no here is disconnected and I don't know if you see this guy but here is disconnected and here I mean if this is not a disc I can take try to take the biggest disc and then try to do things there but you start the modular group acting on the outside but it's not a group yeah if you expand your ideas a little bit and allow not things which aren't quite groups you can do all sorts of gluing together we do have part sorry well yeah what did you mean by not if you extended the group is it like a groupoid or no it's a correspondence to be quite honest it's got some critical points in it oh a group who doesn't have if it's a group generated by murbius transformations there are no critical points right right each thing is just a murbius transformation but if you allow in in various limited ways you allow in critical points then you can start gluing together all sorts of different objects and and this theory has some you have some examples beyond the modular group where you can I could give you examples where instead of the modular group acting on the upper half plane you take the free product well it's it is the modular group as an abstract group but acting in some other way for instance a circle packing acting as a circle packing group rather than a books in group take a look at this paper uh to share it's uh by by Sean and Bill Harvey it's electronic research announcement of AMS of 2000 and there they are actually making considering a family of correspondence of two parameters which are actually in some sense mating of polynomials and planning groups hyperbolic planning groups I mean the the point here that basically the only thing that they figured out is that the parabolic you go to with parabolic and the hyperbolic go to hyperbolic in the sense that I mean the modular group is as parabolic as you get that you have a derivative that is one of one generator so I mean try to meeting it with something hyperbolic because polynomial sorrow hyperbolic infinity super tracking is not the best idea I mean parabolic's fit well with parabolic and hyperbolic with hyperbolic I mean the moral of the story is just this yeah I was actually thinking about that because in your picture of the you know the the famous picture of the fundamental domain of the modular group you know you can have a family of hacker groups yeah where the the two vertical line yeah so if you look at the the two vertical lines of the shaded region and you keep shift you keep sliding them and make them wider right so at some point those lines are going to touch the bottoms of that circle they're they're lying on and then if you push them even further then you get like a shot key group right like if those vertical lines leave the unit circle right push them to the left and the right you degenerate it and when you take the hacker group the dynamics well you've got the dynamics of the hacker group rather than the usual continued fractions but we work in the same way except that you have to make them with Chebyshev polynomials of higher degree that's what takes the place of quadratic maps oh wow Chebyshev polynomials have the the same dynamics if you hunt back a few years I did a lot of work on those with Marianne Freiberger who worked with me as a postdoc you'd have to hunt in the literature for my name and you know I look for it yeah so Sean I think you've really you've really opened our all our horizons on this stuff yeah this is amazing I'll stop thank you thank you I've tried to tell people about this before but Luna is much better at it than I am well it seems like the collaboration works very well yeah well the recent stuff that we've done she has had fantastic ideas which have solved the questions that I posed 25 years ago now fantastic well it's starting 2011 one of these so I mean it's 10 years 10 years is nothing Luna yeah some of us have been working on problems for 25 years made no progress yeah the way lately we are being kind of judged is something like what did you produce in the last year you take me one year for taking a pain in my hand that was super slow I feel like that yeah thank you very much so let me just remind everyone that this that this is recorded and it will be on the youtube ICTP mathematics youtube channel I noticed that Emmanuel has posted the link in the chat if you want to see it but you can easily find it on the search so you can you will find within a few days the recording of this workshop of this seminar and also of the previous seminars as well as lots of other material recordings of conferences in the past at ICTP so there's quite a lot of videos for that well I assume provided our speaker agrees right is that okay by you yeah yeah it's fine it's fine it's fine have you recorded with your agreement okay now we have a video statement that she agrees so it's kind of a loop thing that she agrees okay so thank you again very much Luna for this it was a pleasure okay thank you why and you see we have sunshine in London great is that just is that real sunshine or is it just your your screen man watch it you're the birds singing behind okay bye guys