So this is the idea, clearly, we're most concerned about opportunity, right? If you're in this room, it's because you wanna make the best of your life. You're not really concerned with, hey, how do I not have a complete disaster of a life? But nevertheless, we are concerned with what happens in the worst case scenario, right? What happens if I reach a certain age and I just can't work, I can't support myself, what's gonna happen to me? And the whole idea behind social security is, all right, you have a platform of supporting yourself. Now, this is kind of a weird notion of economic security, first of all. So economic security in any rational world, so the way that it's usually treated is, how do you guarantee that you have resources even if you haven't produced or saved for them? That's the idea behind this idea of economic security, but that's not a realistic idea of economic security because the fact is the government can't guarantee economic security. It can't guarantee that those resources are there. It can only guarantee that, hey, if people produce them, we're gonna take enough from them to guarantee that you'll have some floor of protection. Any rational idea, any rational conception of economic security is really going to be that if you do create something, if you do invest it, if you do save, nobody can come and arbitrarily take it away. Now, that kind of conception of economic security doesn't really pretend well for the welfare state because of course the basic idea there is the government gets to say, hey, whatever we wanna take from you, if we think it's in the public interest, we're gonna do it. I mean, how economically secured do we feel knowing that we're $205 trillion in debt, which by the way, does anybody have $205 trillion on them? Because if you do, let's go out for drinks after this. No, it's more wealth than exists in the entire world, so that's kind of a problem. But let's look even deeper. You might say that's kind of nice wordplay, Don, but what happens if I'm old and I can't support myself? Support myself. So the kind of argument that's usually given is that if we don't have something like Social Security and Medicare, the elderly will starve in the streets. Now, there's an interesting test case for this. What happens when people don't have old age welfare programs? And it's called America for every year until 1935, which by the way, if you can do the math, is longer than we've had Social Security since 1935. Now, did the elderly starve in the streets? And it turns out that amazingly no. Amazingly, people figured out a way to live without handouts for over 100 years, and it's pretty fantastic when you think about it. They were very entrepreneurial in their whole approach to life. And so I'm just gonna go through some of the mechanisms that they took in order to not, as they say, starve in the streets. So first of all, people worked. It's kind of funny today where we think of like, ah, work's a real burden. But actually, the people pushing for a welfare state complained for many years because the elderly, those darned people refused to retire. The fact is that they found their lives more meaningful, more meaningful, more valuable, more enjoyable with work than without it. And they saved. Who has a savings of more than 0%? If you do, you are the 1% because most Americans don't. People at that era saved between 1 eighth and 1 ninth of their income, which is, I mean, that's impressive. Credit, obviously, if you couldn't support yourself, various forms of credit were available. Turning to friends, family, community, informal ways of helping people, the number one, does anybody know what the number one thing people did? So first of all, before industrialization, before people were really dealing with retirement, what was the number one retirement plan? Anybody know? Close, that comes later. The first retirement plan was actually death. We worked and then you died and then you didn't have a retirement plan. But now we have industrialization, people are living longer and yeah, that's exactly what they did. The number one thing people did is they lived with their kids. And when you think about it, social security is in effect like an indirect way of us all taking care of other people's grandparents. Like it's in effect, the government wheels over, somebody else's grandma shoves her into our lawn and says, here, you watch her for a while. But no, what most people did is they took care of their own parents if they needed it. There were also things called mutual aid societies. Now, anybody heard of like an elk's lodge or a moose lodge or something? I'd love to see these re-emerge, they're kind of cool. But these are in effect the descendants of mutual aid societies. And these were community groups in effect where you would pay your monthly membership due and you would be entitled to certain kinds of then support from the club. And it might be things like unemployment insurance. It might be things like private health care. You'd have a lodge doctor, you could see. Or it could be a retirement thing. They had often very, they had retirement homes where if you couldn't support yourself, you would go and you'd be able to stay at the society's retirement lodge. And finally, there was a charity, private charity, which in this era was abundant. Again, we're talking about pre-1935. And people per capita were giving more to charity than they did today. And so what were the results of this? Was it people starving in the streets? No, the fact is that what was happening with immigration were people fleeing the shores of America, trying to get away from this wretched area where people were starving in the streets and going to those countries that had welfare programs. Anybody know what was going on with immigration at this time? This is the biggest wave of immigration to America. Indeed, the people pushing for welfare state programs, which they started doing really in the 1880s, model after Bismarck's Germany, they said, well, we don't have to worry about old people in America. They're doing just fine. We have to worry about poor people. So they spent 30 years trying to create welfare programs for poor people.  Let's focus on old people because most of us one day aspire to being old. So let's do that. And then they eventually succeeded in 1935, as we talked about. But the best statistics we have in terms of what's going on then, and we don't have great statistics, so take these with a bit grain of salt. But the best ones we have tell us that by 1935, only 8% of the elderly need any form of formal charity, whether it's private or whether it's the kind of minuscule government level programs going on. And we're taking a different look of something that we've been diving into in the last couple of years. Now, we have Don Watkins right here. How social security is sabotaging the land of self-reliance. Bring it on home. Good stuff, man. Thanks, man. Yeah, you got it. By the way, if you can do the math, it's longer than we've had social security since 1935. Now, did the elderly starve in the streets? And it turns out that amazingly no. Amazingly, people figured out a way to live without handouts for over 100 years, and it's pretty fantastic when you think about it. They were very entrepreneurial in their whole approach to life. And so I'm just gonna go through some of the mechanisms that they took in order to not, as they say, starve in the streets. So first of all, people worked. It's kind of funny today where we think of like, ah, work's a real burden. But actually, the people pushing for a welfare state complained for many years because the elderly, those darned people refused to retire. The fact is that they found their lives more meaningful, more meaningful, more valuable, more enjoyable with work than without it.  And this is looking at the welfare of the group rather than the well-being of the individual, and says, what's good for, quote, society as a whole? How should we distribute society's resources in a fair way? I think the better way, the more rational way to think about political issues is to focus on the individual. And I think if you think about what's good for the individual, you come to a very different conclusion. So the two essentials in my judgment of what makes us happy and successful life, and I think this came through in all the talks today, is thought and effort. This is Isaac Newton and Steve Jobs. It's...


Alright, what's up everyone? Um, it is currently day 23 in quarantine and obviously I'm occupying my time by sitting alone in my room, talking to my camera, which somehow seems a little bit more normal now, given the current world situation. Anyways, I realized I've been, you know, somewhat absent from the YouTube space for the past school year. Um, and you know, I've been getting a little sentimental over the couple of days, you know, scrolling through my camera roll, seeing all the memories of a year gone by, and you know, I thought I would talk about it. One, so I can update all of you guys that are curious about what I do with my life when I'm not on YouTube. And two, I kind of want to reflect back on the memories and vent a little bit given everything that's going on. Obviously this whole situation is affecting everyone differently, so I really do hope you're staying safe and healthy and your families are staying safe and healthy as well. Um, but I thought I would reflect on some of my personal good memories from this year. So one of the biggest changes for my junior year of college was the fact that I moved off campus. I am no longer living in the hallowed halls and cramped dorm rooms of Timothy Dwight College. Now I can talk about that whole decision process of moving off campus if any of you guys are actually interested, but I'm now living in a house with eight of my best friends and there's honestly no complaints. And when I say off campus, take that with a grain of salt because I'm literally across the street like Davenport College is right there. I can walk out on my front porch and see the bookstore. So I'm pretty much pretty much on campus still just not in a Yale sponsored dorm.  Um, it's provided me a lot of flexibility that the Yale dorms haven't and I've also been able to save a ton of money because I've been cooking most of my own meals. Although the whole residential college system is great and I pride Yale and kind of upholding those community values and everything. Um, it does have its problems. I if you have friends that aren't in your residential college, it's extremely hard to live with them. Um, and most of my best friends do not happen to be in my residential college. So that was one big change. But next, my junior year was kind of a big year for my development as a computer science student. Specifically this past fall, I took a legendary class at Yale CS323, which is our systems programming computer architecture class. This class has a reputation of being one of the most difficult, hardest time consuming classes that Yale has to offer. And let me tell you it did not disappoint. As I talked about in some of my previous CS related videos, most of Yale's core CS curriculum is taught in the basic programming language of C. So that was fun. I think I logged 43 hours of time for the first problem set. Um, that's a little over a work week for something that I had about 12 days to do. It was definitely a shock to the system. The class was only five problem sets, but most of them took at least 25 hours, if not more, which is just a heinous amount of time in general. Looking back on it, though, I do think I enjoyed it. We had some really cool problem sets, everything from implementing some basic Linux commands like make or history, which I'm sure you've used before, if you've ever coded and see. And then we even built a low level latech. And the final two P sets were implementing the limples of Welch compression algorithm and then building a bash shell from scratch, like an entire bash shell, which yeah, that was fun. This last was the main reason this past semester was a complete blur. So I'm so happy I'm done with that. And speaking of last semester being a blur, it wasn't completely due to CS3-23, but a combination of a lot of different factors. If there was a phrase I could use to describe last semester would be work hard, play hard. It was a semester of flip flopping between a week of all-nighters where I'm staring at a computer screen for hours looking at seg faults and bugs that just won't go away. And then the next week I'd be, you know, popping off enjoying Yale social life and kind of making the most of that on a week where I don't have as heavy a workload. I feel like as a junior my core group of friends finally solidified and there was no more wandering around looking for things to do on the weekend. We always had plans and a lot of times they're a lot more low-key, which you know as a junior I finally became okay with. I settled down for a bit and it was nice. Looking back on it it's hard to pick out specific weeks or weekends just because it was there was so much to do and you know it was like it really was a blur and I'm okay with that. As I moved into my junior spring this semester I pretty much finished most of the computer science major. I just have three electives and a thesis left so I made it made it through all like the more difficult core requirements. So I finally had the opportunity to pretty much take whatever I wanted and let me tell you I took full advantage of that. I think this junior spring semester was the first time I could like definitively say that I genuinely enjoy and love every single one of the classes that I'm taking. Like for example I'm taking a medical software design class where I'm working on an iPhone app that uses a machine learning algorithm to make predictions of urinary events and patients with neurogenic illnesses that are struggling with urinary incontinence. So you know bedwetting basically machine learning for bedwetting that's that's pretty cool. I'm taking a cybersecurity seminar in the Global Affairs Department that's cross-listed with the Jackson Institute and law school where I just spend two hours every Wednesday morning debating the effects of cyber warfare and cyber warfare policy on national security and strategy which is just fascinating. In my wireless technologies class my final project is a drone that acts as a Wi-Fi repeater so you can so I can extend the Wi-Fi range of my own house using a drone and connect to the Wi-Fi coming from a drone. I mean it sounds crazy just talking about all this cool stuff I've been doing this semester in my classes but it's even been more fun implementing it. If you guys are curious about any of that I'd love to make specific videos about the projects I've been working on especially since I have so much time on my hands these days. With all these cool classes I also had the opportunity to meet so many new people and for some new relationships that I couldn't have even dreamed of freshman year. This year has been an absolute ride and that's kind of one of the main reasons I'm I'm disappointed that it was it was cut short. Obviously I can't even imagine what the seniors are going through because being a second semester senior all the all the senior specific events that are going on graduation senior week all of that gone in addition to the fact that we were on spring break when school was called so we didn't even get a chance to say our goodbyes. Fortunately I'm still in New Haven off campus self-isolating writing out this pandemic together with a couple of my friends which which has been great. We're just trying to live up the rest of the semester you know even though we're here at Yale it doesn't feel like Yale because I feel like the people that the people make the school and when there's no one here it's just it's just so sad. I can't be more thankful for all the people that I've met the experiences that I've had this year at Yale and although the semester was cut short I know the energy on campus is gonna be unheard of when we hopefully return in the fall. So that was a brief recap of my junior year at Yale University which it's not quite done yet obviously I'm still doing online classes and finishing up those we have a couple more weeks left of that but it's it's crazy to say that I'm three quarters of the way done with college. I just don't know what to think of it it's all gone so fast and it's been such an amazing experience that I it's it's sad it's sad. If anything that I talked about in this video interest you guys you'd like to me see me make a specific video on it obviously I'd love to with all this time and I have a lot of cool stuff that I'm working on that I'm interested in. Obviously I saw all of you guys' comments on the last video I've taken those video ideas into account I think this was one of the suggestions some I'll definitely be putting out some more you know quarantine related stuff trying to stay productive stay busy that's something that I'm personally working on and and struggling with myself. Anyways please drop a like on this video supports me and helps channel a lot more than you think subscribe if you're new and this is the first video that I've seen it's a little on the more mellow side but like I have a lot of other cool videos about Yale school CS magic all the stuff that I'm interested in on this channel that you should definitely check out comment down below any other video suggestions you have anything you liked about this video that you want me to talk about more and with that stay safe stay healthy


Hello and welcome to the Chart of the Week video with me, David Madden. Today's date is Thursday, the 11th of October, 2018, and the time has just gone 12 o'clock for the summer time. This week's Chart of the Week is the Hang Seng, or as we call it on our platform, the Hong Kong 50. The big picture is that the market has been broadly moving lower throughout 2018, and if you take a bit of a closer look, we can see that since about May, the Hang Seng has been a classic example of the downward trend, a series of lower lows and lower highs. In fact, today it actually fell back to a level not seen since May 2017, so give the indication of how bearish sentiment is on the predicted market. Taking a look at the MACD indicator, the MACD histogram down here, we can see that as the markets are pushing lower, there's been a steady increase in negative momentum. So the MACD indicator confirms the downward move that we're seeing on the market itself. Essentially, the momentum is with the sellers at the moment. If you do continue to drive lower from here, we could look at taking out 25,000, and if you go south of 25,000, the next level to keep on that for will be this area here, which comes to play at 24,334, like I was saying, it's a level not seen since the early May 2017. If you do see any kind of pushes higher in the market, resistance may come into play in around this area here, 27,000, it's a big psychological number, so it could be an area of potential resistance. Notice how we have a series of lower lows and lower highs, so if you do have another size of a bounce back, 27,000 could be an area potentially of resistance. If you do head above north of 27,000, the next area to keep on that for will be this area here, the late high of September, which comes to play just north of 28,000, 28,030, that was the September high, and if you go beyond that, we could be looking at it up towards the August high, which comes to play at 28,583, and if you take off that, then the entire kind of recent downward trend could then be going to cause an index iteration, and if you go north of that, of the August high, we could really get back to the July high, which comes to play just north of 29,500, sorry, 29,053, if you are going to be trading the Hangsang, please keep in mind overnight China have trade figures out, and this is going to be important because there's obviously quite a negative trading relationship between the US and China at the moment. Is there any indication that China continues to widen or President Trump feels that China continues to take advantage of the current trading relations that they have? We could have additional fears about potentially new tariffs being announced from President Trump. Before I go, if you have any comments to make on this video or any of the other videos we've made here at CMC Markets, please feel free to leave a review on good reviews, and that's all for me this week. Thank you very much.


Greetings. Let us start our morning session with the talk of Arthur Avila from CNRS, Poincaré and renormalization series. Poincaré series and renormalization. Thanks for the opportunity to talk in this special occasion. I was a bit concerned about how to make the presentation and I hesitated a little bit finally. I waited a little bit to mention to show something of Poincaré and just kind of it would be easier to talk a little bit about renormalization thoughts so that you are not too tired when I get to it and in a few minutes, not longer Poincaré would come in. So the basic objective that I am going to consider, I am not going to introduce renormalization in general but just the concept of renormalization fixed point will be necessary in this talk. So for us, it should be just an all-morph function, f, so f is the all-morph function satisfying a function equation which is, so you have some all-morph function f in some place that will come to that. So some iterate of it applied to some, there is some kind of scaling factor lambda. But it's given just as f is lambda. So if you look at, so lambda is some factor less than one. So if you want to look at, you have some iterate which is kind of in a smaller scale of that. It is the same as the first iterate in the original scale then you rescaled it. And then, so this is just some all-morph function. Of course you have to define the domain. It's not going to be considered the maximum analytic extension that you could have of such a function but we assume that it has some specific configuration somewhere. So there is a special point that's zero since the scaling is around zero. So there's a critical point that's zero. And there is a, the transformation takes some disk, some topological disk around zero and goes to a bigger disk as a degree two map. So there's a critical point of degree two here and then there is a degree two proper map. So you have discovery.  And this would be a real normalization fixed point. So you are going to be, you may ask whether such a real normalization fixed point exists and it's possible to show that for every p bigger equal than two, there is at least one real normalization fixed point. For two, there is a p equals two, there is exactly one. And for a larger p model conform to a fine change of coordinates, for larger ones, for all the z. So this is a functional equation that lambda is some scaling factor that's fixed for some. So the transformation in a small scale behaves exactly as the same under this kind of thing. So real normalization is the operation that makes this rescaling thing and this iteration. This says that the real normalization is exactly the same as the original map. So what's interesting, those are very interesting for several reasons. We can try to understand what's the repercussion of all these kind of copies of the own dynamics of large scales that appear in smaller scales and this gives a very rich picture. Traditionally, everything started, white people started analyzing this was because they just discovered the arising connection with some universality phenomena that was first seen in real dynamics. And those are going to be complex extensions of those real dynamics, which is the Fegenbaum-Coulet-Resset phenomena. That's kind of, is explained in this picture that maybe some of you have seen, which depicts in a family, here is a family of quadratic maps with some parameterization. In the horizontal x you have the parameter and in the vertical x you have the actual phase space, the space where you iterate and it shows the, the attractor of the system, where the orbits of the system, as if you iterate they are going and originally in the left side you have a very simple attractor, it's just a fixed point. Then it gets kind of mildly more complicated, becomes a period of two. There is a tractor that has two points and then it gets more and more complicated and then it's very complicated after some, there's some initialists one, two, four and so on. And this is a finite set and suddenly it's kind of much more complicated. This moment where it's kind of becomes infinitely complicated for the first time is related, it's something that's corresponding to a dynamic, that's a deformation of a renormalization fixed point and the behavior, the quantitative features of how you approach this special parameter are related to precise the properties of the renormalization fixed point, in this case of renormalization fixed point of period two. Okay, so now, so this is in real dynamics, to understand those real dynamics you are led to understand the complex dynamical systems or you can be interested just by themselves, this would lead to some pictures of the Mandelbrot set that I'm not going to show, but if you look at the complex dynamics you are going to be interested on, so here you have map, so the domain is not the same as the range, so there are points that escape and then you cannot iterate anymore. So what you define are the set, the actual dynamics happens at the set of points which are some complicated set of points here that I'm going to show soon, are the points that never escape under iteration. So formally you have this kind of Julia set that's going to be the intersection of the pre-image of the, so you have domains u and v in the pre-image of u. So you have this Julia set that's where the actual dynamics is going to take place for all times. And so I have pictures of this kind of close to the Julia set for a renormalization fixed point or at least as close as I got from it. Here's one picture of this renormalization fixed point at some scale. So it looks like this, actually you only see a little bit from the outside. The actual thing should be in black in that picture, but the colors get kind of changed as you get closer to that. So this is one picture that you get and then it's some scale and then if you look at the next scale you get another picture in a smaller scale. You get a picture like this, maybe I don't do the whole procedure of getting a full slide but you get some picture like this with more complications and you see the colors getting very approaching there. And then you look at the next scale. Oops, I jumped from one. So this should be the intermediate scale, so there is a kind of still a little bit tame thing. And the other one was the slightly smaller scale, not this one, this is the final scale where it's already very, very complicated. So this behavior that you saw that as you kind of zoom in, you get more and more complications, it's normal because when you get closer to a critical point, you actually have interaction of the dynamics in several different scales. So you have copies of the dynamics, of the same dynamics in any scale because you can kind of repeat this equation for different lambda. So at every lambda n you are going to have a new copy of your dynamic. So you have some kind of interaction between the dynamic consistent at all scales. And this reflects, for instance, in a theorem of McMullen that says that when you get, you're going close to zero, it's getting more and more complicated, meaning that if you amplify the Julia set and try to see in a little bow around zero, you're going to get things that get more and more dense in a complex plane. So as you do amplification, you get essentially the Julia set starts spreading everywhere. So this kind of complication is the first hint of the complicated geometry near zero for this kind of dynamics and it comes from this interaction of dynamics in different scales. Now the basic questions that are going to be interesting here is kind of very natural, is what's going to be the geometry of the set, the set that comes from kind of non-scaping dynamics or which actually is a dynamic that contains several copies of itself, and the most basic questions that you get once you get this kind of fractal set is what's the back measure first? Is it possibly back measure? And what's its dimension? Is it possibly of dimension two since it's very complicated and still with the back measure zero or maybe it has dimension less than two? So those are kind of very basic questions once you have a fractal set like this and that's what we try to... And it's when we ask this question that you get to the other part of the talk that we get to have to understand what the Poincare series of those maps and that's where we come to. So let's see what's Poincare series but starting from how they were introduced. So I went to get... It's not very good quality, I took some scum that was read in the internet from the archives but so anyway, so this is the... I tried to look where it's defined, it's in the first paper that Poincare did about flux and functions. So it's a paper that precedes the paper about flux and groups that he did because he was interested in flux and groups because of those flux and functions and not the other way around. So anyway, here is... In this article he defined those Poincare series and you see here the motivation, so that's why I put here and you see that it is... that he's looking for functions that are analogous to elliptic functions with the goal of integrating linear differential equations with algebraic coefficients. And to do this he comes to the need to construct these function functions that he's going to define and what are those. So we are going to see... So here's the motivation, why he's going to be doing this and in particular why he's doing the Poincare series and I give you... In the same article, in the third page now, he's going to get... So in this third article, in this third page, he just comes to this, so those flux and functions that he wants to define, he defines quickly as being the functions. So you are going to have a group of actions, so you have the Poincare disk and you have a group, discrete group of symmetries. So you have the Poincare disk, it is a model for the hyperbolic plane and you consider hyperbolic motions in this plane and that generate a discrete group and he's looking, so you have this kind of gamma, a group of symmetries and what he's going to call flux and functions are functions in D that are invariant under gamma. So that's just what is a flux and function. So just f of gamma z is equal to f of z. That's what he's going to define. It's a little bit... It's in the first page of the article that he does that and here he defines something else which is a function that he calls a data function, data function function which is some kind of function that it will transform, it will not be invariant but it will transform according to some rule, right, which is going, well, the rule is written there, which is, we recognize the rule as being some rule, so actually the data function transforms as a differential form. So it involves some power of the derivative and it is what is required. So a flux and function is a function that's defined on the quotient of the, by the group and a data function function will be something that of weight, so the data function function has a weight that's an integer m and it should be something that transforms that defines a holomorphic form or meromorphic form in the quotient surface. And then this kind of interesting situation, natural, we would be interested perhaps forced on the geometry of such surface, his interest on the function that you can define there and just to recall that it's certainly very interesting to consider because this kind of functions because actually almost all Riemann's surface arise as quotient by a Fox young group, so that there's the uniformization theorem that tells us that and so he's describing actually something very general at the level of Riemann's surface over there. So the points that just after, so that's just the definition of data function and then in the next paragraph it says how to construct a function that has those properties and he writes that formula which is a Poincare series. So that's what appears as a Poincare series. It's kind of summing, he considers a sum over the objects of the group. So he starts with some function a, I write in a modern language and then you apply elements of the group. I write like this and then you divide by some power n and he sums over all elements of the group and when you apply some element of the group to z then you see how this will transform and immediately it comes the property of how it's changed under those coordinates. So that's the way he defines a Poincare series and a little bit after, so this kind of how to construct those kind of invariant forms and to recover the Fox young functions he takes the quotient of two Poincare series of the same weight. So that's a way to get from all this to finally get some functions also. So that's what it's doing. I'm not going to continue very much on how those things develop but he looked at then, of course has to show that things are actually converging. So he writes the series, the group's infinite there's a question of whether this converges or not and it turns out that it's easy to see that those d gamma of z it's a sequence that tends to increase so it's bigger than one except for finite many values of finite many gamma. So the question is the lower the n the easier the convergence and he considers the condition n bigger or equal than two and he's doing everything here allomorphic functions so everything's restricted to integers it's an integer and he proves convergence in this case he actually proves convergence of the absolute Poincare series which is what you get if you put absolute values here so this would be the absolute Poincare series same thing so we get something like this so he proves this what he gets when you do this at this level and let's put here m equals two in the other case that he considers it is in this case we're coming from a quadratic differential to an aria form and basically the argument for convergence is just saying that the Poincare disk has in the Euclidean metric it has finite aria so that's kind of not very difficult to do to establish this convergence so here is the absolute so here's basically what how we would understand this and okay so that much for his original then he did many things so he wrote several articles on this so that later on he decides that he actually has to understand a little bit better the groups in order to say a lot of things about the functions he writes a paper about the theory of the Teohidi group Fugtian and which he says explicitly there that motivation there is preparation for his next article that's about those function functions so he develops this quite lengthy but I'm going to jump over here well first just the point is that Poincare series was a way to get some kind of olimorph function and schizo-miromorph that transform in appropriate way so in this case by this kind of power of the derivative and then 90 years later let's keep all this a Patterson he considered the problem of creating other geometric some other not any more functions but measures that transform in appropriate way also and he wanted to construct this associate to a Fugtian group and in order to construct those functions he came with using again a Poincare series so you're not going to be any more interested on olimorph objects but only on so you're going to be interested now not on this kind of objects so you're going to consider absolute Poincare series only so what he considers is so we have a Fugtian group and you see let me show a picture of a Fugtian group I just take some of those pictures from Wikipedia they're not kind of fantastic so here's one simple one so here this very simple group that's acting on the Poincare disk and it gives a little bit notion of the orb so as you iterate you more or less follow this kind of complicated network and what he considers is so everything kind of is going to infinity as you iterate so he wants to put a measure on the circle at infinity and to do this he starts putting point masses at an orbit with certain weights so he puts point mass at a fixed orbit with weights that are related to the Poincare series I would take just the Poincare series weights he for some reason he considers a different weight because he's looking at a slightly different problem but let's put the Poincare series weight and well then you get a measure that's defined in the disk he wants a measure that's defined at infinity then he considers so of course you have a choice of weight and now since now you're considering point mass at a fixed orbit so weight of those will be some d-gamma of so weight at gamma z you put let's see 1 over d-gamma z to the m where now m can be real he allows real weights so you put this and he gets interest in in order to get something that escapes to infinity he considers the critical exponent so now so he considers the absolute Poincare series again there's one here so it's kind of more similar so you look at this Poincare series and you look at the moment where it's kind of barely infinite so it can be finite it can be infinite so let's look at the moment where it's kind of a little bit undecided so choose m critical so that for so let's call it let's call this delta so that for n bigger than delta you have convergence and for m less than delta you have divergence so then for m bigger than delta you get some kind of finite mass here that you can normalize so then you can normalize if the Poincare series the absolute Poincare series converges you have a measure on the disk now as you approach the undecided parameter those measures sometimes with a little bit of trickery you can make those measures escape to infinity and give rise to something that's supported at the boundary so at the critical exponent exponent you'll get something get measure at infinity that's how Paterson did there are lots of things to discuss here but the point is this was quickly generalized to the setting of Kleinian groups by Sullivan went to high dimensions considered so Fox and groups arise as kind of hyperbolic motions in an hyperbolic plane and now you can consider an hyperbolic space for instance so if you consider the same so high dimensional case so if you consider this what you get is, well you can do it in dimension but if you consider in dimension three hyperbolic space there's something that's called the Kleinian group which is this I took the opportunity to look here so here's the paper where Kleinian groups are kind of named so basically what you have is that you have Fox and groups that are kind of there are motions that have the form so the hyperbolic symmetries have always this form and he considered since it was the situation where it had to preserve the unit disk and now you remove this restriction the circle is no longer some condition relative and fundamental so now the circle is not anymore there and you consider all transformations of the Riemann sphere and you consider again all possible motions that you have and since the Riemann sphere is the boundary now of the hyperbolic space this corresponds precisely to that one dimension up from what is the Fox and case so he extended this theory of Patterson to some kind of considerable there's considerable and this is something that was developed through many years to kind of understand as well so basically you're going to be interested on how this kind sorry I should have read my notes that I didn't say why what you can do with those measures so basically the point was already in Patterson that by analyzing those measures you get to understand things like the dimension of the limit set so the orbits here they are going to some limit set at infinity and I saw some in this drawing it seemed to have the limit set that was everything but it could be something else and you get in general some kind of fractal set at infinity and you can ask exactly those questions of dimension and those questions they are going to be related to study this it's interesting to consider precisely those geometric measures they give you some information about the behavior there and particularly the critical exponent is a way to compute dimension at least in several cases so what he so but then what happens in the case of Kleinian groups how you actually say something about the critical exponent and about those measures turns out that it depends a lot on the geometry of the tree manifold so if you go one dimension up you are going to have now the quotient of the hyperbolic space by the Kleinian group will give rise to a tree manifold hyperbolic tree manifold and it is the theorems that are shown that depending on the geometry of the tree manifold you are going to get different properties of the critical exponent and of the geometric measures so the first results are kind of by Salivan and Takya which give an estimate on the point ahead so I am in the Kleinian group case well, separately and they show that the critical exponent was less than 2 when the surface the manifold of simple in simple case which are the geometrically finite case I can't give very much idea of how the geometry comes in but frequently it comes down to properties of Brownian motion in the recurrence properties of Brownian motion in this manifold that are going to be exploited in the kinds of results the one that I really looked at a little bit more which is kind of closer to what we have here is the theorem about what happened in the geometrically infinite case so this is kind of relatively recent because it combines many things but we know now so geometrically infinite case actually you know that delta is equal to 2 the critical exponent so this is kind of a very complicated result that's actually due to B-Shop Jones plus the Alphos Conjecture it's kind of a funny thing I said that the critical exponent is related to the dimension and basically it is but there is a problem when the Lebesgue measure is of the limit set is positive so the Alphos Conjecture is covered by I think and and this is from 6 years ago or something like this so this Alphos Conjecture is the statement that the Lebesgue measure of the limit set is zero so we have this fractal set and to establish first that the Lebesgue measure is zero and then B-Shop Jones had shown that in all cases where the Lebesgue measure was two and this kind of so here they covered this zero and basically the point of this argument where the Lebesgue measure equals zero appears is because it ensures that some well basically the problem that should be created by positive Lebesgue measure would be that the Brownian motion would run away through the geometrically infinite end of the manifold if this happens it should create and it should create some problems with the critical exponent so but they block this this possibility because of this Alphos Conjecture and you can say that you understand essentially everything about at least the critical exponent in the case of Kleinian groups so that's kind of the situation now you ask what about the renormalization fixate points that were in in the beginning so there is something that's called Sullivan's dictionary that tries to create a parallel between Kleinian groups and so Kleinian groups are some kind of homomorphic dynamics of course but of a group on one hand each element of the dynamics are relatively simple because they may be transformation so it's not very complicated when you iterate but you have a whole group of them now if you have a single homomorphic map that's actually really nonlinear then just by iterating it you already get lots of complexity in the beginning so the point was that there is this dictionary that kind of tells you some equivalence between both theories and that sometimes they send to the level of proofs you can kind of translate one result to the other and hope that you prove some result in one part and then you translate it back to the other case and now you can try to get lots of theorems just by filling lines in this dictionary and it works very well in some case like so we won't talk about he just translated got immediately several interesting parts but in some other case it's less clear so you can try to look at for normalization fixed points there is a line in the dictionary that answers for it and then you see that according to McMullen there are some that are going to be what McMullen was implementing kind of things and he wrote a book at least that was called renormalization and three manifolds which fiber over the circle so there is some class in it you get that there is some class of manifolds that are going to be geometrically infinite which so basically I'm saying that renormalization fixed point will correspond to some kind of geometrically not a general geometrically infinite manifold specific type geometrically infinite manifold and accordingly to this idea you can hope that you can get a proof that the geometric properties of the Julia set so the dictionary says that Julia set of course it corresponds to the limit set for a client group so this is a clear line in the dictionary so you can imagine that the geometric properties of the Julia set will kind of reflect will get the same answers that you get in the case of client groups so then you can go on and try to translate this thing but there is a problem because in client groups you have a three manifold and for rational maps or polynomials or whatever you don't have a three manifold so you cannot run those arguments in one dimension bigger and particularly where you're going to run your ground and motion in part of your arguments so there was some hope that you could construct three dimensional objects that would be so kind of pain that you'd allow to run those arguments that work on three manifolds those would be some laminations it kind of gets more complicated but you have laminations by three dimensional manifolds with three dimensional leaves so it construct those guys and with the hope of getting there but those have very complicated behavior and technically it was not possible to make the argument work you try to translate the proofs at least several years ago which was not developed enough to do that ok since this looks a little bit complicated we can try to do it a little bit differently so after all we don't need I kind of took you all for some kind of hide at this client group but now you forget them and look again at the point of high series so what's the point of high series here point of high series you just write the same thing instead of having all those gamma you just have a semi-group generated by a single map so you're just going to have now dfn so you run over all the math guys that go there fn of y equals w equals z and then sum over n and now you put w and then you put gamma delta here so look at this point of high series you get convergence again for delta equals 2 and you can try to show that you investigate first if it also converges for some delta less than 2 but you have a series here you can try to just kind of make it converge by hand problem is that as you have this kind of self similarity you have copies of the dynamic series things get very complicated as they get near 0 you don't have some properties that are kind of good like hyperbolist and so on more and more complications and it's unclear how to deal with what's happening on 0 if you kind of only had things that run away from 0 then you'd be fine there's some way to kind of put your hands around this and kind of say a little bit of things the fact that it's converging somewhere we like it to perturb a little bit and get to convergence and the point is that we have to use that self similarity here so the functional equation gives some kind of has a reflection on the internal structure of the Poincare series so you can see that the Poincare series if you start expanding terms using the chain rule you are going to see copies of the Poincare series inside itself because you just expand here and an orbit can be the composing two orbits so you have some kind of here that multiply another everything's multiplicative so everything looks fine so you can start expanding terms and the composing orbits in several ways and exploring the functional equation and you come up with some kind of Poincare series that sees itself which gives the recursion it's not an equality not exactly formal but it's an inequality so in that estimate you have a recursive estimate that you get from saying the Poincare series on both sides which has the form Poincare series of f let's say there are kind of inequalities in both sides but it has some coefficients for some reason it looks quadratic b delta p delta f squared so there are coefficients a delta b delta c delta which appears as truncations of this so it's some kind of partial information here that avoids a neighborhood of zero the copies of the Poincare series you can see that the whole structure comes to that so that's that's a formula that we computed in several years ago about 10 years ago me and Michal Ljubic and with this formula we were able immediately to get a first counter example first thing showing that the dictionary is not going so well at this moment because for instance we can so you have to say something about those coefficients and let's see okay so let me give a connection with probability those coefficients here also have some scale the way that I produce this recursive estimate is associated to consider some scales so coefficients coefficients involve truncation some scale so there are several possible scales of truncation as long as you truncate you get something that makes sense but you can get different recursive equations involve different things and so when you analyze those coefficients for delta equals 2 you see that the coefficients for several scales for different scales they are kind of computed basically and the way they behave it just reminds us some parameters in a random walk so you have a random walk on the line looking at things that behave like probabilities to jump from one side to the other side after many steps without passing through some other place so there are things that say that things are going that direction or things are going that direction and the coefficients for different scales are exactly the probabilities of doing certain things from one last point to the other so you get again some probabilities things for delta equals 2 and the idea to take delta equals 2 because you are only trying to decide the dimension 2 or not at first and you can try to compute perturbatively what's happening to this Poincare series at that value and then if you get some good control you are going to at least decide this question anyway, there is kind of reflections to this random mechanism that appears also in the Kleinian group case but anyway to govern this you need this kind of Poincare formula and you can ask them alright so there are those coefficients it's just an inequality what's actually happening it turns out that while in the Kleinian group there are some kind of mysterious reason that makes everything get a universal answer what happens in the in the case of renormalization fixed points is that the initial geometry here the initial coefficients corresponding to some truncation will determine the whole behavior of the random walk so you can see that as I understand there is some accidents that may happen corresponding to some scales I don't see why those things are very important because what matters really is what happened near the critical point but some things that happen in kind of outer scale far away from the critical point influence and determine what's going to happen as we start putting all scales together it becomes more complicated and change the behavior and it seems that here it is possible so we didn't get this right away what we proved was that it's possible so it was published in 2008 that it is possible to have house of dimension equal to the critical exponent less than 2 which says that this path is already not going through this line of the dictionary so this is done by actually checking that those coefficients can behave in a certain way and then we kind of said several things about geometric measures and so on but what we couldn't decide at that time was so this corresponds basically to this random walk that I mentioned be drifting in one direction there are two directions here you imagine that there is this tower of scales and there are points that live in different scales that move up or they can move down they can move close to the critical point or away and you have to see what kind of drift you have for this random walk and you check that it was possible to have drift away from the critical point and in this case this led to delta less than 2 you can imagine that a way is like the geometrically finite and inside the geometrically infinite and just I'm going to finish so we recently proved that it's possible so now we now know that it's possible to construct things so to get some kind of, there are only countably many parameters that you have because there are countably many fixed points but it's possible that there exists some case where the back measure is positive so also the analogous of the alpha conjecture for those manifolds it's not going to go through for the renormalization fixed point and just to say so this is done I'm going to just well maybe it's not necessary I have some pictures that show a little bit how this random walk changes as we change the parameters I just say that this is the actual proof of this we will involve another renormalization theory also that is a different one that allows us to control what happens to the parameters as we make some parameter go to infinity and you can see the generation of the drift in one direction so anyway so that's the conclusion but just so basically that's kind of completely central this absolute Poincare series to get all those results are there any questions or comments? the random walk well it's in our paper that you do this analysis of Poincare series it appears some formulas that are for those coefficients so you see clearly I don't know explicitly say this but you recognize that you see that the parameter we give the formulas relating the parameters at different scales so there's a sequence of parameters and and you see that just by the formulas some of them are kind of decay exponentially while the other is bounded or is the opposite or otherwise there is no drift and then they decay harmonically so you identify those those probabilities that I mentioned so there are three case actually I wanted to mention this that I mentioned that it's possible to have house of dimension less than two this corresponds to two kinds of drifts you may wonder whether it's possible to have house of dimension two and the back measure zero this would correspond to a random walk that has no drift and my opinion is that this is kind of a unknowable problem because in my view these kind of coefficients have some kind of accident on them so those are accidents that happen for the large scale that have nothing to do what's happening at the critical point of some universal behavior and it's like asking whether a given real number that comes there like I don't know the fine structure constant or whatever is it a rational number it's kind of completely so there is this open question but I'm quite convinced that it's not at all noble yeah yeah you can get it in my it has to be in our paper because it's the only place that's done so there's our 2008 paper where there is this but in the client group case there is more things you can see actually the Brownian motion here it behaves like a so you have points that are basically what is the space in which there is a random walk well we just have in the plane so in the plane you have a random walk on the plane but under the dynamics so the dynamics takes points from near zero so you have the back measure and now you have points that maybe close the critical point not asking how you define it but asking the space where it is defined it's on the plane because there is a random walk on the plane which has some symmetry well it's highly dependent on the condition because a random walk anything can be a random walk but you're asking what's the probability that you have points that are for instance close the critical point what the probability that they come away from the critical point after some iteration so that's kind of questions that are you say it's some symmetry associated to the dynamics so you would get a symmetry if you look at the limit of the dynamics so you can kind of consider there is a tower construction that you have this dynamical system here the upper scale here you can kind of minimalize the dynamics and make this kind of outer disk becomes larger and then you're going to have something that's defined on the whole that has a dynamics in all scales also up and now you have the towers of dynamics and you can ask for points moving in this tower as you iterate going up or down in this so this would correspond to the limit that you get closer and closer to the critical point but then you just have points with respect to the back measure and you ask what's the tendency of what they do do they have a tendency to go inner in the scales or do they have a tendency to go up or away from the critical point but to make it really a random walk on the line instead of just half line you have to do this procedure to convert this to a tower that said we don't kind of formally introduce random walk language because it's not necessary we can just estimate things but to remark that to understand what is the meaning of what you are doing I think in terms of those random walks Thank you the speaker again


By special recording, General Mills, makers of Wheaties, breakfast of champions, and Cheerios, the old cereal ready to eat, presents The Lone Ranger! Speed of light, a cloud of dust, and the Hote Hio Silver! The Lone Ranger! With his faithful Indian companion, Toto, the daring and resourceful mask rider of the planes, led the fight for law and order in the early western United States. Nowhere in the pages of history can one find a greater champion of justice. Return with us now to those thrilling days of yesteryear. From out of the past come the thundering hoofbeats of the great horse Silver! The Lone Ranger rides again! Lone Silver! Let's go, big fellow! Lone Silver! Yes, sir! Get on your way! Get on your way! Get on your way with Wheaties! It's inspiring to know champions are made, not born. Just aim for the top like Bob Davies did in basketball. He's the great set shot artist for the Rochester Royals. Small Bob worked hard at basketball, determined he would top them all. He learned to make that fast break play. And here's what helped Bob on his way. Wheaties for breakfast, most every day. Today Bob sinks that payoff shot and Wheaties still help him a lot. A Wheaties kid, a Wheaties man. That's Bob Davies. Wheaties eater, 22 years. Real man food Wheaties. There's a whole kernel of wheat in every Wheaties flake. Okay Bob, lay it in there! Hey! Hey! Hey! He's on his way! On his way! He's on his way! On his way! Get on your way with Wheaties! Cause champions are made, not born. Yes, sir! Get on your way! Get on your way! Get on your way with Wheaties! Breakfast of champions! I am hanging over Pepper Kneller's press flickered gently as the door at the rear of the shop opened and the lone ranger stepped into the poorly lighted room. As the elderly stoop-shouldered printer turned from his work, he reached for the twist of tobacco on his workbench. Biting off a chew, he adjusted it in his cheek and looked up to greet his visitor. Howdy, mister. Good evening to you. Good evening, Mr. Kneller. Yeah, take it easy. Thanks. Thanks, mister. What's wrong? Your mask. I'm sorry if I frightened you. You scared me so I swallowed my tobacco. I can see you never chewed, let alone swallowed a chunk of the stuff. If you're here to rob me... I'm here for some information. Oh, what kind of information? Did you print this mask? Let's see it. Here. Where'd you get this? Never mind that. Just tell me whether or not you did the printing. It's my work. How many of you printed? 50 on the first order, 25 on the second. And right now I have another order from the same customer for 50 more. But I don't savvy why you're interested. The maps are being used to swindle people out of thousands of dollars. Did my customer swindle you? No, but I've talked to several of his victims. So he fleeced friends of yours? I want to see Wink Tatum. Well, if you're gunning for him, why come to me? You know where to find him. What makes you think I know where he is? The top order on your spindle is for Wink Tatum. Oh, that's for the 50 maps he wants me to send him. Where will you send him? Well, I'll have to see what the order says. I'll take the order, Mr. Kneller. All I want is the address. The order says you to send the maps to Wink Tatum. Care of the postmaster in Blue River Junction. He paid in advance for the work. He's due to pay a lot more. Thanks for the information, Mr. Kneller. Now wait! You didn't tell me your name! Or explain that mask! Gone, eh? Well, I reckon doesn't matter. There was a lot I didn't tell you, Mr. I've printed enough reward notices to know how it's done. I'll print one more. Mail it to the sheriff at Blue River. Dodge and the law will keep that mask man too busy to interfere with Wink Tatum. After leaving Pepper Kneller, the Lone Ranger joined Tato at Clarabel Hornblow's ranch. He learned that Thunder Martin Clarabel's top hand had been swindled by Wink Tatum when he had bought one of Tatum's printed maps. The mask man said, Tato and I are going to Blue River Junction to give Tatum a dose of his own medicine. How do you plan to do that? We have several good friends there, Clarabel. I think they'll help us. Will Thunder get his cash back? I hope so. Well, I'd sure like to go to Blue River Junction with you and Tato. I counted on you to come with us. You mean it? Yes, Thunder. I just give me time to settle my horse. Settle mine while you're at it, Thunder. What, Clarabel? Don't argue with me, Oversize Glue. Now I aim to do all I can to help the Lone Ranger help you get your cash back. Several days later, Sheriff Conley in Blue River Junction received an unexpected package. His deputy, Hank Dixon, watched him open it. What is it, Sheriff? A reward, notices Hank. Who's wanted? Let's see. It says $10,000 for information leading to the capture of a tall, slender, educated mask man who rides a white horse called Silver... Gosh, Sheriff. What in Thunder did he do to get a price like that set on his head? I don't know. Who sent the notices? Sheriff Miles in Copper County. Is that note from Miles? Yeah. He thinks the mask man's heading this way. I met him the last time he was in your office. Judge Kennedy was with us. That's right. He admires his local if he thinks that man's an outlaw. I wonder who put up the reward. $10,000 for the Lone Ranger. What are you going to do, Sheriff? I'll wire Miles for more information. I hope you get an answer pronto. With that kind of cash at stake, someone's bound to capture the mask man just to collect the reward. Late that night, as Sheriff Conley sat working at his big roll-top desk, he heard a light tap on the rear door of his office. A moment later, the door opened. The Lone Ranger entered, followed by Clarabel, Thunder and Tonto. How are you, Sheriff? All right, glad to see you, Mr. Howdy, Tonto. Oh, Sheriff. Sheriff, these are friends of mine. Miss Hornblow and Thunder Martin. Howdy, Sheriff. Mr, I sent a message to Sheriff Miles about you this afternoon. What kind of a message? I asked him to explain why he sent me this note and hand-bill. Now read him while I draw the shades so no one will see you in here. All right. The rest of you folks, sit down, make yourselves comfortable. Thank you, Sheriff. Sheriff, have you posted this hand-bill? Fundoration? No. I didn't want anyone to go gunning for you. Well, why would anyone go gunning for the Lone Ranger, Sheriff? To collect the $10,000 reward, Lamb. Reward? Yes, Clarabel. But Sheriff Edmiles knows you. Why, he wouldn't send out reward notices for you. Are you friends of Edmiles? We both know him mighty well. In that case, you might be able to tell me if this is his handwriting on the note with the hand-bills. Do you recognize the writing? Well, Edmiles didn't write it. Well, that writing doesn't even look like his. Sheriff, compare the printing on the hand-bill with the printing on this map I brought with me. Let's see it, Mr. Sure. The man who printed the map knew I planned to come to Blue River Junction. The hand-bill and the map were printed on the same price? Yes. Maybe printer worked with Swindler. I think you're right, Tutto. Who's a Swindler? A man named Wink Tatum. Wink Tatum. Tatum's in town. If you have proof he's a Swindler, I'll jail him pronto. I don't have proof, Sheriff. But I have a plan to make him repay what he's stolen. Does the plan include getting a sidewinder who sent me the letter and the hand-bills? Yes, sir. Good. What do you have in mind? They're a farmer near town who wants to sell his property. Sam Dawson wants to sell out and go east. But his place is run down. It's not worth much. Oh, what's his price? $3,000 for the house, the land, and the livestock. But it's no bargain. I'll put up the cash, Sheriff. Well, hold on, ma'am. Have you seen the place? No. And I don't think you'll want it. I'm not buying it for myself, Sheriff. I've heard the Lone Ranger's plan to outsmart Wink Tatum, and I think it'll work. Glad, pal. At $3,000, there's a lot of cash to lose if anything should go wrong with the plan. Nothing will go wrong if you do your part, Thunder. Yeah, but... Besides, if I lose the cash, I'll just take it out of your wages. You're Dawson! He'll take me forever to pay you $3,000. Oh, calm down, Thunder. I'm not worried about the risk. The Lone Ranger's plan won't fail. Now, Sheriff, if you'll take me to see the banker, I'll make arrangements for my bank to get the cash here pronto so as the deal can go through. All right, ma'am. But I don't see what buying Sam Dawson's place has to do with catching Tatum and his pal. We'll explain that, Sheriff. You see, Thunder Martin signed a partnership agreement with Tatum. The curtain falls on the first act of our Lone Ranger adventure. Before the next exciting scenes, please permit us to pause for just a few moments. This is Mel Allen's porch announcer, and I've been in this business long enough to know that champions are made, not born. Isn't that the way it should be? Gives every youngster a chance. Take Betty's shallow figure skating beauty with the famous ship stads and Johnson ice follies while she was learning to cut figure H years ago. Been eating wheaties since she was only chin-high to her breakfast bowl 20 years ago, that was. Or Otto Graham of the Cleveland Browns who can thread a needle with a forward pass. Otto made himself a champion, practiced hours every chance he had, and been really going for that wheaties breakfast for 23 seasons. There's so long gone much honest energy packed into wheaties. Nothing you pick at and push aside for wheaties are downright goblin good. Let me say again what champions know. There's a whole kernel of wheat in every wheaties plate. And don't you lose track of this fact for a minute. Champions are made, not born. Get on your way with wheaties, breakfast of champions. Now to continue. While the Lone Ranger and his friends busy themselves setting a trap for Wink Tatum, the well-dressed prosperous-looking swindler was also busy. With all the participating acquaintances, he planned to victimize. Early one afternoon as he entered the hotel lobby. Well, who were... Well, look who's here. My old partner, a thunder... Thunder... Of course. Have you found gold yet? Well, no. I bought a little place a couple of miles north of here. It's not much of a farm. I thought you were going prospecting. But I bought Sam Dawson's farm instead. Oh, when did you buy it? A few days ago. Places all stocked. I bought his chickens and everything. Well, if your heart isn't in prospecting, you're better off doing something else. That's right. I've still got the map you sold me. It's here in my pocket. And here's the partnership agreement we signed. See, Daisy, I... Hey, what's that you dropped? Nothing, nothing, nothing at all, Mr. Tatum. It's a gold nugget. Let's see it. Now, listen, that nugget's mine. Had it back, Tatum.  Yeah, I came to town today to deliver some chickens to the hotel. Ask the cook if he don't believe me. Where did you get this nugget? Well, that's my business. That's clam in the hotel cook. Hey, where'd you get the chickens you brought me? Where do you suppose I got them? On the farm, of course. What's wrong with them? Are you sure they came from your farm? Of course, I'm sure. Oh, yeah. Oh, those chickens are good enough for anybody. Look what I found when I was cleaning them. Look, Thunder. Well, I'll be all gone. Thunder, you're rich. The gold must be laying on the ground at your place. Gold laying in the gravel. The chickens must have picked out as... There's no use trying to act surprised, Martin. But I don't... We're partners, remember? We're no such thing, Tatum. You signed an agreement with me. I didn't go prospecting for the gold you told me about. We're partners. Whatever you find, we split 50-50. But I didn't find any gold.  You knew all the time there was gold on that land. That's why you bought it in such a hurry. Oh, now simmer down, Mr. Tatum. He's trying to cheat me. But that's a lie. The nugget in your pocket proves it. The nugget doesn't prove a thing. I have a written agreement saying where partners, I'm holding you to it. You try. I'll sue you if I have to. All right, Judge Kennedy's in town, Tatum. We'll take the argument to him. That suits me. Come on, we'll go to see him right now. A short time later, Wink, Tatum and Thunder Martin visited the Lone Ranger's friend, Judge Kennedy. Wink showed his copy of the agreement with Thunder and said... Martin's trying to swindle me, Judge. It says here in writing that we split 50-50 in any gold he finds. The farm is nowhere near the country shown on the map you have sold me. The agreement doesn't say where you've got to find the gold. I haven't found any gold. That's a lie. The nugget was gift to me. The gold's so thick on your farm that the chickens are eating it with gravel scratch. Now, just a minute, gentlemen. You came to me for an interpretation of your agreement. That's right, Judge. Well, according to this paper, Tatum, you're entitled to half of any gold Thunder Martin finds. It doesn't matter where he finds it. There, Martin. You just prove I found gold on the farm. Prove it. Oh, you can't bluff me any longer. I said prove it. I don't have to. Try to set foot on my property. I'll be there with witnesses. I'll have friends there to meet you with six guys. You don't scare me. Tatum, you can't trespass on Martin's property. You can't set foot on the land. Why, he's trying to cheat me. You can't set foot on the land if Martin doesn't want you to. Unless, uh... Unless, of course, you want to buy it from him. And you'll not do that, Tatum. So you see, you're stopped. I'll not give up that easily, Martin. What do you want to do? Buy my place? How much do you want for it? Oh, you wouldn't pay what it's worth. Dame of Price. Well... Put a price on it. All right. Their price is $10,000. I'll take... You what? I'll pay you the $10,000 and take that land. You'll have to pay in cash. I've got the cash. I'll be here with it as soon as I can get my money from the hotel safe. Judge, you draw up the papers. When the sale was completed, Wink Tatum left town hurriedly to investigate the property he purchased. Thunder Martin went to the sheriff's office, where the Lone Ranger and Clarabel were waiting with the sheriff. The big ranch hand reached into his pocket and drew out a fistful of money. Ha ha ha! Here it is, folks. $10,000 cash. Judge Kennedy drew up the papers and witnessed the sale. So Tatum bought that worthless farm? Yep. That grab-a-pulled cat was so sure he was buying gold-bearing land he couldn't get the papers signed fast enough. Good. Ha ha ha! I told you the Lone Ranger's plan had worked. I'll hand it to you, mister. You've hoodwinked Tatum with the slicker deal than he ever pulled. Hand over the cash, Thunder. Now, wait a minute, wait a minute. $500, that's mine. Take your $500, Thunder. All right. Yeah. And here's the $3,000 you loaned me to buy Dawson's place, Clare. Well, thanks. The rest of the money will go to Tatum's victims. They'll be downright glad to recover what they lost to that swindler. What's going on outside? Oh, the stage from Copper County's arrived. Sheriff Miles will be aboard. And so will the printer, Pepper Kneller. Tutto's waiting at the station to meet Kneller. Well, dog on. Ha ha ha! I'd like to be there to see what happens when Tutto talks to that crew. As Pepper Kneller dismounted from the Concorde coach, he automatically reached for the twist of tobacco in his pocket. He bit off a generous chew as he watched Sheriff Miles get out of the stage and stride toward Sheriff Conley's office. Pepper was wondering what brought the lawman to Blue River Junction when Tutto approached him. You, Pepper Kneller. That's right, Indian. Let me have message for you about Wink Tatum. What about him? Him thinking find gold. Yeah? Where'd he find it? On farm. Not far from town. Dog gone, eh? Say, Indian, will you take me to his place? Ah. Me get extra horse. Take you from him. Ed Miles had just entered Sheriff Conley's office when Tutto rode past with Pepper Kneller. Moments later, the two lawmen, the Lone Ranger, Thunder and Clarabelle, followed them. Oh, Scoundrel. Oh, oh! Here, farm, and there, Tatum fellers. Yeah, I see them. Steady, man. Hi there, Tatum. Pepper. What are you doing here? I came as soon as I got your telegram. What telegram? The one you sent asked me to come to Blue River Junction. I didn't send you any telegram. You did so. I got it right here in my pocket. A fellow who sent message from now. Huh? What are you talking about, Redskin? You'll find out, and he soon. What is this, Wink, that mask man? He came to my place asking about you. There's Thunder Martin. Two lawmen and an oversized female. Hello, Mr. Kneller. Uh, howdy, mister. What are you doing here? I came to see you. And so did I, Pepper. I think you know Sheriff Miles. I traveled all the way from Copper County with him. Sheriff Conley sent for me when he received the note and handbills you sent him over my signature. Uh-oh. But I didn't write any note to Sheriff Conley. Well, I... Don't admit anything, Pepper. The law hasn't got a thing on us. The law didn't have anything on Kneller. Until he started impersonating a lawman, printing false reward notices and sending notes with my name on them. Uh, I'm not admitting that, I said it. We can prove you printed the handbills, just as we can prove you printed the maps. Take them so.  The print and press in your shop will prove it for us. That's right, Kneller. And a handwriting expert will identify the handwriting on the note you sent Sheriff Conley. Wait, do you mean I'll go to jail? Maybe not. If you have cash enough to pay a fine. Well, Tatum, Tatum, you have cash? Well, I better. He gave me his cash for this farm. Well, I'll pay the fine, Pepper. If they'll all wait till I find the gold on this land. I told you there was no gold here, Tatum. But that nugget you dropped. Someone gave it to me. Well, there's got to be gold here. The chickens were eating it. Oh, that. Well, you see, I heard the little gold dust makes chickens mighty prime. Well... I paid in cash for the dust that chicken ate. No swindler. No cheating, swindling, theming, crook. We've been outsmarted, Wink. Can you pay the fine? No, of course we can. Then you'll go to jail. $10,000 for this run-down worthless farm. Cheer up, Tatum. I didn't keep your cash. It's going back to the folks you swindled with here. No account maps. Oh, no. Sheriff, I gave you a list of the names of Tatum's victims. Right, Mr. I'll see that they get their money. Then tell them I'll be on our way. Say, Mr. if you're going our way, thunder and I'll go with you. It's time we got back to the ranch. All right. We'll travel together, Clarabel. Adios, Sheriff. So long, Mr. Anne. Thanks. We wouldn't have accomplished anything without your cooperation. Easy, steady, make sure. I'm mighty proud to have cooperated with the Lone Ranger. What? Oh! Take it easy, Miller. Slap him on the back. What is him? He must have swallowed his chew on the back. What's wrong with you, Pepper? Oh, if I'd only known that mask man was the Lone Ranger. And fights hard and fair. So in the ring, you kids beware. He's dynamite because he knows. He's got gold power. Real shaped like little letter O's. You know, Cheerios is made from oats. And every delicious spoonful of Cheerios and milk is a real muscle building food. Each spoonful contains vitamins, minerals, and proteins your body needs. Yes, the good things in a Cheerios breakfast do good things for your body. Help you have healthy nerves, good red blood, strong bones, and muscles. Have Cheerios every morning. Then you'll hear people say, He's feeling his Cheerios. Ranger, a copyrighted feature of the Lone Ranger Incorporated is created by George W. Trendle. Produced by Trendle Campbell Muir Incorporated directed by Charles D. Livingston and edited by Fran Stryker. The part of the Lone Ranger is played by Brace Beamer, your announcer, Fred Foy. The Lone Ranger is brought to you by General Mills every Monday, Wednesday, and Friday at this same time. Be sure to listen.