This is covering the spread part of the Fandall podcast network There are a few better feelings and going to Fandall sportsbook once again and seeing suddenly Majorly baseball money lines for actual games posted once again I know spring training a lot of you can be good at that if you have information and stuff like that not for me Personally, I want to go to the actual games that count. We finally have them for tomorrow We got money lines over at Fandall sportsbook No props posted as of yet, but we're gonna dive in to the money lines across these games Let you know where my model is seen value and my favorite best for Thursday over at Fandall This is covering the spread right here on the Fandall podcast network and number fire calm My name is Jim Sonness. I am a senior writer and analyst for number fire calm here to break down Thursday's opening day of major league baseball action Letting you know where my numbers are seeing value Across the opening slate of the year as I mentioned no strikeout props up yet I have models for money lines and strikeout props personally So those would be the primary things we discussed here on the show when we record Typically, I'll go through both in the morning because strikeout props are up then for tomorrow Nothing up as of yet So focusing just on money lines and one total that I like across Thursday's games for today But again a more expansive discussion coming up later on as we get more markets up for other podcasts throughout the year before We dive in to opening day though want to remind you this looking for cement basketball tournament discussion We talked to with dr. Ed Fing about the final four got his read on both those games Talked about the futures market talked about much more to get you ready for those that it's up now on the covering the spread podcast Also over on the Fandall YouTube page check that out there tomorrow. We're talking to Justin Carter of rotor baller getting his read and her hoop stats Getting his read on the women's final for talking about two really fun games LSE Virginia Tech We got South Carolina taking on Iowa that should be a whole lot of fun So we'll talk to Justin tomorrow get his read on how to bet those things how to research where to find good data and His read on those games futures markets and much more that'll be up on the covering the spread podcast feed I'll probably closer to noon or so then over on the Fandall YouTube page after that So go subscribe and if you like what you hear leave us a five-star rating because that does help us out a bunch Grand slams no hitters and double plays are back and there's no better place to get into the action than Fandall America's number one sportsbook. 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That is going to be an ESPN at seven o'clock That is the white socks and the astros and I actually do have some thoughts in this game in terms of betting It's Dylan seeds versus from Breval des the one thing I like in this game as of right now over a fandall sports book I like the astros money line at minus one fifty six the implied odds there are sixty point nine percent whereas my model Has the astros at 64% to win there are a couple reasons why I am higher than the astros in the market Right here. The first one is I think from Breval des is extremely good We saw what he did last year very good year for Valdez He kind of proved himself as being kind of a an elite tier guide despite not being a high strikeout picture now With that said Valdez did get more strikeouts across the second half of the year He did that by throwing more sliders across his final 14 starts in that time Strikeout rate didn't crease his skill interactivity array was 2.87 and he still allowed just a 17% fly ball right in that time Now he's facing the white socks here. The white socks are a very good team against lefties They had a 121 WRC plus Against lefties last year based on the guys who are currently on their active roster So it's a tough matchup for Valdez, but I think given the fly ball, right? He is a guy well positioned to handle this kind of offense Dylan sees today a fantastic job last year I was suppressing hard contact and that is a skill that is something we can bank on he is good at it And that does matter a lot But the white socks as a team struggle elsewhere Their defense is not that great. The bullpen is not great It's actually in it's worse than the Astros there and the Astros defense is good as well They get a bump being at home So I think that minus 156 in the money line here for the Astros I think that makes a lot of sense So the Astros to me one of my favorite bets of the day for opening day Minus 156 the money line there and a good spot to me to find some value So the Astros one of my favorite bets if you're tuning into them For the national TV game on Thursday night Across the other 14 games on Thursday There are five other spots where I see value in the money line That probably feels like a big number and it might be so I want to get that disclaimer other disclaimer here is that? Baseball was a struggle at times last year. You can find my record over at Action Network You can see the bets are recommended and it was not a great year for me by any means So I want to be transparent in that so you can decide Listen to what I'm saying decide. Do you agree or not? If you don't agree? That's fine. Totally understandable You know, I want you to make your own decisions listen to the rash now listen to the data behind it and decide Whether or not you agree but again the caveat here is baseball is not great last year Did get better second half of the year made some tweaks to the model at the all-star break Tried to refine that as a year went along August on did go very well Then made some additional tweaks to the to the model across the offseason So my hope is it's better this year, but I want to be up front in the fact that last year Definitely not my best year in terms of betting baseball Let's go through though where the new model is showing value for Thursday's games and again just keep in mind that this is not me you have to bet it But this is where I am seeing value personally One of my favorite bets is for the Padres against the Rockies the money line for the Padres in this game Is currently minus 210 and I like the Padres right there and that has shortened a bit from opening Because I believe it was minus 196 it open It's not minus 210, but even at the new number I am still showing value the applied odds of minus 210 67.7 percent whereas my model has 71 percent and That's primarily due to the non starting pitcher elements of the Rockies I don't mind they're my Marquez think he's a decent pitcher especially on the road But this bullpen for the Rockies may wind up being one of the worst in all of baseball The offense of the Rockies not great once you take them out of course field now The copy up there is that they're facing a lefty in Blake's now They're definitely better against lefties than they are against righties They have a minor or a 105 WRC plus against lefties again based on their current active roster And that helps in their face in snow is a guy who can be Super erratic so laying minus one or minus 210 on a guy who is volatile can Definitely come with risks, but I think the good thing here the thing that helps Snow and helps the Padres case here is that the Rockies are not a team that puts the ball in the air a ton They have like a 33 percent fly ball rate against lefties. So his issue is hard contact and fly balls Not gonna get a lot of those in this Rockies offense again The Rockies 33.2 percent their fly ball rate versus lefties last year based on the current active roster So it's tougher for them to exploit Snell's biggest issue I'm not sure they're the team to take advantage of Snell where he struggles most and that's why I'm okay Laying the minus 210 on the Padres here at home in their game against the Rockies That's the first one I want to recommend and probably my favorite across the entirety of the day Second money line and one to touched on was the Baltimore Orioles They are plus one away taking on the Boston Red Sox the Pitching matchup here Kyle Gibson against Corey Kluber and I do like the Orioles here at plus one away and a lot of that's because I'm pretty skeptical Kluber not his best year last year does keep locks and check Which means he's not gonna put a lot of free runners on base, but His contact suppression good But I think it needs to be a bit better to let up as many balls of play as he does to Truly get by about a 40% fly ball rates over his Final seven starts last year we cut back in his change of usage didn't see a lot of positive movement for Kluber Even if he made that tweak 4.2 skill interactive ERA in that time Contact suppression again good, but not great. I think the Orioles are a decent offense I'm not super high on the Red Sox offense personally So I think that this one should be a bit more towards the Orioles I've got the Orioles as like the slightest of favorites here basically a toss up 50.1% for me for the Orioles versus 48.1% implied at plus 108 I think that's a big enough edge for me to take it So I do like the Orioles here plus one away in the money line against the Red Sox against skeptical That's Red Sox offense skeptical of Kluber a bit higher and Kyle Gibson personally That's enough to put me over the edge and feel good about them there One game that I feel pretty good about is the Rangers against the Phillies right now looking at the the markets for that game Got the Rangers money line at minus 130 for the total of six and a half And I like the Rangers money line at minus 134 And I do like the over as well at six and a half and it might feel weird because it's Jacob DeGron versus Aranola don't expect a lot of runs in that situation Couple factors push me towards the over here The first one is that Jacob DeGrom not fully stretched out yet in his final tune-up star when 54 pitches He's around 34 in the star before that So we're not gonna get the full Jacob DeGrom experience here because he had that injury to kick off spring training The Rangers bullpen is fine I'd say they're below average more so than hideous which is why I'm not scared off the money line of the Rangers Despite the fact I don't expect a full DeGrom in this game Other reason I like the over in this game and also like the Rangers money line is that Aranola struggled a lot during spring training Now that could be a lot of things in spring training. Who cares? The reason I care a bit about it is because Nola had the fourth longest pace between pitches among qualified pitchers last year And that might mean that with the pitch clock being in play It could be a bigger adjustment for Nola than it would be with a lot of guys now That factor is not in my model because I could be wrong Maybe Nola just adjusts well that could very well happen So I'm not including that in my model But it definitely does not hurt to see guys with longer pace Potentially struggling to open up the year. So looking at this one right now I think that the Rangers should be a bit more heavily favored in this game again Minus 134 on the money line for the Rangers the implied odds there 57.3% I've got their win odds 59.3% is not a huge edge about two percentage points of value there But that's enough given how tough major league baseball can be to bet to take that the Concerns around Nola the concerns around the pitch count with him and the concerns around DeGrom's length are why I like the over Six and a half at minus 105 I think the roof will probably be closed there tomorrow because of coming rain It was 68 degrees outside should be 70 with the dome closed. So That won't affect things too much But I think the over six and a half at minus 105 and the Rangers money line both solid ways to go for the Rangers and the Phillies Final two money lines that I like here for opening day are both on underdogs The first one is the Cardinals there at plus 102 right now taking on the Blue Jays The other one is I might have to sell you on this the A is at plus 194 Against Shohei Otani. Let's start things off with what I think should be the easier sell here And that is the Cardinals money line at plus 102 against the Blue Jays facing off against Alec Minoa the Cardinals facing or starting Miles Nicholas here Nicholas I don't think it's a bad pitcher. He's been he's gotten good results his entire time the big leagues largely by Suppressing hard contact now last year. I think that he wanted to shift his approach up He wanted to get more strikeouts So he changed things up and he did it did hurt his ground ball rates But his strikeout rate did increase which means he's letting a fewer balls in play fewer or less variants involved there That is a good thing and I think that although I prefer hard contact suppression I'd rather let up no contact. So it's a step I think it's kind of a neutral step for Nicholas But at least it was not a step backwards to go with a bit higher strikeout approach versus letting up more balls in play as far as Minoa He really leaned on his sinker down the stretch across his final 20 starts his hard hit rate allowed 34% that's pretty good But it's not good enough for you think that is 2.24 era across the full season is Sustainable his expected era which factors in the amount of hard contacting I think does the best job of factoring in the amount of expected amount of hard contact Expected era 3.31 more than a full run higher than his actual era And that's part of this but also Minoa has a very slow pace He had the second slowest pace in the majors last year among starting pitchers So again, I'm not factoring that in but it doesn't hurt when I already show value on the opposing side This Cardinals offense. I think it's very good. So is there defense with no one are not about there So I think that We should be a bit higher in the Cardinals here. I think it should be favored I had the Cardinals 53% to win the implied odds at plus 102 Under 50% so I didn't the Cardinals are a good bet here at plus 102 on the money line to take down the Toronto Blue Jays Again, the final one is the A's. I understand if you don't want to go here. They're facing Shohei Ohtani I do not enjoy betting against Shohei Ohtani. It is not fun It is typically not super profitable either to do so because he's a very good pitcher a very good player But Ohtani actually had the longest pace between pitches in the league last year So again, that introduces more uncertainty and when the A's money line is plus 194 uncertainty is a good thing Volatility is a good thing because if there's more uncertainty It's tougher to have a big number on the money line at plus 194 the implied odds there are 34 percent I have the A's win odds at 38 percent 38 percent is not high and I think that's worth mentioning for sure But I can live with that gap Kyle Muller opening day starter for the A's kind of got trashed for being the opening day starter a lot of Twitter hate on Kyle Muller asking, you know, why is this guy the starter? It's not his fault You know, the A's don't have the best rotation but Muller. I do think it's pretty talented He made three starts in the majors last year did not go well But good numbers in triple a both last year and in 2021 So I am very okay with this even if it's not fun to bet against Ohtani again The number there plus 194 on the A's money line. I have 38 percent implied odds there 34 percent Think that is a a fair way to view things So to recap the places where I'm showing value for tomorrow are the Aster's money line at minus 156 I got an edge on the A's a plus 194 Cardinals a plus 102 the pythons a minus 210 the Rangers minus 134 along with the over in that game Six and a half at minus 105 and the Orioles add plus 108 in their game against the Red Sox Again, it is a delight to have money lines to look at once again on Friday's show We'll be back here talking baseball once again I'll have strikeout props to that one usually strikeout props last year Fand will going up pretty early in the morning I would open things up When I get back from the gym in the morning, they typically be posts are ready So we should have strikeout props to discuss In future shows just not here for today because they weren't up quite in time to get them on the show Once again, if you're looking for some college basketball talk on the men's and women's side The men's one already up over on the covering the spread podcast feed in the Fando YouTube page Women's one coming tomorrow with Justin Carter getting his read on those games Make sure you are subscribed to covering the spread to get those as they are posted And if you like what you hear leave us a rating or a view again five stars We appreciate those of you who have done so already if you've got any questions for me I am on Twitter at Jim Sonnis J. I. M. S. A. N. N. E. S. You can also follow the Fand will podcast network at Fand will podcast baseball is back It is a delight to be talking about it with you once again here looking for doing it throughout this entire MLB season This has been covering the spread right here on the fan duel podcast network


OK, good morning. So we continue with our big diagram of the relation between renormalization and topology of maps, measure theory, and geometry, and bifurcation of systems. So last week and yesterday we discussed the relations for circle diffeomorphisms. And today we are going to part two of the course, which is about unimodal maps. And most of it is going to be a little bit historical. It's going to be relatively soft today. OK, so what are we going to look at? We are going to look at a unimodal family, and they have t from the unit interval to itself. And like it's something like this, like a family with sort of unimodal maps which get higher and higher up to the end and say, if the family is labeled by how high you are. So this is just a full family. You move from the bottom to the top. And we are interested in the boundary of chaos. And in this case, the boundary of chaos is reached by a pair of doubling cascades. So what happens is, you have seen this picture, even in the first class of our course. So if you write down here the parameter, so we are going to move up our family, then what you will see is for small parameters values, you will see if you take like this parameter, you will see that the dynamics is controlled by an attracting fixed point. You can easily see this in the picture. So if you take like this one, and you see like there's a fixed point here, and points will converge to that fixed point. And so for very low values of t, you see that the dynamics is controlled by an attracting fixed point. But then what happens is, at a certain parameter value t bang, this attracted fixed point bifurcates in a pair of two points. So you will see that the fork happening. This is something like that. So if you take a parameter value here, the dynamics is controlled by a pair of two attractor. So the original fixed point will continue forever. And then the same thing repeats. And so there will be a little later parameter value t2. And there will be, again, a pair doubling bifurcation. And what you see is if you take a point a little higher than this bifurcation moment, you see that there is a period four attractor. And again, this periodic two point continues. And then this process continues forever. So you will see another bifurcation at the moment t3. And it will continue like that. And you will see the sequence converging to a t infinity. It's like that. And so now what you see is that on this part of the bifurcation pattern, you see that the dynamics is very simple. The attractors, they are periodic. And so this is definitely not chaotic. So we are in the simple dynamics. But then if you are sitting on the other side, what will happen is that on this area, you will see that the number of periodic points of period n is like e to the power, something like that. It grows exponentially. So suddenly when you are beyond this t infinity, you will see an explosion of periodic orbits. And the rate of how many there are is h. And this h is positive. And it is called the entropy. So that looks bad. It looks like chaotic. And indeed, this is the chaotic. So what you see, let's draw the same picture. But let's forget about all the details. And there is this moment t infinity. And here you see no chaos. And here you see chaos. So there is a sharp moment where the behavior changed dramatically. And now the historical part comes, you can call this a phase transition. It's a phase transition. This is terminology from statistical physics, that you have a parameter and you change the parameter. And suddenly, the physics is very different before and after. Exactly, yeah. Exactly. As you see, here the attractor has period two. And here you will see the attractor has period four. So each bifurcation, it doubles. And then after this, you will not have all periods. But there will be many. And if the period grows, it grows exponentially. And so this growth rate starts with zero and then slowly it starts to go up. And so there is an exponential growth of periodic orbits. But it is small, just that. The endopic growths are continuously. Eighth depends on t, yeah. Thanks. That's a good way to say it. Eighths of t. Thanks, yeah. That's the right way to say it. Yeah, when you get to this thing, like the full map, then if you look at how many periodic orbits there are of period n, that is like 2 to the n or something. I never know. I think it's like it grows like 2 to the n. So for this guy here, this is the worst situation. The number of periodic points grows like the end of p is log 2. Ah, yeah, OK. It's log 2. So it grows like 2 to the n. And that is the worst you can get in this type of dynamics. But it starts very slowly. So let's go back to the global picture of what is going on here. So we have our moment t, which is at the accumulation of period doubling. You call this moment, as you'd have said, this is the accumulation of period doubling. And in that moment, you see a dramatic change in behavior from cyclic behavior to chaotic behavior. And let's call this phase transition. And this terminology comes from physics. And I would like to tell the physics story. Because the physics story inspired three people to introduce renormalization into dynamics. So this point of view, that there is a phase transition, is like a crucial historical moment. So let me tell you a little bit about this phase transition stuff. So this is called spontaneous magnetization.  And then you cool it down, cool down. And then you make a diagram. So here you write the temperature. And in this axis, you write the magnetic force. This is just a regular piece of metal. Like something in the chairs. So that means for room temperature, there is absolutely no magnetic force to feel. That is just a piece of iron. But then if you get to what they call a critical temperature, then suddenly the thing spontaneously becomes a magnet. Something like that. And it jumps up, sort of like a root. And then what happens here, and that is sort of amazing and very surprising, that this magnetization at the temperature is proportional to something like this, 0.3, when the temperature is smaller than the critical temperature. So it's not amazing that it behaves like a root. That can happen. But what is amazing, if you take some other material, which has the potential to turn a magnet, and there are many of them, this whole class of these materials, which show the same behavior. You start to cool it down, and suddenly it becomes a magnet. So there are many of those things. And then the amazing thing is that for each material you choose, the critical temperature will change. And the critical temperature depends on the material. But then this number is always the same. So there is some sort of a universal. There is some universality. I should not make it too much a mess. And so what you see is clearly there is a phase transition here, so from not magnet to magnet. And apparently, there is some universality. I should write this big, because this is really important. And unbelievably. So what you see is the critical exponent is independent of the material. And the name had this independence. This independence refers to universality, universal, in parameter, the dependence. So this is the point. This is the point. It's not, yeah, let's keep the, this is the picture. This is the picture. There's something going on there. So there's something very special going on in this phase transition. And let me give you a very rough and maybe a very naive idea of what is going on here. And the idea will lead to renormalization. So the study of this introduced renormalization in physics. So let me give you an idea of what is going on there. So you know that if you have a piece of iron, the iron atoms, they are organized. They are sitting in a grid. So it looks something like this. It is a grid of, and we are very naive today. So let's make it two-dimensional picture. So you see something like this. And in each corner, there is an iron atom sitting. And this is under a super microscope if you look in this piece of metal.  So it has a little magnetic direction. That's one ingredient. So there are two parts. Each atom has a spin, a little magnetic direction. And suddenly each atom is shaking. How do you write shaking? It's vibrating. And so if you look just at this one little atom, the guy is sitting there. It has a certain magnetic direction. And the thing is moving. It's like shaking like that. And the shaking, the amount of shaking, or how wild it is shaking, is proportional to the temperature. It's related to the temperature. So the higher the temperature, the thing is shaking like crazy. And you slow the temperature down, then they start to shake less. But then if you look at this neighboring atom, maybe it's spin was like that. And if they are shaking a lot, the spin interactions are very weak. And you can forget about it. But if the shaking gets sort of slow, then they can align. So energetically, less has less energy. So spins bond to a line. But if they are shaking too much, they cannot align. But if you scroll down the temperature, they start to shake less. And suddenly they align. And apparently, if you go to this temperature, the shaking became so weak, it's almost nothing, that all these guys are aligned and all of them are pointing in the same direction. And then the thing becomes a magnet. So low enough temperature. So you get alignment of the spins when the shaking, shaking, or temperature gets low. So that is the picture. So now how do you describe something like this? And now we are getting to the renormalization thing. So the picture is something like this. We have our grid. And here we have our atoms. And they have their spins. And the way they model this is as a random process. And the model is random. And the spins are assigned randomly. But of course, if you remember your probability courses, there is something like an independent random process. And so that is the way they assign the spin here is completely independent of the spin here. And of course, that's not true. So this random assignment has to involve some interaction between the pieces. There has to be some interaction. And so this is not at all an IID process. There is some dependence in it. And that makes it complicated. If there would be no dependence, then you would get sort of a central limit of the theorem behavior, like something like the bell curve. But now the claim is that once you get to this phase transition, there is one universal law which describes this random assignment of spins. And so at phase transition, there is a universal law assigning the spins. And now this universal law, a universal in the sense that if you model iron, you get something. If you model another atom, you would get still the same random process. And this universal law and this law, this universal law, is a fixed point. You will not be surprised. It's a fixed point of renormalization. So let me explain you what this renormalization is. Because this is where historically our story began. So what you can do is take these spins and let's take four of them. And they have randomly assigned their spin. So there is some common distribution of this. So what they do is they take four of them and they replace it by one spin. So they group them together and replace these four spins and the distribution of those four spins to a distribution of a bigger piece. This consists, if you look inside, you would see this picture. And this operation is called renormalization. And this acts on the space of distributions. So there's a law describing this. You block and you will get a new distribution. And renormalization is acting there. And now the claim is that this process has a fixed point. And with whatever material you begin, you will see that the statistics on small scale is controlled by this one and the same universal distribution. OK. So this was the naive and very simplistic description of what is going on in phase transitions. This is not complete. There's one difference.  So they zoom out. And the physicists, they zoom out. That's the only difference. So somehow, you see, they take this thing and group it. So they zoom out. OK. So this happened in the late 70s. And then something nice happened for us. So there was Coulet, Pierre Coulet, and Charles Trecer, and independently, Feigenbaum. So Feigenbaum was already an established physicist. These were young kids, students, but in physics. And this was in the 70s. So and then this is considered a spectacular story in physics. So they got inspired by how this universality observed in phase transitions. And then they thought, OK, in dynamics, have you also have a phase transition? No chaos. Chaos. So it's a phase transition.  Whatever it's going to be, you have to observe at the boundary of chaos at the transition moment some form of universality. So what they did is, so this is a computer, of course. So what they did is they made the picture I erased. So they took, say, the quadratic family as, let's say, Qt. Qt of x is, I think, 40x1 minus x. So a very specific family, the quadratic family. And they drew the picture. So you see the bifurcation, the first bifurcation moment, the second one, the third one. And this continues like this, up to t infinity. So in this family, they calculated this bifurcation moment. And then they observed. And that is the equivalent. Oh, you remember, I erased it, there was the 0.3. So this is what they observed is. The first thing is that this tn converges to the limiting value. That's like the critical temperature where the phase transition happens. And they observed that this rate, this is exponential. And the rate is like 1 over 4.669 and some other numbers. And then they did the same picture for another family. And maybe something like something with sinus, like something like t sinus pi x or something. And for this thing, they made the same picture. And they measured the moments where the bifurcation happens. And they observed that it converts again exponentially with exactly the same rate. So this is what they call parameter universality. And there is some sort of geometric organization in the bifurcation space, in how the parameters move. There's something going on. And that is equivalent to what you should compare that like if you had the magnetization. And here, there was this power x to the 0.3. And so this 0.3, you should compare to that one. It is something which is universal in the parameter dependence. So then they did another measurement. And remember that if you take the attracting fixed point, then at some point, it flips and becomes a pair two attractor. But this orbit continues. So it's like somewhere here, p1. So and then you look at the pair two attractor, and you look at like this one. And when it bifurcates, it still continues. And this you call p2. And these are the periodic points which are survived. And you continue like this. This one continues p3. And so this has period 1. This has period 2. This has period 4. And it continues like that. So this sequence pn converging to some p infinity. This is the positions of the periodic orbits. And what they observed is that this pn converges exponentially fast to p infinity. And the rate is 1 over 2.6, I think 6 squared. And then, so this day first, they did it for the quadratic family. But they were hoping for universality because they believe in physics and they believe. So they did it for the sinus family. They did the same picture. And they calculated the positions of this periodic point. And they saw again that it converges exponentially fast. And the rate is the same number. And so this is phase space universality. Universality of dynamical geometry. And so this starts to smell a little bit. Remember yesterday, we constructed the conjugation between a circle diffeomorphism and a rigid rotation. And it had something to do with the cutting place. Had to be on the same place each time. And that created rigidity. And so the fact that these periodic orbits, we will come to that later, the fact that these periodic orbits are at a very specific place, asymptotically, is an indication that we are going to get rigidity, something equivalent to the Hermann theorem. OK. So these are the observations. And then, the Coulain-Tréssère, they were physicists and quantum physics. So they know they have to look for a fixed point. Somewhere in some space, there exists a fixed point. And that fixed point is describing what is going on here. And we are going to discuss precisely how that works. So for today, what we will do is we will discuss what renormalization is in this context. And then tomorrow, we will see how it connects these dynamics. And so of course, it has something to do with spirit 2. So let me say how it works. So let's be a little bit precise. So our space of maps are unimodal maps. So they look something like that. There's some critical point here. And let's say that this behaves, this is 0, 1. This is 0, 1. And let's say that it behaves as a C3 map. And let's say that at this point, the second derivative is negative. So the critical point is not degenerate. It has a nice sort of quadratic shape on small scale. So this is the space on which we are going to look at renormalization. So inside here, there is a space of renormalizable maps. And let me tell you by this picture what these maps are. I shouldn't screw up. So a renormalizable map. So renormalizable. So it looks something like this. It looks something like that. So this goes here. And then this goes here. And the critical point is here. So you look at this interval and you look at this interval. So what this is, you have here the critical point. Then this point is the critical value, which is value 1. Then this point is the second critical value. And then this is the third critical value. And then this is the fourth critical value. And the map is renormalizable if these four critical values are in this combinatorial situation. So the third here and the fourth between the critical point and the third. So these are renormalizable maps. So what is important about them is that this interval is mapped to that one. So this one here is mapped to that one. You see it happen immediately. And this one is mapped back. So you see here an interval of period 2. So this was our map F. Let's make a picture of the second iterate. So you remember, if you renormalize, you have to choose a domain and you have to look at the first return map. That's always the scheme. But the delicate thing is, which domain do you choose? So what we are going to do here is we are going to choose this interval as our domain of renormalization. So let's call this v. We might use it natively. So there's a u-interval and there's a v-interval. So let me draw them here. Here is u. And here is v. And you can sort of check easily that the picture of the second iterate looks something like that. No, that doesn't look right. No, no, no. That the picture looks something like that. Sorry. You can check it. So now you see that this thing here is the first return map to u. So this is our renormalization. So rescale. So you rescale this one. And you get a new map. And that you call the renormalization of f. And so it is a r of f is a rescaled version of the second iterate restricted to u. So it's all everything we used to do. OK. I'm not sure how much time do we have. You know, let me finish with the difficult picture. So it's about how this operator behaves. And you will not be surprised. So there's a theorem. And it says that r has a unique fixed point. And that is some unimodal map. And it has some properties. It's an hyperbolic fixed point. The unstable manifold as dimension is 1. And the stable manifold has code dimension 1. And what is important, that the stable manifold is exactly the class of infinitely renormalizable maps. And this is a topological class. Because having this order of critical points is a topological condition. And so you know that this is, again, we identify, like in the rotation case the other day, we identify the infinite renormalizable guys, which is a topological class with a stable manifold of renormalization. 15 minutes. Oh, it feels like Lundstein. You know, OK. 15 minutes. Wow. And this insight of Coulain, Treser, and Feigenbaum happened in the late 70s, like 77, 78. And now we have this theorem. It took a long time. The proof of this theorem has a long history. And let me name some players behind this theorem. So of course, Coulain, and Treser, and Feigenbaum, you are behind this picture. Oh, I should make the picture. And so it looks something like what we saw the other time. It is something like that. Stable manifold and the unstable manifold. And so Coulain and Treser, Feigenbaum, are behind conjecturing this picture and using this picture to understand the boundary of chaos. So Bidendurg is quite a list of people who helped establishing this in some little parts, some very large parts. So the first proof of this theorem was by Landford and Epstein. This was a computer-assisted proof. So then Sullivan came in. And he introduced Teichmuller's theory behind this renormalization operator. And he got convergence. And then there was MacMullen, who understood the expansion of renormalization and many other things. Then I helped in constructing fixed points of renormalization and periodic points of renormalization. And then the final of the present picture and a very beautiful picture is obtained by Avila and Lubitsch. So this is the place where you should go for understanding this theorem. No, no, no, no. You can answer that yourself. And you know, that is a very good, that is a very, that's a such to the point question. So let me first answer why it is no. So if you start with a polynomial, then you have to look at the second iterate. So the degree becomes like double. If you have a polynomial, you put in another polynomial. The degree gets much larger. So this fixed point of renormalization is definitely not a polynomial. It is an analytic function. It is analytic. But it is quite a mysterious thing. And tomorrow when we co-discuss the geometry of the attractor of this thing, you will see that it is a very rich thing and something you can clearly not make by hand. It is a very rich object. It's very specific. And that shows that in this context of unimodal maps, the situation is going to be painful. Because remember, when we did the circle diffeomorphisms, like the first observation was that the attractor are rotations. So you know, like the first observation, you know already the limit of renormalization. And these are just affine maps, so something very simple. And the fact that the limits are so simple allows you to develop a renormalization theory for circle diffeomorphisms with elementary tools. So in this context, the limits are complicated object. There are some analytic functions. So this is an indication that this is going to be quite a story. And I think this is, I would say, among the most beautiful pieces in dynamics. So the next part would be to have exactly what we did in circle diffeomorphisms, to use this picture to go through all the parts of dynamics. And you will see again that we can understand everything in terms of renormalization. And the most interesting part, so let's go home. And let's go have lunch, and then tomorrow we start with the blobs. It's come here, so let's go home. Bye.