That's Rachel Fukumoto. Hi, Rachel. Hi. Nice to see you again. Yeah, nice to see you. Yeah, Rachel's from Hawaii Energy. And she wants to talk about Honawha School this time, what Hawaii Energy is doing there. So give us the poop.  We started off on Monday at a school assembly. And so there were 700 students in attendance. And so what Hawaii Energy did for Honawha School, they're out in Waipahu. We provided an incentive for energy efficient lighting. And so they took that. And they're a pilot school with Abterra energy, too. So through the Kahe program, they really just did energy efficiency first. So they worked on their lighting, their air conditioning, really brought their load down. And then they put PV panels to make up the rest of it. Is this a drill or a real thing? It's a real thing. Real thing, all right. Yeah, it was incredible. And so hopefully this is the pilot school. And other schools around the state can use them as a great example. So what kind of reactions do you get from the kids? Oh, it was wonderful. We really got them involved. A lot of them did do home energy audits. They took it home, what they learned at school, and then they brought back the information back to the classrooms. And so actually two of their students, they wrote essays. We want to give a shout out to Shannon and Brianna. They were wonderful. And so Shannon and Brianna, OK? Yes. And so they actually received home energy retrofits for their house as well for doing that and participating. You know, the thing about kids is they talk to their parents, most of them. And most of them, parents listen, actually. Kid comes back from school and he's all excited. Was she all excited about energy? Tells the parents. All of a sudden, the parents are learning something. So you guys are very clever. Using the parents, the children is a way to get to their parents. You know, it's all about market transformation. And so with Hoy Energy, we provide these behavior and market transformation programs, really providing the education at the school level. And so hopefully, we can really allow these kids to, like you said, take it back to their parents. And hopefully, their parents can take it to their workplace too. Yeah, filtering that information into the community, disseminating it. The idea is to change the way people think about energy and about clean energy. So it's a project that Hunter White done? Yes. Everything is completed. And so now, I think the Cahe project, they're in the woodworks of finding more schools to participate and just really using that school as a great example. Yeah, so now you have a template. Yes, exactly. So use it again and again. So can you tell us, or is it a secret, as to the next school you're going to be visiting? You know, I'm not sure. But we have to stay tuned. OK, you have to come back and tell us more about this adventure, this most excellent adventure by Hawaii Energy and Rachel Fukamoto. Thank you so much for coming there. Thank you. Yeah, aloha. Aloha and happy New Year. It's 2017. Please keep up with me on Power Up Hawaii, where Hawaii comes together to talk about a clean and just energy future. Please join me on Tuesdays at 1 o'clock. Mahalo. Hi, I'm Tim Appichella. I'm the host for Moving Hawaii Forward. And the show is dedicated to transportation and traffic issues in Oahu. We are all frustrated by sitting in our cars in bumper-to-bumper traffic. And this show is dedicated to talking with folks that not only we can define the problem, but we hopefully can come to the table with some solutions. So I invite you to join me every Tuesday at 12 noon. And let's move Hawaii forward. It's me, Angus McDuck. Wishing you to welcome and join us to see us on Hibachi Talk on Think Tech Hawaii. 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You can watch me live on Thursdays, 3 p.m. to 4 p.m. each week. Aloha. So, do you have a big party yet? Oh, we're back. I was asking Scott, too, if you had a big party yet, because he's now the vice president of public affairs of a wine electric. What do you think? Party? Every day's a party. Every day's a party. Public affairs, it's all a party. Congratulations. Thank you. Thank you very much. Congratulations, vice president of public affairs of a wine electric. And Meena Marina, former chair of the PUC, and the chief cook and bottle washer at Energy Dynamics blog, which you write all the time and is very well informed. Well, I mean, I kind of backed off for a while. I got to get back into it. OK, we'll be looking. We'll be watching you. OK, thanks. So February is a very exciting month. Yeah, let's talk about it. So the Hawaii Energy Policy Forum is kicking off its series today on the utility of the future. And having both the Hawaiian electric companies and Kauai Island Utility Cooperative participating in various discussions on a number of topics this month. Utility of the future, all February. Every show in February on Hawaii, the state of clean energy. So what is the utility of the future? I mean, just define the term for me. What does that mean? Well, I'm going to define it from the terms of, I guess, how our customers would experience it. So what we are all talking about, and when I say we, it's not only us at Hawaiian Electric. There's many, many conversations happening about this, which is that in the future, our customers are all going to be playing such a critical, active role in terms of not only how they use energy, which is what we currently do, but then also how we play a role in actually allowing this energy system of ours to actually operate and do so in an efficient economic manner. It's a little bit different because normally, the traditional utility system is, well, we are there. When you turn on your lights, we figure out, how do we get the power to you? That utility of the future means the customer and the utility are actually working side-by-side, dynamic, real-time. It's going to be a much more complicated system, frankly. But at the same time, ultimately, the customers get more value from it. So I think that's a pretty squishy description, but it is really a different dynamic. It's a story of transformation is what it is. Different technologies, different roles. I think basically what you're looking at is the utility as a platform that can either provide services or receive services to have this transactional relationship with the utility. So good for you, good for one electric, good for the state. Wish you well in this. I think it's going to require a change in public thinking. It's one thing for the utility and for the industry to understand this, but the public have to understand that it's a two-way street and that it's a handhold, a collaboration as never before. That's pretty exciting, actually. How are you going to do clean energy at the same time? Well, I mean, clean energy is actually a part and parcel of that utility of the future. We're all moving towards the 100% renewable energy future as well. So it's not only, I mean, there's the clean energy piece. There's the figuring out the role of our customers and the utility and our other stakeholders, our regulators, and very critically, which is really the topic that I'm here to talk about. How do you do this so that throughout thick or thin, your utility system, your energy system is going to be there to function and deliver the power to you? We're talking about reliability and resiliency. That's interesting, because if you look back at the old model of utility is, we're here. We'll do it. Don't worry about it. It's going one way. And we will be responsible to get it to you on a reliable basis. Now it's more complicated because it's two ways. And the assurance of reliability is all the more important. And there are so many things that could get in the way of reliability when you have a two-way street. So this is an exciting time. Or as they say in China, an interesting time. So I think one of the questions that we want to ask Scott is, since today's topic is grid resiliency in an era of climate change, so I think one of the things that we want to start off with is what is the definition of grid resiliency? And that's really important, so we all have a common understanding of what we're talking about. Well, these are three of us anyway. Yeah. And it's actually a very good place to start because, I mean, I tell you, even in the energy industry, sometimes we start off with different conversations because we have a different framing of the meaning of reliability versus resiliency. So the simplest way. And that was the second question. What's the difference between resiliency and reliability? So the easiest way that I think about it is that the reliability is something that we have all talked about very frequently over these past several years. And it talks about, OK, well, how often might I experience a power outage? What is the duration of those types of power outages? And the utility industry, we have metrics where we track our performance. So again, it's the frequency of outages and duration is typically how you might measure your reliability. It's every day, right? Real time, every day, there may be a car that hits a pole and we have a power outage somewhere. So how do we actually manage reliability and do what we can to improve it? Resiliency talks about more from a, I would call it a broader view from the energy system as a whole. What is the capability of our energy system to be able to withstand the big shocks to it, whether it's from a natural hurricane event or a cyber security attack or something really bad happening? And will our system be able to basically bounce back? How can we minimize the damage that takes place in the first place? And then how can we bring our system back and recover? So those are the nuances, reliability versus resiliency. OK, so let's talk about reliability first, OK? Am I right to think that there are additional reliability considerations and issues when you move toward clean energy and you move toward this collaborative handshake in terms of the concept of the community and the utility? Yes, absolutely. And I think maybe the simplest way to think about it is, like you said, Jay, in the old days, the utility had its handful of power plants. We had our lines and poles going out into the neighborhoods. And essentially, we could easily tell when something went wrong. And I think the main thing here is the flow of electricity is just in one direction. Right, right, right. In the future, when we talk about having, literally, thousands of customers playing a real time active role in producing electricity, obviously consuming electricity, now you're talking about how do you manage the reliability when there are so many different inputs and factors. And little things can go wrong over here. Over there, you have a resource to compensate for that and so on, so it is much more complicated. One of the, I guess, the fundamental tenets that we have to think about as we're designing this future utility grid is that all of these resources, even though they may be relatively small by themselves, they can all play a key role in either supporting reliability or also harming it if they haven't been designed correctly. So yeah, it is going to be more complicated. I think my overarching message is that all of us, all of our little resources, will play a very important role. It's very important that we all have those resources, though, to serve as tools that the system operator can use. It is going to be more of a challenge, I think, if we start to lose the control of certain resources. Certain resources may be more flexible and provide a lot more value. What's a resource? I'm sorry. When I say a resource, that is my generic term to consider that a customer's load, your energy use, can be a resource for me. If I'm the system operator, if you are a generator, you are actually producing electricity, that can also be a resource for me. The information that you may have in terms of what you are planning to do with your usage can be a resource for me, because that can help me actually plan ahead. So all these different, these are different types of resources than what we have traditionally thought of, like a resource being a power plant, for example. The inputs into the system. That's right. So the question is, how do you do that? Little black box, about that big. Well, I think that's one of the challenges for the Hawaiian electric companies. I'm having visibility within their system so that they can make these kind of split second decisions to balance the system because it always has to be in constant balance. And so not having the tools like advanced meters. What are the tools? I mean, by this time, you must have an idea about what tools are necessary in terms of function and who's making them and where can you get them and how can you replace them if they fail and all that? Well, I would actually correct you on that. I think we have a general sense of the types of tools that are going to be needed. But as far as knowing, OK, this particular manufacturer provides this piece of equipment, and it'll cost specifically this amount of money, that's where, depending on the type of technology you're talking about, the jury is still up. Is it jury out because there are multiple possibilities and that, I mean, I'm really interested in, is it that you can say, Jay, you can use A, B, or C, different brands, different just follow these specs? Or is it that you totally can say, look, Jay, you better use A because A is the one we like best and that's the one that's safest. Which way do we go on that? I would say it's more of the, and let me use an actual example to sort of see our thinking. Let me use, as an example, a plug-in electric vehicle that actually is capable of not only taking electricity from the grid as it's charging, but also perhaps delivering electricity back. Sometimes people call that a vehicle to grid type of system. That's a great resource because, hey, if I need somebody to use electricity at a certain time a day, good, charge that car. And if I need somebody to give me some juice back to the system, OK, good. That car can provide that to me. I am not at a position today to tell you, Jay, that, well, OK, already the technology is robust enough that manufacturer A is the one to go with. Because although I know that certain car manufacturers are already basically demonstrating this capability, it's not enough that they've been able to really bring it to market. And so I can't really tell you, or I wouldn't want to tell you or recommend, which vehicle to go with at this time. It's just still developing. But yet we know already that in the future, that's the kind of technology, a consumer, almost product that is going to be real critical to making this whole system work. I'm really curious, though, when we get there, and we know all that we need to know, and we have pilot programs to rely on and all that, are you going to say to me, Jay, A is the one? Or are you going to say, Jay, you have choices. You can do A, B, C, or D. It's sort of like the telephone company. Originally, they had their own phones. They owned the phone. And you rented the phone. But then there was a flood of phones on the market. We all know this before cell phones. Is it going to be like that, or is it going to be just one? I highly doubt that it's just going to be one, because these types of technologies, I mean they're not going to only be valuable here in Hawaii. They're going to be valuable on the mainland and other countries. So I would really expect that there is going to be a good competitive market for different types of products that will meet certain standards. Do you see that also plug-and-play? Yeah, that would be the best. But part of the challenge is all of these standards have to be developed. And that alone is a safe process. Well, they have national standards for it, international standards for it. Do we not have standards for plug-and-play now? Well, see that's a little bit tricky, because this great 100% renewable is that Hawaii is launched forward. And given that we have relatively small island grids, they don't have, frankly, they don't have the national standards in place which would really necessarily govern what we need here in Hawaii. So Hawaii, we are actually on the leading edge of actually identifying the needs. What should the standards be for our systems? And then ultimately, that will be what we are targeting for these technologies. One of the challenges, though, of course, will be that if we come up with such nuanced standards in Hawaii that these big manufacturers say, well, that's really, really going to be expensive and it'll only be good for Hawaii, then it may take some time for them to actually give us the types of technologies that work here. Well, we could be leaders in this. Oh, yeah, absolutely. We're going to end up being in the center of establishing this standard because we're willing to go the extra kilowatt. Yeah, I mean, when people talk about adopting best practice, I see Hawaii as developing the next set of practices. How do you see Hawaii going forward to a new utility?  I've been thinking about this, I know you have, of the future. Say 10 years from now. I'm not talking 2040 just yet, but 2045, I know you like 2040, but whatever it is, how do you see it in the next, say, 10 years? What's it going to look like? Well, I think eventually, it's really moving to the utilities being the balancing agency in maintaining the grid and monitoring the participation on the grid for services. But I see that as a long time coming because- How many years do you see it? Oh, God. I love asking these questions. That's what they pay me the big bucks for, yeah. You know, it's how fast we can move on developing the right rate structures because if you're looking at adopting a market approach, the rates has to be right. You know, the right price signals have to go out. And that is a slow process. How slow do you think it is, Scott? Well, you know, I think there's part of the challenge is that there's a number of different areas that have to actually transform. I mean, a lot of times people just talk about, well, what's the utility got to do to transform? You know, we can transform all we want, but if other sectors are not transforming as well, so technology development sector. Regulatory structures and how we are regulated needs to transform. Our customers' expectations and the market, that all has to transform as well. So when you put all those together, yeah, it can be a long time. I do think that- Five years, 10 years? Yeah, 10 years, 10 years maybe? I don't know, I'm not. I have no secret crystal ball about that. It does really come down though to having to have a broader systems approach. You know, when I've just rattled off, you know, these three or four or five different areas that need to transform, that's gonna be critical.  Yeah, this is hard business because it's a moving target, all these, and you don't have control over everything. You have to negotiate with these resources and people and agencies and what have you. Well, you know, going back to what Scott was saying earlier about, you know, he can't tell you what technology to buy right now as, you know, they're being developed and tested and nor should they, you know, if we're really looking at a market approach for various services. So, you know, the big question is what kind of investments do we make today that we will have no regrets? Who's we? All of us. You know, what? The rate payer, the utility, the state. Where do we put, where do we put the subsidies? Where do we put the taxpayer money? Where do we put the rate mayor pay? She says it's a big question, Scott. So, let's see, so you're asking me what's the big answer? Well, I mean, that's what you're struggling with with the PSI key, right? Yeah. Who's gonna pay for this? The rate payer. Well, ultimately the rate payer, but here's the other aspect. Okay, the other area that needs to transform is we as a state need to be smarter in terms of how do we actually monetize all of this good work that is happening in Hawaii? You know, how do we actually... The know-how, the best practices. The know-how, the intellectual property value.  And meanwhile, we're actually helping these folks develop their technologies and then they go off. Yeah, that's true, that's a good point. So, but you're talking about intellectual property protection, what? Well, I mean, that's just one component of it.  What about as we design and test the new piece of equipment on our electric grid here? What about that company and whether it's Hawaiian Electric or, you know, other estate agency actually having a discount? We have to be mindful of that because if we're gonna be out there in the front leading this and doing all these innovative things that help other people elsewhere, we should have some sense of control on it. But let me go to the other point that we need to discuss before we run out of time and that is resilience. Resilience in the face of climate change and sea level rise and extreme weather, all those things. How do you make, how do you plan to make? How can you make a grid? The grid of the utility of the future, you know, protect itself and be resilient against those things? Yeah, well, so that's where it really, we have the opportunity, I think, as we are trying to design this future system to make sure that we are thinking about those scenarios that we might have to face as a state in the future. You know, so if, for example, our system of the future is gonna be much more distributed, for example, where we have distributed generation, distributed energy storage devices all around our island, how do we make sure that we are able to, let's say, compartmentalize sections of the island? You know, let's think about microgrids. What could microgrids, what role could a microgrid do on our island? It's like a ship, we have compartments. What a safety compartments. So if one compartment gets flooded, the other one doesn't, same idea. Right, right, right. And you know, this is the part of the discussion that I think, frankly, I think we need to elevate more because I know all of us are so ingrained in talking about the 100% renewables and what specific generation devices are in there and have you. Let's not forget that if we don't design resiliency and reliability into that system, then it's sort of, you know, waste time, right? Because if I can't rely on my power being there, then, you know, all the work that we're doing to try and build the cost effectiveness and clean into our system will quickly be sort of overtaken by not having a functioning energy system for economy. So.  Right, right. They gotta have them both, they gotta work on them both at the same time. Yeah, and I think, you know, we have to be very cognizant that even though there's a really big push to move towards distributed energy resource, distributed generation, roof top, there's, you know, a lot of chatter about that, the majority of the customers still get their electricity from the electric utility. What, 70%? It's probably gonna stay that way too. Yeah, yeah. So it needs a systems approach, is what it means.  You know, we are still talking about perhaps hardening of our system in certain areas. Let's say that there is a critical substation or a set of poles that actually need to be strengthened. Right, that's not the kind of stuff that, you know, you see, you know, in the headlines all of a sudden or we don't talk about those things at conferences, but yet those are actually very critical to really being the backbone of this energy system. And so I do think it's gonna be important that we are mindful that those things are still very important. We didn't have a real time to talk about cybersecurity. I think everybody is mindful that that is becoming more and more top of mind. I think the challenge generally with cybersecurity is that also as we move forward, and we have now thousands of entry points or end points, as you will, we have to make sure that any one of those points isn't somehow a way to somehow bring the entire system down. So, you know, it was interesting.  I mean, is that something you can share? Let me say this. The problem about building a system that deals with the handshake and the collaboration with customers at the same time is resilient against climate change. And at the same time has cybersecurity protections in it. That's not easy. And all three, not all, just two, but all three have to be dealt with at the same time. And I agree absolutely. Going forward, that is a big issue and will become a bigger issue. Just look at Mr. Putin for a moment and you will see what I mean. Anyway, thank you, you guys. It's Scott Hsu, vice president of public affairs and also a systems engineer. Yes. I mean, Marina, former chair of the PUC. Thank you so much. Thanks.


OK, thank you. So now I will continue with this series of examples of partially hyperbolic systems. But as I announced this morning, now we will switch to give examples of partially hyperbolic systems in a weaker setting that I recall again. So we said that F was partially hyperbolic if there existed a filtration. A filtration was something like this. And all of these open sets were attractive neighborhoods, meaning that F of UI closure is contained in UI such that the maximal invariant sets in this neighborhood is partially hyperbolic, splitting this meant that d lambda i splits as a stable plus a center plus an unstable direction. OK, so you might say, and these filtrations were something like this. You have this open set, first is the whole manifold, then you have an open set over here, an open set over here. Each one is mapping inside itself. And what you're looking at is this lambda i corresponds to the set of point that remain here in this region over here. And now what we are asking is that the dynamics might not be globally partially hyperbolic, but inside each of these sets, we have a partially hyperbolic splitting. And essentially, this, in principle, provides a larger class of examples. So let me tell you that, in fact, it provides a much larger class of examples. And by this, let me just recall a very old theorem by Schubert's Mail that says that hyperbolic diffeomorphisms are dense in the C0 topology among diffeomorphisms. So why is this result interesting for us? Because this morning, we had to work a lot in order to obtain examples of partially hyperbolic systems. And each one was kind of difficult to get. So we had these algebraic or geometric examples, which were the starting points. And then we had to start with them to construct new ones. And we have no other way of constructing examples. And in principle, as in the example I mentioned this morning, and Silvan also said that this theorem by Franks and Newhouse says that being hyperbolic in the whole manifold gives a lot of constraints on the topology of the manifold. However, here, this theorem is telling you that whatever the manifold is, and whatever the isotopic class of diffeomorphism you're looking in, you can approach it by a hyperbolic diffeomorphism. And so this local property, the fact that we are asking hyperbolicity only in the recurrent part, gives a lot of liberty to create examples. So now we will look more to localize kind of examples. So the first type of examples I will do is the ones I promised in the morning, which are skew products. Let me say what a skew product is. So you have F from M to M, a diffeomorphism. And you have K inside M, a partially hyperbolic set with splitting TKM. OK, so now we are not asking the whole manifold to be hyperbolic or partially hyperbolic. We just need a compact invariant set, which is partially hyperbolic. So for example, you can choose the horseshoe that Silvan explained the other day, a hyperbolic attractor, or even a periodic orbit or whatever. And so now we will choose she to be a function from M to the space of diffeomorphisms of another manifold, N. But we will require that the contraction and expansion that this diffeomorphism have in the direction of N is smaller, is dominated essentially by the expansion here and the contraction here. So how do we write that? We write it like this, that for every x in K, if you are outside K, we don't matter. It's OK. So we have that the ES is smaller than the minimal contraction of she here. So she is a diffeomorphism of N, so we don't have to restrict to anywhere. This is always smaller or equal than the she, x, and the norm here, and we require this one to be smaller than the lowest expansion here. We wrote like this. So now, what are we going to look at? We're looking to look at this diffeomorphism times N such that f of x, y equals f of x, she, x of y. For example, an example we did in the morning, so you just take she to be constant equal to the identity, and so you will always have this condition. And the point is that you have the following proposition that tells you that the set K times N is partially hyperbolic FBF with splitting PK times N, N times N. And we can say a little bit more. We can say exactly who this bundle is, and we can say more or less who these bundles are. So what we know is such that the bundle EC hat is exactly the bundle EC that we have here times the tangent bundle to N. And these bundles, we don't know exactly where they are. They need not be the same bundles that we had for F, but at least we know that they will project into these bundles. And if pi from M to N to M is x, y maps to x, then the derivative of pi sends each of these bundles into the correct one. So when you multiply here by the identity, it's an exercise to show that the new bundles are exactly the stable or unstable bundle. You have times 0, so there's no contribution into the new part of the tangent bundle. However, when you make something more complicated, which is mixing, then it's possible that the thermomorphism changes the angle of this. But by a simple cone criteria, you can show that this is partially hyperbolic. So the proof of this is really easy, and we are leaving it as an exercise. So before I continue to explain why I wanted to do skew products over general sets and the motivation for this, let me first give an example of a skew product, which is not so nice in the sense that the manifold is not a product. So it's a very beautiful example, due to Bonati and Wilkinson. So the example is older than this, but the way to see it appears in a paper by Bonati and Wilkinson. So let me explain this, because it shows how you can construct non-trivial skew products. So the idea is you start with this manifold, T2 times the circle. And this is a trivial product manifold. But now you want to construct a circle bundle over the torus, which is not a trivial one. So how do you do this? You cut. So it's related to the surgery thing we talked in the morning. So cut T3 along a circle. We have this T3 over here. So this is T2, and here we have S1. And we cut here along, we make like a tube here, and we remove this tube we have here. And we glue it again by pushing fibers when you do a circle. So what we are doing is remove a solid torus and glue again, preserving the fibers. So you have now T2 minus a disk times the circle, and you remove D times the circle. Minus, sorry. You have these two manifolds now. The boundary of each of these manifolds is a torus. This one, the boundary is equal to a torus. And this one, the boundary is a torus. And each one of these torus in this boundary is a circle. So you have a foliation of this circle. Let me, this represents the boundary of D, and this represents the circle. So I cut these torus away, and then I glue back. But now, instead of using the identity to glue back, what I do is I start rotating in each of the fibers. So this fiber, I glue it correctly. But then this one, I glue it a little bit pushed away. And when I make one turn, I did one turn of the rotations. So if you like a formula, you glue, like if X, let me use another S, parametrizes this, and T parametrizes the circle. So you glue with S, T, moves to S, T plus S, or M times S. When you glue this, you can show. So exercise, the new manifold is not T3. In fact, it's what's called a nil manifold. And now, what we are going to do is to construct a partially hyperbolic diffeomorphism here, which is, OK, but this is still a circle bundle over the torus. You have a projection, you have not changed the basis. So you can project to the basis, and you get a torus. And so now, you wish to construct a partially hyperbolic diffeomorphism, which projects to an anose of diffeomorphism. So let me explain how to do that. If you don't believe in this construction, so let me say what do we have? We have n, which is a manifold, which is a circle bundle over the torus, meaning that the preimage of each point here is a circle. And this manifold is not a torus. And what we want is to define a function f from n to n, which is partially hyperbolic, and such that if you project, so if you do f composed with p equals 2, 1, 1, 1 composed with p. We want to make this example. And so to have it partially hyperbolic, a nice way to do this is to preserve the circle fibers. So p goes from the manifold to the torus. And I'm assuming this is acting on the torus. So essentially, what this means is that this f will preserve the fibers. And it will move them as an anoso. And the easy way to guarantee that this will be partially hyperbolic is to know that the action on the fibers is an isometry. If we manage to do that, it's easy to see that it's partially hyperbolic. It's essentially the same idea that this construction. These are also called SQ products. So let me explain the trick. So choose U and B in T2, such that you have the following design. So this will be U, for example. And B will be something like this, the outside part. So from the way we constructed the example, we can choose trivializations of the neighborhood. This is a circle bundle. So you can trivialize. So let's choose trivialization phi 1 from U times S1 into N. And phi 2 from B times S1 into N. So these are diffeomorphism, preserving the fibers, or something like this. So that if you look at phi 1 composed, phi 1 minus 1, composed with phi 2, which goes from U intersected with B times the circle to itself and fixes the fibers, then these are all rotations. It's exactly the way we constructed the manifold. So you are cutting, making a turn. Here you are just making translations. And then you are gluing back like that. So you can choose these trivializations. OK, if you don't believe me, that is the argument. Otherwise, you can believe that this is possible. And now what we are going to do is to define one diffeomorphism, which is supported here. One which is supported here so that the composition will be nice. So exercise show that 2, 1, 1, 1 can be written as C1 composed with C2, such that C1 is the identity where C1 restricted to B is the identity, and C2 restricted to U is the identity. How do you do this? So that's why I draw U very big and B very small. So you just choose a diffeomorphism, which is equal to a nozzle here and glues to the identity outside.  So you have the identity in each one. And so once you have this, you can define C1 from N to N, such that C1 equals identity on B, and equals, how do I write it? Phi 1 composed with C1 times identity, composed with phi 1 to the minus 1. And C2 equals the identity on U, and equals phi 2 composed with C2 times the identity, composed with phi 2 to the minus 1 on B, on the rest. So doing this, you get that the diffeomorphism F obtained as C1 composed with C2. This one has exactly this property, because in the basis, it's the composition of C1 and C2, which have this property, and it acts as isometries on the fibers. Because here it acts as isometries on the fibers, here it acts as isometries on the fibers, and the change of coordinates are isometries. So it's an exercise to show that F is partially hyperbolic. So this was just to give an example of a non-trivial skew product, so that the topology might change when you do this construction. So let me just quickly mention some other reasons why studying skew products may be interesting. So this will be just a list of places in mathematics where skew products appear, but maybe some of you are interested in these other things. So in general, there are several people which are interested in what's called iterated function systems, which essentially require, if you have a finite number F1 Fk of diffeomorphisms of n, then you want to understand the dynamics of the semi-group generated by these functions. And a nice way to study these objects is to construct a skew product, which is partially hyperbolic. So how do we do that? You take a horseshoe with K different legs, and you make a product which is equal to F1 in this leg, F2 in this leg, F3, et cetera. So when you make this skew product, what you obtain is that the dynamics on this set K times n is, resembles a lot, the dynamics of this iterated function system. And so you can always choose, and this is a very interesting remark here, that independently on the strengths of these functions, you can always create a horseshoe for which the contraction and the expansion are stronger. So you can always embed this into a partially hyperbolic system. So OK, this is partially hyperbolic. And essentially, this idea appears in many other contexts in mathematics. For example, when people look at random dynamics, which is like choosing at random a diffeomorphism and applying it to a manifold, usually the notion of randomness is very related to the notion of a shift in a certain space, which is a horseshoe. And you can always recreate this type of study by making a skew product over a horseshoe. The same goes for co-cycles of diffeomorphisms. And this is how a very typical hypothesis appears, which is sometimes called fibered manching. This hypothesis that appears a lot in the study of co-cycles has to do with the fact that a skew product will be partially hyperbolic. So now in the rest of this lecture, I would like to focus on one thing that will be very important for us, which are attractors. So let's speak about these attractors. So when we discuss these filtrations, so if we have a partially hyperbolic diffeo, we have these filtrations, which are mapped inside themselves. And this separates what we call the chain recurrence classes. But sometimes understanding all chain recurrence classes might be a very hard thing to do. And sometimes we focus on a particular class of chain recurrence classes, which are interesting because they are attractors. So at least a lot of points, we know that a lot of points go to those classes. And so let me just introduce a notion that will reappear mainly on Friday's lecture, which is the notion of quasi-attractor, definition. Q is a quasi-attractor if two things. On the one hand, it is a chain recurrence class. And on the other hand, it admits a decreasing basis of trapping regions. This means that you can write Q as an intersection of Un such that f of Un is contained in a particular class of quasi-attractors. Attractor, if not only it is, but the same open set, you can obtain the set A. So if it is a quasi-attractor, and A equals the intersection of fn of U. You can find the particular trapping region is just one open set. But why do we focus more on quasi-attractors than attractors? There are mainly two reasons today. One is because it's easier to guarantee that something is a quasi-attractor than an attractor. In general, we don't know when we take an open set, we send it inside. If it will stop somewhere and have new points, or it will continue forever towards our attractor. And this is related to the fact that these ones exist, and these ones not always exist. These sets always exist. And they have very similar properties to attractors in a certain sense, so we prefer to study these ones. No, the example would be something like this. So take the dynamics on the line, and take a sequence of periodic points converging to this point, and make something attracting, repelling, attracting, repelling, attracting, repelling. So this point here, on this side, you put attracting. And so this point here has a basis of trapping neighborhoods that defines it. But it's not an attractor, because whenever you take an open set and you iterate forward, it stops somewhere. This is, of course, very non-stable. But in dynamics in higher dimension, this phenomena can be quite persistent. And so it's important to treat these kind of examples. So just to mention some examples, let me explain generalized solenoid, which is an example due to Bonati, Mingli, and the way Yang. So maybe you already know the solenoid. In any case, I can repeat it. You consider a solid torus, and you make it C-mash so that it makes two turns around the circle. And so now the usual construction of the solenoid makes this to be a hyperbolic set, a hyperbolic attractor. If you know this, you can construct this in order to preserve the foliation by disk here. And so you have an expansion in this direction, but and a uniform contraction in this direction. However, you can imagine to do whatever you like here as long as you preserve these disks. And as long as you don't expand in this disk more than this in this direction. So if you want, you can think about this example as a skew product over the doubling map in the basis. You can think about it as a skew product. So each disk maps to another disk. But the mapping depends on the point. And you want that the expansions inside the disk are less than the expansions than two, for example. So this map, you can write it in d times s1 as a mapping of the form x t maps to f t of x to t. I'm seeing the circle as 0, 1 quotient. And so what we require is that the derivative is smaller than 2. And if we have this, we will have here an invariant cone field that will give a partially hyperbolic splitting for the maximum invariant set here. So what I would like to explain is an argument due to Bonati, Li, and Yang, which shows that if we do this correctly, we can ensure that there's at most one quasi-attractor here. If we do it freely, we can make things here to be quite wild and that many classes appear here. In fact, what they do is they construct examples which have infinitely many chain recurrence classes inside here. But the point is to show that there's at most one quasi-attractor. So let me explain that. For the moment, no restriction. Now I will put one hypothesis that will guarantee the existence of a unique quasi-attractor. The derivative of f of t is less than 2. And another condition is that f t of the disk is mapped inside the disk so that we have this drawing. We require this to be an embedding. So we are not working in a global manifold. We are just defining an attracting region. So the map is a map big f from here to here, which has this form. And what we require is that the disk is mapped inside the disk. But it changes one disk to the other with the rule of the doubling map here. And then we ask for this so that the maximal invariant set here will be partially hyperbolic. So no other hypothesis. In fact, that's the reason they do this, to be able to play with this. And then Bonati and Shinohara have continued to play with these examples to get more interesting properties. But the only property I will show is that under some assumption, you have only one quasi-attractor just to explain a mechanism. So proposition, assume f of 0. So the map f of 0, 0 is a fixed point of this map. And we will ask that the map f of 0 is a contraction. It's a real contraction. It has a unique fixed point, and everything goes there. It's a uniform contraction. Then f has a unique quasi-attractor, the proof of this. So the reason we are asking this hypothesis is to know that f has a unique fixed point, p in d times 0. The unique fixed point is given because this disk is invariant, and it's a contraction, so you have a unique fixed point. And the idea is to show any quasi-attractor contained in d times s1 will contain this point p. And since chain recurrence classes are disjoint or equal, this completes the proof. So we will show that if q is a quasi-attractor, then p belongs to q. And this is enough for our purposes. So to do the proof, I will explain it. It's very easy, but I will explain a little bit. So because this type of argument will be very important, mainly on Friday's lecture. So why? So choose any trapping u which contains q. And now, if we want to show that p belongs to q, it's enough to show that p belongs to u for any trapping u containing q because we have this property. q is the intersection of every trapping set. So it's enough to show that p belongs to u. So how are we going to do this? So let me make a drawing here. So this here is d times 0. And we are choosing an open set u which is a trapping region. So in principle, it does not intersect here. But well, I won't draw u because we don't know where it is. But let me take any open set inside u which is inside this set. So take this ball b. And so what we know, since u is a trapping region, is that every iterate of b will be contained in u because it's a trapping region. But now, if you look at what's the dynamics here, you get that the projection by 2 of b in the second coordinate contains an interval. So if the projection contains an interval, as we start iterating, we are iterating the doubling map. So eventually, we will intersect this region here, probably over here. This will be fn of b. But as we intersected here, at this point here, as we iterate it, it will converge to the point b. So what we have proved is that p belongs to the closure of the future iterates of u. And the future iterates, we can start by the third iterate if we want, so n larger than 1. And so this is contained in u as we want. So this proves this proposition. And I think just to finish, I don't have time to explain the rest of what I plan. The other thing I wanted to do is to show other types of attractors, which can be constructed more or less in the same way as we did Manier's example today. They are called derived from Anosov examples. But then it's possible to play with which eigenvalues you change in order to have different types of interesting attractors. So since I have no time, in the notes, there's some explanation on how to do these examples. So I finish here. Thank you.


Um, so, hello, and indeed goodbye. I've put together some slides, they're not very long or fancy, but we can maybe go through them, and then if someone still wants to say something after that, we can do so. So, first of all, thank you to everyone. And with everyone, I mean everyone as was possible. Firstly, thanks to our MiniDebcon for online stuffies. Um, I think this is the fullest natypada crop of tumbleweed azim, a very high-voltage abec Jonas. Oh, let's devote to Gnaw and Acute, and if I missed someone out, very sorry, I'll add it to the slides, let me know. Also, thank you to our speakers. Without them, obviously none of this could have happened. Both JP are not Antonio, Christoph, Andrea, Paul, Ben, myself, and Dileb, who just finished talking a few moments ago. Also, thank you to you who have been watching and participating. Without you, all our effort would be kind of pointless. In RSC, I think we had just over 115 people participating. According to our CDN logs, it seems like at least 281 people have watched for at least 20 minutes consecutively. So I guess it was close to 300 people in total that have watched our streams. And then also, thank you to our sponsors. Even though we didn't have particular sponsors for this MiniDebcon, we used infrastructure that we paid with the usual Debcon money that does come from our usual Debian and Debcon money. So thank you to all our usual sponsors. And maybe in the future when we figure out a whole stack, which I think we will get pretty close to after this MiniDebcon front line, then it would be nice to get some sponsors as well, even if it's just for some loose things that we might want to buy like t-shirts. Or maybe we could do things like having a sprint in the morning, like something worldwide where people get together once we can do that. And then in the morning, people in different cities can get together to hack in the morning. And then we have talks streaming everywhere in the afternoon where we get some pizza and beer or something. So these are some ideas I think we can play with that. That's nice. What's the next slide? I know it's an upwards. Yeah, we did learn a lot this weekend. We got lots of feedback. We figured a lot of things out. The stack we used this weekend was very minimum viable product. We wanted to get something that works reliably. It wasn't very fancy. But for the most part, it did get the job done. And the biggest problems we had was when people had internet connections or power cuts. And it seems like Murphy Law strikes when it does the most damage. But that's how things go. And that's it for the slides. And our panel is pitchless. I wonder if there's any more. I guess thank you to everybody. Thanks for attending. Thanks for bearing with us through various mishaps. Luckily, there weren't too many touchwood. Thank you for attending and hopefully see you again sometime.  And this is something else. Is there one more thing? Doesn't look like it. Okay, bye-bye. Oh, I guess there'll be the virtual group photo in the cafe. I'll post the link on our see if someone doesn't have that yet. And then we can get a quick group photo. See you there. Okay, see you there. Bye.


And here are some other words, boys and girls, waterfalls. We learned that one of our insects lives by a waterfall. Does that name help us out? What is a waterfall? Waterfalls to the ground. When waterfalls onto the ground, usually in a river or stream, the water flows down. And here's our word zigzag, and woods, and speedily. We learned that speedily is very fast. And slender, who remembers another word for slender? Who remembers another word for slender? Yes. Thin and skinny, great job. OK, boys and girls, we learned a lot of facts about dragonflies and butterflies. Now we're going to take all the facts that we learned, and we're going to compare and contrast them. OK? Who remembers what we call this type of chart? What do we call this type of chart that we use? Bernardson. I think we call it A. What do we call these two circles put together? What do we call it when we compare them and see how they're the same, and then we contrast them and see how they're different? What do we call that? Can you help him a month? A Venn diagram. A Venn diagram, very good. So let's think about it. Who could tell me one thing that they have that's the same?