Monique is one of the best-known female African-American comedians in the USA. She ventured into live stand-up comedy and starred in movies, a talk show, and streamed comedy specials. Her career began when her brother suggested she try stand-up comedy and she took the stage at an open mic night at a comedy bar in 1988. Encouraged by the positive feedback, she started performing at comedy clubs in Baltimore, Atlanta, and along the East Coast. Eventually, she decided to leave her job as a customer service representative for a telecom company to pursue a full-time career in stand-up comedy. Following her stand-up success, she opened for musicians and featured in TV specials like Russell Simmons' Death Comedy Jam and Comic View during the late 1990s. Notably, Monique gained recognition through guest appearances on the TV show Moisha in 1999 and 2000, leading to a spin-off series centered around her character. The sitcom, The Parkers, became a five-season hit from 1999 to 2004. From 2004 to 2008, her venture into film commenced with various roles, spanning from the gritty portrayal of inner-city life in Baby Boy to the comedic yet controversial soul plane. In addition to her stand-up performances, Monique joined the Queens of Comedy tour in 2000. Transitioning to acting, she gained attention for her role as a drug addict and shadow boxer, voiced a character in Fars of the Penguins, and starred in the romantic comedy Fat Girls. However, it was in 2009 that Monique received significant critical acclaim for her powerful portrayal of the violent and sexually abusive Mary in Precious, based on the novel Push by Sapphire. In her role as Mary, Monique's acting was praised by the audience. Critics commended Monique for her skill in portraying the ignorance and desperation driving her character's inhumane actions. Her outstanding performance earned her the Academy Award for Best Supporting Actress. While acting in film, Monique also worked in televised shows. She hosted Monique's FAT Chance from 2005 to 2007, a beauty pageant celebrating full-figured women, and brought her assertive style of humor to the talk show Arena with the Monique show on Black Entertainment Television. In 2015, she portrayed Ma Rainey in the TV movie Bessie, a biopic about Bessie Smith. Monique continued her television ventures by hosting a comedy special Monique and Friends Live from Atlanta in 2020. In addition to her on-screen work, Monique authored Skinny Women Are Evil, notes of a big girl in a small-minded world, a candid exploration advocating for overweight women. She followed this with a similar cookbook, Skinny Cooks Can't Be Trusted. Her career was not that smooth as Monique also got entangled in different controversies. Monique has faced challenges, revealing in a 2008 Essence magazine interview that she experienced sexual abuse from her brother Gerald Ims during her childhood.  In 2009, when Monique took on Precious, directed by Lee Daniels, she received a payment of $50,000. As the film gained critical acclaim and awards attention, she was asked to promote it at the Cannes Film Festival by executive producers Tyler Perry and Oprah Winfrey, along with the production company Lionsgate. Monique, however, declined, citing her contractual obligations to the film's director Daniels. Following her Academy Award win for best supporting actress, Monique expressed gratitude to the Academy for recognizing the performance rather than political considerations. She later revealed that Tyler Perry had apologized to her for how she was treated during the film's promotion. However, in a 2020 stand-up segment, Monique criticized Perry, Oprah and others involved in the feud, expressing her devastation at Oprah's communication and describing it as malicious. In 2018, Monique accused Netflix of racial and gender bias for offering her $500,000 for a comedy special far less than her peers. Netflix cited perceived market value rather than her resume. Monique called for a Netflix boycott and filed a 2019 lawsuit alleging discrimination and retaliation. By June 2022, they settled, and on July 19, 2022, Monique announced a new special with Netflix. Despite her successful career, Monique faced financial challenges, including a reported $620,000 in back taxes and legal issues with the IRS and the Georgia Department of Revenue. With a net worth of $13 million, it is a no-brainer that Monique will have some of the best properties in her portfolio. Let's take a quick tour of Monique's properties, which every normies dream of. Welcome to Monique's extraordinary mansion in McDonough, Georgia, a colossal 10,000-square-foot haven on a vast 20-acre property. This is a perfect blend of modern and traditional styles, welcoming you with a touch of country charm that suits the Georgia landscape. Step inside and you'll find yourself in a world of luxury and comfort. The dining room exudes timeless elegance, while the living areas boast warm hardwood floors, creating a cozy atmosphere. The kitchen with its old-school vibe is the heart of the home, a place for family gatherings and grand entertaining. But that's not all. The basement is a hidden gem. It's an entertainment hub with a game area and a full kitchen for added flair. The mansion goes beyond the basics with a basketball court for recreation, a movie room for cinematic experiences, and even a tennis court and swimming pool for various interests. Outside, the grounds are as charming as the interiors. The tennis court awaits a friendly match and the swimming pool beckons for moments of relaxation under the Georgia sun. Monique's mansion isn't just a home, it's a lifestyle. It's a perfect mix of sophistication, comfort and recreation, a place where luxurious living meets thoughtful design. Whether you're enjoying the views by the private lake or entertaining in the grand living spaces, this mansion embodies the essence of refined living in the heart of Georgia. Monique's real estate portfolio extends beyond her Georgia mansion, encompassing properties in various desirable locations. One of her notable residences is a stunning Mediterranean-style villa nestled in the hills of Brentwood, California. This three-story house boasts elegance with six bedrooms and eight bathrooms. Beyond the luxurious living spaces, the villa is equipped with state-of-the-art security systems and spa-like facilities, providing a blend of comfort and security. In Maryland, Monique owns a property that holds special significance as her childhood dream home. While details about this residence are limited, its sentimental value adds a unique layer to her real estate portfolio. Additionally, Monique has a city loft in New York City, characterized by modern design and panoramic views of the bustling metropolis. This urban retreat offers a contemporary living experience in the heart of the city that never sleeps. In Miami, Florida, the comedian and actress enjoys the pleasures of a beachfront villa. This property comes with private beach access and an infinity pool, creating a serene oasis with the rhythmic sounds of the ocean and breathtaking waterfront views. Monique, the multi-faceted entertainer millionaire, also boasts a car collection that mirrors her diverse tastes and appreciation for both luxury and performance. First in the lineup is the BMW M3, an emblem of precision engineering and high-speed performance. This sporty marvel isn't just about acceleration. It combines cutting-edge technology with opulent interior features such as lavish leather seating, creating an exhilarating driving experience. Transitioning to the GMC Yukon Denali, Monique demonstrates her affinity for grandeur and spaciousness. This full-sized SUV transforms the road into a lavish sanctuary, offering heated seats, top-tier sound systems, and an abundance of technological amenities. It's an indulgent journey reminiscent of a private VIP lounge. Adding a touch of Japanese sophistication is the Lexus sedan. Renowned for its commitment to luxury, Lexus sedans redefine opulence on the road. Monique's Lexus likely exudes refinement with plush interiors, advanced tech features, and a gracefully smooth ride. Completing the ensemble is the Chevy Tahoe, representing practicality without compromising style. This large SUV strikes a balance between utility and luxury, featuring comfortable leather seats and sophisticated tech elements. It embodies the fusion of functionality and elegance. Monique's car collection is akin to a symphony, each vehicle contributing a distinct note to the overall composition. From the adrenaline-inducing BMW M3 to the expansive luxury of the GMC Yukon Denali, her choices showcase a sophisticated palette, affirming that her journey is not just about reaching a destination, but doing so with unparalleled style and flair on the roads. Monique's lavish lifestyle is a celebration of her hard work, resilience, and smart decision-making throughout her remarkable career. From her early days in stand-up comedy to achieving critical acclaim in films like Precious, Monique has forged a path that reflects both success and challenges. Her luxurious properties collectively represent the realization of her dreams and aspirations. Her diverse car collection from the sporty BMW M3 to the spacious GMC Yukon Denali, the refined Lexus sedan, and the practical yet elegant Chevy Tahoe symbolizes her journey through various stages of success. Monique's life is a testament to her commitment to her craft, entrepreneurial spirit, and authentic approach. Although we all dream of a life that is opulent or at the minimum can raise our family with dignity, we should all remember that every individual has different paths to success. We can't copy what she did, but of course her success could be an inspiration for us. Before we wrap things up, I'm really curious to hear your thoughts. What do you think about Monique's mansion and how do you feel about her rise to success? Let me know in the comments. Also, if you loved this exclusive peek into Monique's luxury homes and lifestyle, don't forget to hit that like button and subscribe to my channel, ring the notification bell, and be the first to see my newest content. Once again, this is Cara the Vampire Slayer. Follow me on Instagram to chat. Bye.


Well, we'll put them on the list. So in summary, this node.js, and quite frankly, mobile and cloud and social in general are rapidly changing and disrupting the architecture of how firms are organizing their IT infrastructure and service provider infrastructure. The benefits to the business line is driving top line revenue and also reducing costs, which is the key to business. Obviously the big players that take advantage of this that are positioned for success, HP with the three-power acquisition, EMC gearing up with project lightning. We expect a big announcement this month or next month coming out something new there. Recently gone public fusion IO. Upstarts like SolidFire, Verident, and Cleversafe all positioned perfectly for this massive new surge and should drive a ton of revenue and a lot of competition. And that's good for these developers in this growing market. So great input, great manifesto, IT centric, infrastructure, clearly powering the developers. We're seeing that with Joyin and all these guys powering some great solutions. And they need some back-end help. And I think those systems guys will step up to the table and provide that level of expertise because we heard from Theo code breaks and that's normal in their operations. And that's okay, iterate, iterate, iterate. However, to run systems, you can't be down. So I think there's going to be a really nice intersection here between the two. And there's one last thing I'd like to say is that what's going to drive this is that the applications that can be written with this type of technology are going to be completely new. They've been constrained by small amounts of IO. What you can do is consolidate those databases. You can link those databases. You can do your data warehousing at the same time as you're ingesting that data. That's going to lead to a completely new set of applications, the types of applications that we're seeing now in Node.js, for example, new applications, completely different paradigms of developing analytical applications. I think it's going to be the most exciting decade in computing that's coming up. I'm John Furrier with SiliconANGLE.com and SiliconANGLE.tv. And I'm here with David Floyer, co-founder and chief researcher at wikibond.org, laying out his IT centric infrastructure thesis and research ties perfectly into all the thermal trends, mega trends around Node.js, and the rapid, rapid rise of this new type of developer. And it's really going to intersect beautifully with the existing market, great research. And this is all intersecting. It's a perfect storm for SiliconANGLE and wikibond day because we have been covering big data with Hadoop World. We have SiliconANGLE.com. We have DevOpsANGLE. We've got ServicesANGLE. And this is right in our wheelhouse. And we have the Stratoconference coming up around the corner. We're going to hear the big data angle, which again is the whole database, non-SQL and SQL, all intersecting with this real time and on-demand cloud computing storage IO. IO is the key to success. You've nailed it with your thesis. And let's bring in...


In what ways can sisters help the African-American community and African-American Muslims? I'm going to ask a question. And I think that the first piece says is really self-education, about learning the history, the gaining knowledge, and things like that, so that it can help to change the narrative that had been given to us, whether it's in schools, society, wherever we got the narrative from, our own families, is to alter the narrative. And that comes with retraining ourselves first. And the other piece, too, is that begin to change the way that we may look at our sisters, our African-American sisters, and look at that in terms of positive aspects that we are contributors to Islam. We're contributors to our community more so than we may have thought before. So that part is really important. And there is another question about the social justice aspect of it. And I think there is one piece around really also being mindful of yourself and your health with really getting into the social justice movement and things like that, and helping with the African-American community, and it's being mindful, because I was speaking with someone earlier who came to me to speak before. It can be really draining when really looking at a lot of those pressures and the negatives that happen as we focus on Islam, focus on Allah, Prophet Muhammad, Islam, and our health, and knowing that we can't do it all to pace ourselves. So I will hand it over to someone else, and then I'll come back around to me to answer some of the other questions. Thank you, Hamdullah, for having me.


Good morning. I would like to start with an administration remark. There will be lecture notes on the course on partial hyperbolicity. It will be on the web page. And there will also be notes about the renormalization course, but they are just handwritten, so don't expect too much. I think you can use the notes for the renormalization to recall the definitions, but it is handwritten. Okay, so let's begin. So we are going to talk about renormalization, low dimensional dynamics. And we would like to discuss renormalization in any dimension, but it will be very low. We will be discussing one and two dimensional dynamics. Okay, so what is this renormalization? We start from scratch, so don't worry. So what is renormalization? It's a very simple idea.  This is our state space and our map. And then what you do is you just choose some subset and then you look at the first return map. So the renormalization of f is the first return map. And this is the little tricky part. A well-chosen u part of x. So what you do is you take a point x in u and you iterate it a couple of times and then it returns. And that is the how you define the renormalization. So you get the renormalization of our f. There will be a map from u to u. But then it was off because you could take this u very tiny. So if you don't rescale it, you might not see anything. So you rescale also.  And then here we see really our renormalization. So there is a rescaling involved. And just to get back to unit size. Okay. Okay. So you see this is a very simple idea. But in the cases we understand, so in most of the situations we don't understand at all how this behaves. But in the cases we do understand, something nice happens. So renormalization unifies the main aspects of dynamics. So if you here have renormalization, so if you do dynamics, you would also like to understand the topology of your original dynamical systems. And very often, in the cases we understand, renormalization allows us to give a topological description of the dynamics. This is vague but we will see examples and then this connection will become very precise. Another part which is closely related to the geometry. And that is easier to understand.   So renormalization is like a microscope. And a microscope is perfect to understand geometry. And so the first thing about renormalization is really understanding geometry. So then there is another part is measure theory. And so the existence of relevant inferred measures. And another part which is missing is bifurcations. And so in the cases we understand, renormalization connects all these main branches of dynamics. And that is a nice aspect. And a little bit surprising that this very simple idea of using a microscope sort of starts to connect. You can easily say the main branches of dynamics. We are going to discuss concrete examples. But let me clarify these connections a little bit. So let's say something about topology. So if you do, if you choose a piece where you are going to look in, a smaller piece, you associate to that a type of renormalization. And so this what they call this topological information, the type of renormalization, the type. And there is combinatorial information. Combinatorial information. And for example, it could be that here you have our original system. And it could be that our set U and the return map to U is something very simple. It could be that the set U just goes to another set and then it returns. And so it means that the renormalization, here is the renormalization of F, as it is just the second iterate of the map restricted to this U. Yeah. No, no, no. So no, this is like the simplest case. U goes and it returns in one step and the whole thing. This is the simplest of the simplest situations. And in general, it will not be at all like that. In general, this return map will be something complicated. But you will see examples where this will turn out. So this picture illustrates that a renormalization has a type. And this is topological information that says that the renormalization is built in such a way that you take the piece, you map and you come back. That's a specific type. And clearly that is combinatorial information. So now in the cases, we understand. And so if F and G have the same type, a renormalization type, and lately we will see precisely what that means, then F and G topologically conjugated attractors. And so here we have our map F and here we have our map G. And you can renormalize F and there is a certain type and you can renormalize G with the same type.  And then there will be a conjugation between the attractors. And so H of F is, and so apparently the renormalization type implies the topological structure of the attractor. And that is sort of the machinery how this works. If you know the renormalization types, you can figure out what is the topology of the attractor. And again we will see concrete examples of how this works. So let's go to the geometry. Okay, so let's take again, so F and G have the same renormalization type. And then what happens is that if you start to take repeated renormalizations of F and you start to take the corresponding renormalizations of G, then in the C1 topology this converges to zero. And so if you start to mean, what the means is that if you have your map F here and you have your map G and you start to zoom in here, say this is U1, here we have V1 and now we start to zoom in here U2 and we start to zoom in V2. And it continues like this, so you get a very tiny UN.  But that means that if you look in the geometry of what is living here and the geometry of what is living here, these are parts of the attractors so we are zooming in to a very tiny part here and we are zooming to a very tiny part here at the corresponding place, then the pictures are the same. And so apparently this convergence of renormalization implies a certain universality of geometry. And so you get here, it's almost the same on small scale, it's universality of geometry. So this is sort of a key ingredient. And maybe not everywhere but this is at least in the place where we zoomed in, we see that we see the same geometry. So it can be, so this is I should have written this larger because this is really a crucial aspect of renormalization. You see convergence of renormalization gives you universality of geometry but sometimes it goes even further. So in the picture we zoomed in on a very tiny scale here and a very tiny scale here and the geometries are the same. So that means that this conjugation is differentiable in this point. So it might happen that it can be, there's a notion of rigidity, that this conjugation from the attractor of F to the attractor of G is differentiable.  So these are crucial notions of universality and rigidity when you're talking about renormalization and we will see very concrete examples of this. So let's go to the measure theory. So measure theory. So again in the cases we understand the renormalization type, if you know the renormalization type and it will allow you the construction of invariant measures, of relevant invariant measures. And what we want to have is that if you start to average a function and then that converges to the integral of respect to the function. This is called the physical measures. So in the cases we understand the physical measures can be constructed out of only knowing the renormalization types. And again we will see examples. So this connection is also there and about the bifurcation pattern. That is sort of where everything began. You know, if you have a question or if something is clear, let me know and we can try to make it clear. So what happens is, so clearly this renormalization operator, what it does is it takes a system and it produces a new system. And so what renormalization is, it is sort of made to study the space of systems. Because that's where the renormalization acts. So we have the space of systems and renormalization acts on that space.  And in the cases we know, in the very simplest cases, you see another dynamical system here. And renormalization by itself is a dynamical system. And in the cases we understand, it is one of the simplest ones, namely hyperbolic ones. And so what you very often see is this picture, you see, the dynamics of renormalization has a fixed point and it has some unstable manifold and it will have some stable manifold. It's a hyperbolic picture and the holy grail of renormalization is to prove this picture, hyperbolicity of renormalization. And then the key thing is that is, again in the cases we know, this stable manifold of this fixed point is the same as the topological class, as a topological class. And that means that if you know the renormalization types of your system, that will define the set of all systems with the same type. It's a topological class. And it turns out that that has a manifold structure. And this is something that's quite the result. So we discussed very briefly how the main questions of dynamics are related, are connected by renormalization. And in the cases we understand renormalization well, this picture is working fine. OK. So the reason why this picture is working fine is that renormalization comes with so-called renormalization phenomena. And we mentioned already universality and rigidity. And so in this phenomena by itself they are interesting and fascinating. And let me give you an example, a simple example of a simple and surprising example of this intrinsic renormalization phenomena. And I'm sure that you have all seen this. But let's speak about period doubling. And let's first speak about a period doubling verification. That's what we have is we have a family. And what we can plot is we can look at that we can plot the parameter. And what happens is at a certain parameter value, before that value the dynamics is controlled by an attracting point. Like if you take this low parameter value, what you see here is something very simple. Ft. And you're just sitting somewhere a fixed point and everything converges here. This is like the simplest type of dynamics you can imagine. But now you change the parameter and you get to this specific value t0. And then what happens is that this periodic point is attracting, stops being attracting and it becomes something different. It's still a fixed point but not attracting anymore. And it splits into a period two orbit. And so in this case what you will see is if you look at this parameter and you look at the picture that there will be like two points which are exchanged and things will converge to this period two attractor. So this is a period doubling verification. And I'm sure that you all have seen this picture in your first courses in dynamics. And then there's something, the period doubling cascade. And so I'm sure you have seen this picture. It's sort of a famous picture. And so now we have our family again. And what happens is that we start with our attracting point and then there is a first moment where we see the attracting point bifurcating the period two attractor and then a little later the period two attractor will split into a period four attractor and then a little later it will bifurcate by a period doubling bifurcation in period eight. And it will continue like that. And it will be a limiting value. And very often this is the boundary of chaos. And so if you push your parameter a little bit further the system turns chaotic. And so this is the period doubling cascade and it is like the period doubling route to chaos. This is the place where renormalization was introduced in dynamics. And we will get to more details about this but not at this moment. So what happens is, and it has something to do with the bifurcation pattern.  So this is in real world systems with this friction. Then what you see is that you see all these period doubling bifurcations and they will convert to the boundary of chaos. And what you will see is that these bifurcation moments they convert to the boundary of chaos and they convert exponentially fast. And the rate is one over 4.669. Something like that. And this is in real world examples. So you can build electrical circuits and if you turn the voltage up then at some point it will create a period doubling cascade and it will turn chaotic. And then if you measure the parameters where you get the bifurcation you see 4.6. People measure this. And there are tons of mechanical systems with friction. And in electrical circuits they even found something like this in fluid dynamics. This is a little bit more tricky. So this is what they call parameter universality. For obvious reasons. Apparently these moments where you are bifurcating a period doubling bifurcation that the rate doesn't depend at all whether you are looking at an electrical circuit or a mechanical thing or even a fluid dynamics. There is something universal in this parameter behavior. And this is surprising. So apparently you have the renormalization connects the typical questions we know in dynamics but it gives itself new phenomena. There are intrinsic renormalization phenomena like parameter universality. This is by itself a very interesting phenomenon. These are things you can actually measure and people do measure this. Okay. We will come later to the history. More precise history of this fact. So let me show you the plan. So I apologize. So far we have been discussing like in a very informal and very sort of vague blah blah story. I have no illusion about it. You just have to let it come over you. This is just phenomena. But we are going to make this more precise in examples. And we are going to make the list of all the examples we understand well. Most of the examples we don't understand. But we will discuss all these phenomena in particular in this picture with examples. And that is what we are going to look at is the first thing is we look at different morphisms of the circle. And you are going to renormalize those things. You are going to look at unimodal maps. And this is our map. Something like that. Three. You are going to look at Lorentz maps. And you know that it's like the butterfly. You know how is that again? You know the butterfly. And you are going to look finally at hand-on maps. Okay. So this is clearly one dimensional. The dimension refers to the dimension of the underlying space. Unimodal is one dimensional. So Lorentz is... You know you can reduce this flow to a map on the interval. But it is a funny interval. So this is sort of one and a half dimensions. No, there is a serious reason. It's not a joke. Of course. There is something going on here. And then of course hand-on maps. It's really 2D and even 3D and higher. But we concentrated 2D.  And the difficulty is making this picture precise. This is the heart of the matter to prove hyperbolicity of renormalization. And it is something like... That is the easiest... Renormalization of circle-different morphisms is the easiest one. A little bit more difficult is Unimodal. And then Lorentz maps is still very difficult. And hand-on map is a little bit understood. A little bit. And so the complexity sort of goes down. It's becoming more complicated when you go in this way. Circles are simple. Unimodal more complex. Lorentz, you see. How are we doing this at the time? Ten minutes. So let's start to discuss circle-different morphisms. It's a simplest case, but we will need one or two days to discuss this. So it's not that simple. And also probably many of you have seen already the circle case. So it's like the first example, also to refresh a little bit your memory if you have seen it already. Okay, so let's do a circle-different. And so hopefully many of the blah-blah things will become more precise in this discussion. Of all the blah-blahs should become precise in this discussion. So we are looking at maps from the circle, the interval to the interval, and they look like this. This is 0, this is 1. So somewhere is what they call the critical point. And somewhere is the critical value. And it looks something like that. And so you're looking at maps which are something like that. There's a branch here, and there is some branch here. And what we want is that f is piece-wise smooth and monotone. So this piece is something smooth, and this piece is something smooth. So that's important. And then there is a little tricky thing which makes it into a circle-diffumorphism. But you want this. Oh, I made the wrong picture. Sorry. I made the wrong picture. Sorry. Let's do this again. And somewhere here we have the critical point. And somewhere here we have the critical value. And we want the two branches to be separated by the critical point. So then the second property we need, these branches should be a little bit related, and we want that the derivative of f on the plus side of the critical point times the derivative in zero, which is this derivative. It should be the same as the derivative on the minus side times the derivative in one. Yeah, it's really a circle. Yeah, because so let me say it like this. If you start a circle-diffumorphisms, and then of course you just a circle, and you could open the circle, and you get the interval, what you will see is that these brands, and these brands, they really sort of continue. It's a small thing. And this thing here, and this thing here is really the same. So they are circle-diffumorphisms. But now if you renormalize, and you will see the next step will be renormalization of such things. If you renormalize, it will not be like this anymore. It will not be anymore that this thing is just a continuation of the other side. There is sort of a knick. So here there is a knick happening. And also there is a knick happening here. And what we want is, we want to have a space of systems, and our renormalization acts on that space. So now you see if you will be really working with circle-diffumorphisms, if you renormalize, it's not a circle-diffumorphism anymore. So we have to enlarge the space. And if you do this, if you do this, if you use this definition, then that will introduce a space of systems which will be invariant under renormalization. That's a little tricky thing. Not tricky, but that's what you have to do. And you will see why this happens. Okay, so, yeah, this is like zero, and this is one. This is one. Oh, now you take the derivative here, and you multiply it times the derivative in one. I think it's correct. It's like this is that, and that is that. I think it's right. Let me introduce renormalization, and you will see why you need to do this. Okay. So renormalization. Exactly, yeah. No, no, no. And so we want to understand circle-diffumorphisms. So we are going to renormalize them, but the way we are going to renormalize, it will be such that if you take a circle-diffumorphism, it's not a circle-diffumorphism anymore. And then we are lost, because then we cannot repeat the process. So we have to enlarge a little bit the space where we are living, and then this enlarged space of maps will become invariant in renormalization, and then we can start to talk about hyperbolicity. And the enlargement of the space is this. And clearly, if you have a diffumorphism, it matches. So the diffumorphisms are included here. Okay. So let me define the renormalization. There are two cases, yeah, unfortunately. So it can be that C is smaller than... You know, let me give some name. The space of the systems is called D. And if the C is smaller than V, we call it D minus. And then how do you renormalize that? And so here we have C. And here we have V. And so here we have our map, something like that. And remember, when you renormalize, what we have to do is, you have your space, and here we have F, and we have to choose a U. And then here we will look at the first return map to define the renormalization. So how do we do this here? Excuse me? Yeah, yeah, oh, sorry, yeah. We have to define smalls, monotone, and orientation. Thanks, yeah. And like this picture, this picture. So the U we use in this case, our domain that's called the domain of renormalization, is U, and it is just this interval. So we are going to make a picture inside here. So U is the interval zero, V. And now we need the first return map to that interval. Now you see that's easy. This part is already returning. So the first return map on the right is just the original map. It puts it back immediately. And on the right, on the left, you see, you have to do this, then you land here, but then you're back. So you will get something like this, and that is the second iterate. And now you rescale the bottom to unit size, and you get renormalization. So that's how you renormalize. And that is in the case when C is smaller than V, and in the other case, I guess that means to stop. Do you allow me to finish the definition, and then tomorrow we take it? No, you're gentle. I understand. So the other case when C is larger than V, and we call it the V plus case, and we have a situation like that, and it is, this is the picture. So we need something like that, and we have something like that. And you can imagine what to do in this case. The U is the interval V1. So now we have to look at the first return map in this little square. And now you see the scope to be exactly the opposite. On the left side, you can just keep the branch. It's over the return, and on the right side, you see you have to go outside, and then you're back. And so we will see something like the second iterate. It's very similar to this case, just flipped around. And so in that case, in that sense, we have our renormalization operator. We should stop today. Let's continue tomorrow.


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