Hi everyone, Natalia here. Welcome to a special mini episode of Conservators Combating Climate Change. In light of all that has transpired in these past few weeks and then I thought it was important to take a time and directly address how and why social equity is critical to the conversation of climate justice and creating a more sustainable future. This has come up in all of our conversations with guest speakers so far but we see the heightened awareness of the critically important Black Lives Matter movement as an opportunity to discuss what we have learned about why climate justice is social justice. I'm not an expert but I'm going to try and tackle this truly complex topic and I'll do that in three main categories. First, why these topics must be integrated. Then I'll give some examples that will demonstrate how systemic racism is causing a disproportionate harm to marginalized communities both directly from the change in climate and in our nation's response to disasters. I'll end on a relatively high note and focus on one pathway moving forward in our communities and within institutions. The honest truth is that nothing will change unless we take strategic action. Okay so why must these topics be integrated? If you follow US politics you know that many elected officials claim that climate agendas shouldn't conflate issues of environmental regulations and carbon neutrality with quote-unquote unrelated issues like the job sector, health care, civil rights, and so on. The difficult fact that we all need to accept for this conversation to continue transparently is that these problems are connected because they are systemic. Within literature sustainability is often described as the triptych of social justice, ecological integrity, and economic well-being. Others posit that the pillars of sustainability are actually biodiversity, cultural diversity, and human well-being. With either of these definitions we can see that diversity and equity are defining properties of sustainability and cannot be separated. We also know that widespread societal changes need to be enacted and quickly to avoid the long-term catastrophic effects of human induced climate change. To create this future we must address the energy sector which means we need to critically evaluate the transportation, agriculture, manufacturing, industries. Every aspect of society must be evaluated and adjusted accordingly. And since we live in a country where racism and bias are seeped in the foundation of our nation these injustices are likewise seen in all of these different sectors. It's my opinion that if we separate these conversations we are perpetuating the injustices of our system. So let's talk about those injustices. There are a devastating number of examples that show how marginalized communities are being disproportionately affected by the effects of and response to the climate crisis. Specifically within the U.S. marginalized communities means communities of color, low-income communities, indigenous communities, women, and non-conforming youth and elderly persons. As Maddie Cooper said in episode 2 climate change is a risk multiplier so climate change compounds these inequalities. The changes in frequency and intensity of extreme weather events particularly affect communities of color because these communities are already vulnerable due to economic insecurity, substandard housing stock, the list goes on. The change in climate affects geographical areas differently. For example sea level rise is displacing coastal dwelling communities which are disproportionately comprised of people of color. Climate gentrification has begun to push out black communities in these cities and will continue to accelerate as the crisis intensifies. It also affects zip codes differently because poor communities tend to have less trees which means less shade which results in hot spots which within the media are often referred to as urban heat islands. Studies have shown that individuals in these communities often have higher rates of chronic illness that are exacerbated by the hotter temperatures. Along the same lines the NAACP Clean Air Task Force found that extractive and polluting facilities are frequently operated in majority black communities which compromises long and short-term health of these communities again causing elevated health risks. A report by the EPA researchers in the American Journal of Public Health in 2018 found that black Americans were exposed to air pollution particles at a rate of 1.5 times higher than the national average. We've also come to learn how the response to human-induced and natural disasters disproportionately affects black and brown communities. A harrowing fact that most people are already aware of is that black men are incarcerated at a much much higher rate than any other demographic 70% of incarcerated peoples. Inmate labor was used to clean up the BP oil spill in 2010 and was used in the wildfires throughout North and California in the past years and these people are often being paid pennies on the dollar one dollar per hour in California for extreme life-threatening work in the hope of an early release and it is truly disgusting that this is allowed to happen and will continue to happen legally as disasters occur with increased scale and frequency. The reality is that black communities are being affected more severely than white communities because of the interconnectedness and pervasiveness of systemic racism. I'm sure that you too are feeling a mixture of emotions, anger and grief, frustration and personal responsibility. Adhering these facts it will take an incredible amount of work to create equity but we need to try and start somewhere. The NAACP has an incredible climate justice initiative toolkit that they've released in 2010 it's as relevant today as it was 10 years ago. In it they talk about an equitable green economy and this includes jobs that not only improve the environmental health of the community but also pays living wages, benefits to workers, provides safe and healthy work environment, enables advocacy for ourselves and community. In their toolkit they provide a clear pathway for understanding and creating opportunity. So how does this relate to the conversation about museums and higher education and conservation? Well thanks to Henry we know that built heritage is part of the problem and has to be part of the solution. We can invest in these renewable energy sectors within our institutions and it will be mutually beneficial. It's been shown that sustainable initiatives save institution money and let's be frank in a capitalist, corporative society it often comes down to profit. It will also create a demand for green-collar jobs which we know statistically are mostly filled by black and brown men. This is just one optional pathway forward and it certainly won't change the systemic problems that persist in our country. It is a starting point though and we need to start somewhere. I want to thank you for joining me today for this conversation. I hope you feel some empowerment by the increased information provided and are committed to creating change or at least having these conversations within your sphere of influence. I'd also like to thank the entire ECPN and AIC community for supporting MNI in this new venture and to the University of Delaware for financially supporting our guest speakers in honor of Buonapuya. In our next episode we'll speak with Francis Lukasek who worked with an indigenous community in Alaska who is very directly being addressed by the climate crisis. Until next time.


Hello everybody, this is President DeTier, and I can't tell you how excited I am that in a few days, just a few short days, our Immaculata community will be back together again. October 19th, as you know, our classes start. Many of them will be in person, and it is just exciting to think that after seven long months, we'll be back together again. During the summer, and it was a long, hot summer, we did a lot of work to get ready and prepare for a safe campus. I'm here and gave you the library right now, and when you arrive, you'll notice some changes. Changes such as spacing of the tables, plexiglass on the tables. You'll also notice in your classrooms that they've been completely reconfigured. All the seats have been socially distanced, and we're asking everyone to wear their mask in the classroom. We've done a lot of things to make sure you're safe, including a very intensive cleaning and disinfecting schedule. However, your safety, our safety really depends on all of us. We've done a lot of things, as I said, over the summer to ensure your safety. But here's where we need you to help. We are going to need you to wear your masks. We are going to need you to try to keep socially distant. And we are going to need you to use the sanitizing stations we have spread throughout the campus. There is one additional thing, though, that I really would like to ask you to consider. And that is, I'm sure you've heard over the last several weeks, many schools have had to either suspend or expel students because they had large gatherings, parties of sorts. I will tell you I like the party just as much as you do. However, I think it's in all our best interests to defer those for the time being. That's so that you're safe, we're safe, your family and your friends are safe as well. So welcome back. We are looking forward to seeing you and I'm hoping to have a very successful remainder of this fall semester. Stay safe.


What makes for excellence in heart care? It starts with one of the most amazing teams of heart surgeons and cardiologists that I've seen assembled under one roof and the region's leading technology in a new $40 million facility. This is the nationally recognized Heart and Vascular Institute of Eastern Kentucky at Pikeville Medical Center. The city of Pikeville is seeking to upgrade their radio system via the Motorola Solutions The goal is to improve communications of the local police and fire crews across the city. This will help first responders stay in touch while monitoring the streets and call for backup if they are found in any dangerous situations. Our radio system has struggled for some time. We have essentially one tower in the city that provides connection to our radio system. When officers are on the extremity of the city, either the extreme southern end or the extreme northern end, sometimes that those radios do not work quite as well as we would like. Roughly $300,000 is being invested to improve the city's communication lines. This money will go towards replacing the current radio tower as well as adding a brand new one. So we have for some time been working to identify another solution for how to solve that problem. And we've worked with some outside consultants and identified a number of paths. And we have recently invested about $300,000 in our police and fire radio system that will replace all of the radios in their vehicles, their rebel radios. The handheld radios that they have are good. They don't need to be replaced. But it will also replace a tower site and add a second tower site. So that invoice is just one of the invoices as part of that project. The public safety radio system allows our first responders to better communicate, keeps them safe by having essentially a lifeline to dispatch. Should something happen that we can order to other officers or other firefighters, they need to respond. That is their safety is paramount to the city and the city commission. Reporting for Mountain Top News, I'm Nick Colum.


Welcome to Toffee TV, it is the start of the 11th show Everton vs Norwich City at Goulderson Park. A potential banana skin that hopefully ever can get over and get those three points. Get us back on track for the beginning of the season. In goal, Jordan Pickford is available, he is training according to manager Rafa Benita so he is very much back in the team for me this week. There's absolutely no doubt about that if he's fit and he's ready to go then he is back in the side because he's just a much better goalkeeper than Osmio Begovic but also one of the things I think we've missed with him not being in the last couple of games is it's just kicking as well. It's kicking is fantastic, his ability to start attacks and keep possession from the back as well so gives us something a little bit different so he is back in goal. Flatback for me at right back it's going to have to be Ben Godfrey and continuing from last weekend's villa. I don't think he covered himself in glory but needs more shameless communism available yet. John Jo Kenney is not the answer and Ben Godfrey will start at right back. At left back there was a few concerns over Luca Dean but he is training and he is available for this game so he starts for me. Scored on Tuesday night which is great for him getting him back in the goal scoring and hasn't done that for a while. Went off with obviously 10 minutes to go but it looks like it was just a precaution and he is he will be in the team for tomorrow and that leaves Michael Keane and it leaves Jeri Mina to play at the back. Ironically probably probably aren't much solid pay at the moment. Those two will continue at the back and hopefully be able to deal with whatever Norwich Thrawarres which hasn't been much this season from them but you know how it is so we look all right at the back. I mean Ben Godfrey it's not ideal but hopefully they have a good game against this former team. Inter Sentiment Field, Allen and Decoree retain obviously they're our best partnership and I think with those two in the middle of the park we can control the game from there a lot better and really there isn't anyone else out there because Andre Gomez is injured and nobody else even looks anywhere near those two so that's just the way it is but hopefully if those two have good games we should win the game. Head of them I've gone for Alex Oboby again last couple of games hasn't been great but what can we do we haven't got lots and lots of options we really are down to the bare bones in you know every single position or behind that we know the squad's not great and it's about utilising them and Alex Oboby's actually said this week that he feels like he has to start doing something and having a level of consistency and you know it's it's on him so let's see what is against him I don't think he played well against Villar don't think he particularly played well against QPR so you know playing at home probably in a more favoured position as he has so far under Rafa Benitez he's got to start pulling that consistency out every single time he plays forever which will be more and more if he does if he does find some kind of form. On the right hand side Andres Townsend who has been fantastic I think so far this season in terms of consistency and work rate scored goals assists hopefully becoming someone we can rely on yeah I mean that's all you can ask for someone you can rely on week in week out to get a seven out of ten performance and he's done that so far so let's hope that continues on the other side Damari Grey who again is probably being as sharp as player this season the player that you look to to just inspire us a little bit get get fans off off the edge of the seats but he's provided goals as well and assists so he's doing fantastically well and let's hope he you know he can add to that against Norwich because it's going to be needed and up front Salomon Rondon will continue for his third game starting in a week it's not being perfect it's not probably what anyone wanted after only you know after five months not playing football but he's had to suck it up and he's had to he's had to do it and hopefully tomorrow we get a little just a little bit more out of him and hopefully that leads to goals that's all you can hope and listen it's just it is a stopgap but while the Charleston and Dominic Carmelune are out the side it's not perfect got to rely on other players probably to get the goals for us the likes of Grey the likes of Townsend at the moment hopefully a Wobby the Coray they start you know adding to the tally and hopefully every week as he just plays more and more football we get a little bit more sharpness out of him I think that sharpness listen it's never going to be anywhere near a Charleston Dominic Carmelune but I think if you put the ball in the box crosses you'll always have a chance of being on the end of it if he's sharp and that comes with fitness that match fitness so he's starting there's not much about that maybe with another week of training maybe the likes of Ella Sims might be now ready to step off the bench because we're going to need that because the bench is going to be very very light against Norwich City let's be honest so if we do need a game changer let's hope someone like Ella Sims is now getting that little bit extra obviously he's been out for a while as well so let's see how it does let me know your thoughts in the comments if you would make any changes this side I don't know how many changes you actually could make with the pace and then we've got available to us at the moment so there it is that's my 11 for the game against Norwich City let me know your thoughts in the comments give this video a like make sure to check out our match preview make sure to check out our opposition match preview which is available over exclusively on Patreon if you want daily live video which join us on Patreon as well the link is in the description and it's also on the screen now see you later


Duo Fragments' Fierce Attar Re-Review. Hey, what's up guys, it's Josh here. Today I wanna do a little re-review on this Duo Fragments' Fierce Attar. Now I've almost sprayed the full batch of this one. I have sprayed 100% of the original Duo's Fierce. That one's been long gone. And I do have a little bit of Fierce Supernova as well, which I really do like. I like all three of these. And one of the best things I could say about Duo is that they took their time fine tuning this classic fragrance. So it's almost like anything that they do with this fragrance is gonna be at this tip top quality. Now I really wanna do a re-review back in March when I re-sprayed this one. However, I've been kind of revisiting it. Been kind of like getting the last little drops out of it. And I really don't think my review is gonna change too much from the original review. This fragrance is absolutely fantastic. I was actually surprised that a lot of people who like Fierce weren't really a fan of Duo's Vintage Fierce. Well, I mean a lot of people did like it, but there was a few people who it just really wasn't for them. There was almost like a body odor type vibe to it. Almost like a teenage type vibe to it. And some people actually gave the fragrance away, which actually just blew me away because Duo's Fierce was just so phenomenal when it comes to Fierce. Next to the original classic Fierce, it's for sure number one if it's not number one. So this really takes all the strength of the original Fierce and just takes a little bit of that sharpness, a little bit of that immaturity away, but it still smells very, very Fierce-y. I do crave it to have a little bit more of that classic sharpness and that classic slight immaturity to me just because I've smelled it and I think of that in Fierce, but this stuff is absolutely phenomenal and it's nuclear. It's one of those ones where I could be on a hot summer day for sure, spray this two, three times and I'm really almost golden for the rest of the day. I will probably get whiffs of myself, seven, eight hours from now. If I were to hug anybody, they would definitely smell it. If I were to get really right next to somebody, even at the end of the day, I'm pretty sure they would smell it. It's gonna last on your clothes for, I mean, it's probably a few days. It just has nuclear projection and it's just so pleasant and so nice and I really wouldn't say this one is stronger than the original Duo's Fierce. I think that that was kind of the marketing campaign is that the Attars is just like the maximum amount of everything they could put in the fragrance, so much so that it makes the original fragrance different but stronger. I don't really get that one with this one. I get this as more of a less immature Fierce.  All the fragrance notes are just popping out in the highest qualities. I do smell a little bit more lemon. I would say in the original Fierce and the original Fierce or Duo's Vintage Fierce is a little bit more fresh here, but this maintains most of what I like about Fierce. You know, if I were to compare this to the Fierce Intense that I've smelled, the Fierce Intense, now this is the 2016 version. That version is more sweet, more spicy for sure, much more of a nighttime scent, much more of a cold weather scent, but still smells like Fierce ironically. This one, I guess it would still be more of a hot weather style fragrance, maybe a little bit more of those middle temperatures. If I were to compare Central Obsessions Vicious with this one, Central Obsessions Vicious reminds me of the 2009, 2010 Vintage Fierce, not only Vintage Fierce, but that batch of Fierce. Now at that time, we're getting out of the less impressive batches. I would say the most shocking and most impressive batches have been the first two, which is like the red letter and the silver cap, those for me personally, but I got into Fierce around 2009, so that was fierce to me, and it just smells like those classic style fragrances, and they're all good.  So anyways, guys, my updated review, it's probably gonna be in line with the original review of Fierce Attar because I pretty much have the same feelings for it. I absolutely love it. It's a powerhouse, it smells so clean, and it's pretty much near perfect. I would slightly prefer do as Vintage Fierce to this one. I would also probably prefer the actual Vintage Fierce that I own probably over this one only because the original Vintage Fierce for some reason gets like so many compliments. It's just absolutely crazy. So anyways, guys, let me know what you think of Fierce Attar. I know that they've dropped some other ones and I know Alexandria has their version of Fierce, so I'll be interested to try that one out as well. But anyways, guys, we're on the road to 50,000 subscribers, and I couldn't do without any of you guys' help. You guys are the best. I'm having a great day out here. Hopefully I'm having a great day at home. See you all in the next video. Peace.


Excuse me, miss. The other night you took something that doesn't belong to you. Stealing's not polite. Is it stealing if you steal from another thief? You're not brave. Men are brave. You're psychotic. That's a three syllable word for any thought too big for little minds. It's time you learned what it means to be a man. Stay down. If I wanted it, you'd be dead already. I'm really confused why they are fighting. Like they couldn't talk it out for a second. I mean, these are both reasonable guys, right? You're not brave. Men are brave. You say you want to help people, but you can't feel their pain, their mortality. It's time you learned what it means to be a man.  I don't, it's just humble brags because Batman's a man, so that's all it takes. Boys, Bruce Wayne meets Clark Kent. Ah, I love it. I love bringing people together.


Hello, I'm bingo McLean. Like you, I'm an airman. For more than 50 years, I've been privileged to serve as a member of the greatest Air Force in history. First as a cadet, then as an officer, and finally as an Air Force civilian. My career has included C-141 Navigator, F-4 Weapon Systems Officer, Exchange Duty with the U.S. Navy as an F-14 Instructor Radar Intercept Officer, Multiple Staff Tours at TAC, PACAF, UCOM and the Air Staff, and finally as a Doctrine Analyst with the Curtis E. LeMay Center for Doctrine Development and Education. Now, that doesn't make me any smarter. It just makes me older. But in getting older, I've been through many of the issues that you younger airmen will have to deal with in your careers. Been there, done that, got the t-shirt. Maybe a little of what I have picked up along the way can help you in your career. At least I hope you'll think so when I'm done. One of the major issues that all airmen deal with, especially when deployed, relates to the subject to command. Who's in charge of me? Who has the authority to give me a mission and the responsibility to be sure that I am able to carry out that mission? Are we doing things right? Are we doing the right things? The starting point to answering these questions and to gain a better understanding of what is involved in commanding airmen to exploit air power can be found in Air Force Annex 330, Command and Control. The complete annex can be found in the U.S. Air Force Doctrine website at doctrine.af.mil. For airmen, it is time well spent to read and understand the whole annex. But for now, let me just hit a few of the high points for you. Chain a Command, Joint Force, Component Commanders and Command Authorities in Relationship. Been said that a picture is worth a thousand words. Well, this one may come close. In this chart, you see all four of the points I just mentioned and how they fit together for command and control of joint operations. Let's break them down by each element, beginning with the chain of command. The chain of command provides an unbroken line of authority and responsibility from the President of the United States to you, the airmen in the field. It begins with the President, then proceeds to the Secretary of Defense. Below the SecDef, the chain divides into two branches, the Administrative Branch, shown here in blue, and the Operational Branch, shown here in purple. Administrative Branch from the SecDef runs through the Secretary of the Air Force and the Chief of Staff of the Air Force to the Services Command Structure of major commands, numbered air forces, wings, groups, and squadrons, ultimately to you, the airmen. Now, back in my younger days, as a debt, we were taught, sir, the mission of the United States Air Force is to organize, train, and equip air forces for the conduct of prompt and sustained combat operations in the air. Sir! Try screaming that from a position of rigid attention with an upperclassman yelling in your face, and you tend to remember it. It is through the Administrative Branch that the Air Force carries out this mission. But as airmen, we also have another mission, to fly, fight, and win, and that mission is accomplished through the Operational Branch. Whereas the Administrative Branch is responsible for preparing forces, it is the Operational Branch that employs those forces. Now, let me repeat that for emphasis. The Administrative Branch prepares and provides the forces that the Operational Branch employs for mission execution. And within the Operational Branch, the service forces are integrated into a joint force for mission execution. So what's a joint force? Within the U.S. military structure, a joint force is a generic term for a force made up of significant elements of two or more military departments organized under a single joint force commander. Joint forces include the 11 combatant commands, like Central Command, subordinate unified commands, like United States Forces' career, and joint task forces, like Joint Task Force National Capital Region, which provides military support to activities in the National Capital Region, such as presidential inaugurations. But joint forces are made up of service forces, organized and provided as a service component command. From the SECDF, the Operational Branch goes to one of the 11 combatant commanders, then to the service component commanders assigned to that combatant commander. If you look at the chart, the blocks at the bottom illustrate the different service components. As an airman, you are part of the Air Force component under the commander of Air Force forces, also called the COMF4. Notice that the two branches of the chain of command, which diverged from the SECDF, now converge at the COMF4. The COMF4 has responsibility and authority derived from both branches to prepare, provide, and employ airmen to help accomplish the Joint Force Commander's missions. The Joint Force Commander can, if they choose, conduct operations through the service component commanders. However, for many years, the more common and better practice has been for the Joint Force Commander to conduct operations by designating one of the service component commanders as a functional component commander, integrating like forces from all the services. As airmen, we are most familiar with the Joint Force Air Component Commander, the JFAC, or the Combined Force Air Component Commander, the CFAC, when part of a coalition. Although it is the authority of the JFC to designate any one of the service component commanders as the JFAC, historic practice for more than 20 years is that JFC has always chosen the COMF4 to be the JFAC. Air Force doctrine assumes that this will continue to be the case and is written to describe proven best practices and provide advice to the dual-headed COMF4 JFAC. For convenience, instead of always saying the COMF4 JFAC, we refer to this as simply the Air Component Commander. As I said before, the Air Component Commander exercises authority over airmen in both the administrative and operational branches of the chain of command. I just want to briefly point out three of those that have the most impact on you as an airman. Administrative control, ADCON, operational control, OPCON, and tactical control, TACON. As the COMF4, the Air Component Commander uses administrative control, or ADCON, as delegated from the service of administrative branch of the chain of command to ensure that Air Force forces meet Air Force standards and are properly prepared and supported in accordance with those standards to be employed for operational branch tasking. As an airman, you're subject to Air Force ADCON from the time you join until the time you finally separate. Even when you are serving as part of a joint force or with another service as I did with the Navy, you are still an airman under the ADCON of an Air Force Commander somewhere in the chain. The Air Component Commander also has operational branch authorities delegated from the JFC to employ forces for mission accomplishment. As the COMF4, the Air Component Commander holds operational control, OPCON, over Air Force forces. OPCON includes all the authorities necessary to accomplish the mission, including the authority to reorganize the forces, task airmen for specific types of functions to be accomplished, direct specific joint training for those functions, assess and assess the risk to the force and, if necessary, direct and grant waivers to service ADCON standards. In the dual role of the JFAC, the Air Component Commander also has tactical control, or TACON, over Air Force forces and other service Air Forces made available for tasking. TACON is the authority for more detailed, specific mission direction as to timing, type of target, weapon choice, etc. With TACON, the Air Component Commander, through the Air Tasking Order and Air Operations Center, provides specific mission tasking to the Air Forces and executes real-time employment of those forces. It's time for another picture. This one shows the boxes and lines that I just described from the President through both branches of the chain of command, the Air Component Commander and, ultimately, to you, the Airman, as part of the Air Force forces. The discussion we just had sounds like a great potential for confusion as to which orders you, as an Airman, are subject to, and should follow. Am I being tasked under the operational branch or the administrative branch? Is that order from OPCON, TACON or ADCON authority? Well, let me offer one of the tenets of air power as shown in this chart to help simplify it. Airmen work for Airmen and the senior Airman works for the Joint Force Commander. You, as an Airman, are included here in the Air Force Forces box. The senior Airman is the Air Component Commander, shown here as the dual-headed ComF4JFAC. Above the Air Component Commander is the Combatant Commander in the operational branch and the Air Force chain of command in the administrative branch. It is the Air Component Commander's responsibility to know and employ the correct command authority to task you. You either prepare for mission tasking or to execute that tasking. And by the time you rise to the level of the Air Component Commander, you will have years of experience and a well-trained staff to help you keep straight if the orders you are giving are from the operational branch authority of the JFC or the administrative branch authority of the Air Force chain. So, in summary, what key points should you take from this? Airmen are organized, trained, and equipped and provided for operations as part of a Joint Force under an Air Component Commander. The Air Component Commander prepares and employs Air Forces to meet the Joint Force Commander's taskings. The Air Component Commander has responsibility for and authority over Airmen in both the administrative and the operational branches of the chain of command. There are many more details and nuances to commanding Airmen and controlling air power that you will pick up throughout your careers, both from academic environments and practical experience. But those points should give you a starting point upon which to build. I gave you just an overview of some key doctrinal concepts regarding comrade, but if you'd like to know more and study the details of each doctrinal concept, it's as easy as using Google. Literally. Pull up Google and type annex 3-30 command and control of Air Force Doctrine. Back when I was a young captain, yes, I was young once. My Squadron Commander told me the most important job I would ever have was to train my replacement. That was my objective for this discussion. I now pass that charge under you. Have a great Air Force career and enjoy it while you can. Then go out.


Get in losers. We're going for a ride. Welcome to Church of the Chair where we all float down here I'm your host E and today we'll be clowning around It is what it is. But first I have promises to keep this comment is only about four years old But I think it's time to show it off. The first copy. I have is this paperback It's a pretty boring cover But I like it anyway even though Pennywise is more of a mime in this one then we have the movie tie-in cover from the 2017-2016 version of the reboot and we have another version of this This one was sent to me by my friend Sarah by the way, I can't forget this is the 2017 Reboot, but yeah, there's the this the trade and this is the mass market paperback Believe this joke is 1400 pages. It's nuts. Then we have the first edition paperback This is the one that I have read the most if you don't know I have read it by Stephen King 18 times And this one accounted for probably you can see the bow in the in the spine this one accounted for probably 15 of those Used to be on my nightstand as soon as I stopped reading as soon as I finished the book I would just start it all back over again now on to some international editions my buddy Richard sent me this one Which is the Chinese edition of it and then my buddy Kobe? Kobe sorry kobe sent me the Dutch version of it This one's probably the closest to the original American artwork I think yeah hit and the last international edition was sent to me by my friend Wayne Fenlon amazing dude I had been looking for this one for ages and he just sent it to me free of charge And I will always this is also my favorite cover of This novel I mean he sent it to me free of charge didn't ask for anything in return One of the best gifts I've ever gotten and that is the UK edition of it Yeah Next up we have an edition that I started reading After I stopped reading this one for fear that it was gonna fall apart because these first edition paperbacks are kind of hard to find I went ahead and I bought this one This is the re-release that came out just before the 2017 reboot and this one's heavy and I also have the cover I have read this one at least three times and you can see the spine is just Disgusting and last but certainly not least I have not one But two first edition copies of it This one over here has a I it's a much better condition But the reason why it's so much drabber and duller is because I had to have this printed because I found this in great Condition without the dust jacket and then this one is the one that I've had the longest I haven't had it since release date all of my Stephen King books my mother's Stephen King books got lost in a hurricane in 95 But I did find this one. It is a first edition It's got the price code up here and everything 2295 it is in terrible condition But I love it all the same Anyways on to the rest of the video. I know that took a while But thank you for hanging out with me if you're new around here a word of warning before we continue I will be discussing all of Stephen King's books in detail With spoilers throughout this entire series. So if you have not read all of his works, I Suggest you click away now. You've been warned if it if it isn't obvious to you already today We're talking about Stephen King's it my favorite novel of all time Not just my favorite Stephen King book, but my favorite novel of all time first things first I want to read you guys an excerpt from it. Tell your friends. I am the last of a dying race. It said Grinning its sunken grin as it staggered and lurched down the porch after her The only survivor of a dying planet. So that excerpt is important Because that is how I tie in all of these books to the Tommy knockers dreamcatcher the stand all those things Even the books that technically do not connect that I have a theory that they connect like Carrie The shining so on and so forth the most important aspect here is in the opening chapter of it after the flood Pennywise introduces himself to Georgie Denbrough as Pennywise the Dancing Clown or Bob Gray or Is there a or Mr. Gray this is important because the Tommy knockers and The end in dreamcatcher the aliens are called grays There's also the short story gray matter from night shift that deals with an alien shape-shifting presence In a little boys house now as far as a hard tie-in to the dark tower series We have mature in or he's also known as Gan Who's the turtle on which the world the rock the world rides on his back There's also the path of the beam the turtle beam That is a hard connect not to mention and not to forget that Stephen King Was was writing it and the wastelands at the same time and he says in the afterward of Either it or the wastelands I'm not sure which that he was stuck on how to finish it and then lightning struck and he realized that everything Tied together some references to other books that put this directly into the King universe and ties all that together is Dick Holleran was a cook at the black spot Also Belch Higgins after Belch dies He shows up to pick up Henry Bowers in none other than a 1958 Plymouth Christine another note that ties it Into dreamcatcher is the obvious connection that Penny wise lives is spray painted on the loading dock of tracker brothers in Derry and many thanks to silent silo 987 for this comment tying under the dome into it You pause to read but wait There's even more we have this comment from draw your sword for 20. I Like the name again, you can pause to read I definitely gel with the concept that Ardelia lorts from the library policeman is a creature Just like Penny wise and the outsider so on and so forth We also can't forget that both Bev and Richie were in 11 20 to 63 as pointed out by Valperga literature review 96 oh two. Yeah, that's a mouthful. Anyways, here's a comment. I probably already put it up there But next we have Chris one eight seven who said this and And while that poem is All over the place even outside of Stephen King's work. I still think it's important to highlight this comment Thanks, Chris and going back to Richie. We have this comment from Rick Alicia's 92 and it says I Do not remember this But if it's true, very cool. Well, that's all the time I have for you today Hopefully I didn't miss anything this time But if I did feel free to tell me about it down there in the comments If I miss something if you need to correct something if you have your own theories Please leave them down there in the comment section, but until next time feet feet Richie


Okay, good morning everyone. So in the last lecture, we looked at the topological conjugacy classes of linear maps, right? So we have the space of linear maps, AX equals AX, A in R. So R is the parameter space of the space of linear maps, right? For each value of A, you get a linear map. And we looked at the topological conjugacy classes of invertible hyperbolic. So if you remember, there were four conjugacy classes, minus infinity to one, to minus one, minus one to zero, zero to one, and one plus infinity. So we showed that in fact, any linear map in this set, so any linear map except for A equals zero plus or minus one are conjugate to each other. But they're not all topologically conjugate. If you want the topologically conjugacy, the topological conjugacy distinguishes, separates, okay? Maps that belong to these four regions. So what we, interesting question now. So if you remember the difference between the conjugacy and topological conjugacy, it's just that the conjugacy is a homomorphism in this case. So question, what about C1 conjugacy? Can the conjugacy H be a different morphism? In other words, in the same way as when you go from simple conjugacy, in which H is just a bijection, to topological conjugacy in which H is a homomorphism, you distinguish more in detail the properties of the system. And so you split it up into four instead of just one conjugacy class. The question is, if you say, if you want this conjugacy to be C1, is that a stronger condition? Obviously, two maps which belong to different topological conjugacy classes cannot belong to C1 conjugacy classes, right? Because if it's C1, in particular, it's a homomorphism, so they cannot. So the question is, if you take two maps within the same topological conjugacy class, are they also differentially conjugate? Or does this create a finer division into more C1 conjugacy classes? Okay, this is the question we will address today. So I will give, the answer actually is fairly simple, and I will just give you the answer, which is that indeed in general, two linear maps, even in the same topological conjugacy class, are not C1 conjugacy, are not C1 conjugates. And the answer is the, and the reason is the following, the position. I will state it in a little bit more general form. So let F, R to R, and G, R to R, be C1 differentomorphism, which C1 conjugates. So there exists differentomorphism, H to R, such that H composed with F equals G composed with H. Let P, Q be fixed points for F and G, respectively. If they exist, if they exist with H of P equals Q, then the derivative at P equals the derivative at Q. So what is this saying? I stated it in one dimension, actually this is true in higher dimensions too. But basically one dimension, so this is F, R to R. This is G from R to R. Suppose they are conjugate by C1 differentomorphism, okay, this is the assumption. Suppose there exists a point here, which is a fixed point, which is mapped to fixed point Q by this differentomorphism. Then what is the statement saying? It's saying that the derivative of P must be equal to the derivative. Okay, so this is the assumption, right? So the assumption is that there exists the differentomorphism, C1 differentomorphism. And the conclusion is that if there exists two fixed points that map to each other, then this differentomorphism may have the same derivative. So what is the implication for our question here? Why is that relevant to this question? Excuse me, zero is a fixed point. So for all the linear maps, zero is always a fixed point, that's right. So what is the derivative of the fixed point at zero? A, okay? So each of these maps has a fixed point, right? You can actually draw the graph of these maps, right? X equals AX has a graph, okay? This is the graph of the map A of X equals AX. It's linear, so the derivative is always A, right? In particular, at the fixed point, the derivative is A, okay? So if you take two different linear maps, A of X equals AX and B of X equals BX, can they be differentially conjugate? Only if A is equal to B. So if A is different from B, they cannot be C1 conjugate because of this proposition. What if A is equal to B? Do we know that they are C1 conjugate? So this just gives a kind of negative, right? This says if this does not hold, then they cannot be C1 conjugate. This is how we use this. So what we're saying is any two linear maps cannot be C1 conjugate unless A is equal to B. What if A is equal to B? They're the same map, very good. I thought you should get that. So if they're the same map, does that mean that it's conjugate to itself? Is a map conjugate to itself? Yes, but we didn't, I mean this is a little detail which we should have addressed probably right at the beginning, sorry? Is a map in general C1 conjugate to itself? Exactly, you take the identity, conjugates the map to itself. The identity is C1 defilmorphism and it conjugates the map to itself. Sorry? That's right, we didn't, yes, we proved it. So I guess it's part of the definition. We, in principle, we didn't discuss it explicitly in the class, but we implicitly checked that the map is conjugate to itself and we proved that it's an equivalence relation, okay? So the consequence of this proposition is that how many C1 conjugacy classes do we have for linear maps? Infinity many, uncountably many, even, right? Every map is its own conjugacy class. So this shows what the comment I made at the beginning, is that the difference between conjugacy, topological conjugacy, and C1 conjugacy, right? General conjugacy is a very weak notion of conjugacy. All these linear maps are all conjugate to each other. C1 conjugacy is a very strong definition of conjugacy. None of these maps are conjugate to each other. You don't gain anything, you've not really grouped them in any way, they're all different conjugacy classes. So you might as well have the original maps, you know? You've not gained anything in this particular setting by taking C1 conjugacy. Topological conjugacy seems to be a very interesting compromise because it groups them in just four classes, so a small finite number of classes. Each of these classes has some obvious characteristics that they share that seem to be fairly natural and it seems fairly natural that you don't need to distinguish really between different maps in here, okay? So of course there are certain situations in which it's important to know whether maps are C1 conjugated to discuss that. So I'm not saying this is never relevant, I'm just using this example to really highlight the difference between these three notions of conjugacy, okay? I think this is a very good example and we will come back to these three notions in different settings. Yes, the topology on what? So there's two topologies involved here, right? There's the topology on the spaces. So a topological conjugacy means that H is homomorphism from this space on which one dynamical system is defined. And this is a space on which the other dynamical system is defined. So these need to be topological spaces for you to be able to say that this is a homomorphism, right? So implicit in the definition of topological conjugacy is that two systems are conjugate if there's a homomorphism that conjugates the two dynamics. So in particular the two spaces on which the dynamics are defined need to be homomorphic. They need to be homomorphic topological spaces. I'm not sure if that was your question or not, yes? Yes? Okay, so let's prove this. This is really a very simple calculation. I will do it, but I could almost leave it as an exercise, but let's do it. So it's really just an application of the chain rule. So using the conjugacy, we write by the definition of conjugacy, we have F is equal to H minus 1 composed with G, composed with H, right? Because H is invertible. In fact, here it's a diffimorphism, so the inverse is also C1 diffimorphism, right? And so by the chain rule, we just differentiate, right? So what is the derivative F prime of X is equal to the derivative of all of this in the point X, right? And you know what the chain rule, how the chain rule works. So this is equal to the derivative of H minus 1 prime in G of H of X, right? Times G prime in H of X times H prime, okay? So the chain rule just says that you differentiate each map in the point at which it is applied. So now we use the fact that the P is a fixed point. So since P is a fixed point, P is a fixed point, and H of P equals Q, then we have that H minus 1, so we're applying this at P. Okay, so let me write this. F prime of P is equal to H minus 1 prime of G of H of P times G prime H of P times H prime of P. So this is equal to, so H of P is equal to Q. H of P is equal to Q, right? So this is Q and this is Q. So this is H minus 1 prime in G of Q times G prime in Q, H prime in P. But Q is a fixed point for G also. So this is H minus 1 prime of Q times G prime of Q times H prime of P, okay? And now this H minus 1 prime in H of P is equal to, but, okay, so H minus 1 prime in Q is equal to H minus 1 prime, H minus 1 prime in H of P. Q is H of P in H of P, which is just equal to H prime of P to the minus 1. Okay, this is just the inverse. The derivative of the inverse is 1 over the derivative in the image, okay? And so this is the inverse of this, okay? And so this is just equal to G prime of Q. Excuse me? Ah, yes, you're right, you're right. But it cannot be 0 because otherwise it would not be a defilmorphism, right? If the definition of defilmorphism, a defilmorphism cannot have 0 derivative at a fixed point because otherwise the inverse would have infinite derivative at that fixed point, which is not acceptable, right? If the defilmorphism at a fixed point has 0, in fact, not just at a fixed point, at any point it cannot have 0 derivative anywhere because otherwise the inverse would have infinite derivative, okay? So defilmorphism cannot have 0 derivative. Thank you, that's a good point. Okay, so this is just a simple calculation that shows this. Okay, so this completes our discussion of the different kinds of conjugacies. So now to wrap up this section on linear maps, we want to use, apply all these ideas of conjugacies to study the structural stability. So if you remember at the beginning I said one of the most interesting applications of this idea of this kind of equivalence class is to talk about structural stability for one-dimensional linear maps. So do you remember what structural stability means? Anyone? In the same conjugacy class, that's right. So structural stability is the following. I take a system and I say what happens if I make a small perturbation of this system? Does it change the conjugacy class or not? Does the system change or not? Okay, so there's two crucial ingredients that are required to formulate this. One is what do we mean by changing the system a little bit? And the second, what do we mean by the system stays the same? So in other words, let Ax equals Ax, okay? Does the dynamics change under small perturbations? So there's two notions that are involved here. Two notions. What do we mean by one does the system change? And two, what do we mean by small perturbation? So what do we mean? In this case it's particularly clear. So what do we mean by does the system change? Means is the new system conjugate to the original one, okay? And as we have seen, there's various ways to define, you know, there's various conditions we can use to say they're conjugate. So depending on whether we use a simple conjugacy or differentiable conjugacy or topological conjugacy, we will get a different answer as to whether the systems are the same or not, okay? And the second is what do we mean by small perturbation? In this case, it is fairly natural. There's an obvious topology on the space of linear maps. What is the topology? What is the space of linear maps? The space of one-dimensional linear maps is just given by the parameter a. So there is an obvious way to say what it means for two linear maps to be close is just a and b should be close, right? So small perturbation means a small neighborhood of a. That is a small perturbation. So in this particular setting, the question is to formulate the question more precisely than this. So given a in R, does there exist... does there exist a neighborhood, let's call it u of a, such that for all b in u, a of x equals ax and b of x equals bx, r, say, topologically, or c1. This is the notion of structural stability. So I take r. This is... I take it as the parameter space of linear maps. This is 0. This is 1. This is minus 1. So my question is this. I take a parameter a and I say, is the linear map ax equals ax structurally stable? What is the answer? Is it structurally stable for this parameter value a? It is. So can I find a neighborhood of a such that all the maps in here are all conjugate to a? Yeah? Does it depend on where I chose the point a for every a, minus 1 and 1, what? Then what? Then it will be stable if it's between minus 1 and 1. And what about here if it's bigger than 1? Is this structurally stable? Awesome. Let's make sure that everyone is on board here. Annie, do you know where we are? Is the question clear? What about the answer? So what do we know about whether two parameters a and b like this are conjugate or not? Mariam, that's your name, right? What do you think? Are these two maps corresponding to these two parameters are they conjugate? a of x equals ax, b of x equals bx. Are these two conjugate? Do you know? Was that a kind of, OK, 50-50 chance? Why are they conjugate? That's the definition of conjugate. But why do they belong to the same conjugacy class? Did we show that? And so, the same conjugacy class. What kind of conjugacy? Topological conjugacy, c1 conjugacy? Topological conjugacy. What if I'm interested in c1 conjugacy? A and B conjugate? Exactly. So if I'm interested in the c1 conjugacy, does that exist a neighborhood of A such that all the parameters in here are conjugate to A under the c1 conjugacy? No. So is a structurally stable under c1 conjugacy? It's not structurally stable. Structurally stable means you change a little bit and things do not change. According to the conjugacy, you have chosen. So the notion of structural instability depends on which notion of conjugacy you have used. If you choose topological conjugacy, then all these points, as you said, except for 0, minus 1, and 1, they're all structurally stable. Because if you take any point, because remember, these are exactly the topological conjugacy classes are exactly this. This is one topological conjugacy class. You can use different colors. This is another topological conjugacy class. This is another topological conjugacy class. And this also is four topological conjugacy classes. They're open in the standard topology on the real line. Which means that if you choose any point inside here, you can find the neighborhood any point. However close to 0 you are, you can find a small neighborhood that remains inside the same conjugacy class. So every invertible hyperbolic linear map is structurally stable with respect to topological conjugacy. Exactly. When I say invertible and hyperbolic, I use exactly to exclude those cases. What about 0 and minus 1 and 1? It's structurally stable. This map at minus 1. Is it structurally stable? There is a map at minus 1. a of x equals minus x. It's a perfectly well-defined map. No, it's not, but it's still a linear map. So my question is, is that structurally stable? Exactly. And remember, we know that for minus 1, for example, every point is periodic of period 2. When you take a small perturbation, you have only one single fixed point at 0 and no other periodic points. So this map at minus 1 cannot be even conjugate. We don't even have an anthropological conjugate. It just cannot even be just conjugate to any of the maps that have arbitrarily close to it. So the points minus 1, 0 and 1 are not structurally stable in any sense whatsoever. Because if you make a small perturbation, you will get something that is not even conjugate to the original map. But if you take anything outside these three points, then they will be structurally stable with respect to topological conjugacy, but not structurally stable with respect to C1 conjugacy. Because no map is structurally stable with respect to C1 conjugacy. Because with respect to C1 conjugacy, every map, every arbitrarily small perturbation, is a map that is in a different C1 conjugacy class. Because the only C1 conjugacy class is just individual points. So maybe let me summarize this. So proposition. Every invertible hyperbolic linear map is structurally stable with respect to topological conjugacy, but no map. But every linear map is structurally unstable with respect to C1. Structurally stable with respect to topological conjugacy, structurally unstable with respect to C1 conjugacy. I'm not going to write down the formal proof because we've discussed it. It's pretty clear, but you need to make sure that you understand how to prove this. This could be a good candidate for an exam question. I'm saying it's a good candidate. It collects results we've been doing until now. It tests understanding of the concepts that we're talking about and the specific results that we've discussed until now. Complex parameters. Yes, depending on the family of dynamical systems that we have, we will have different topologies on the space of dynamical systems. So in certain situations, there are certain families of maps which are naturally parameterized by the complex numbers. In other families, as we shall see in fact now, we will talk about nonlinear maps and we don't really have a parameter space, but we have a topology on the space of all maps, like the C0 topology or the C1 topology. I will talk about it either today or tomorrow. I don't know if you've done these topologies on the space of continuous maps or C1 maps, no? Okay, I will introduce them. But that's right. So implicit in this is because we're looking at some very simple examples, and the family of all one-dimensional linear maps is parameterized by the real line, then the most obvious topology on the space of linear maps is just the standard Euclidean topology on the real line. But of course, this depends on the topology we choose on the real line. Because the notion of neighborhood depends on the topology. So what you want is a neighborhood such that everything in there is conjugate to each other, all the maps in this neighborhood are conjugate to each other. We will see many examples of this kind. Yes, yes, yes. Try to formulate your question a little bit better. So you see these, the conjugacy classes, each conjugacy class is an open interval. One, two, three, four conjugacy classes. So if a conjugacy class is open, then automatically any element in that conjugacy class is structurally stable. Because by definition of open, any element in there will have a neighborhood that's contained in that conjugacy class. Okay? So in this case, this is another way to see this, is that if the conjugacy classes are open, clearly every element in one of these conjugacy classes is open. In general, this is not necessarily the case. You might have conjugacy class that has a boundary. In this case, the boundaries of the conjugacy classes do not belong to that conjugacy class. But you might have some situations in which the boundary belongs to that conjugacy class, but it's still the boundary. So it means that the boundary belongs to the conjugacy class, for example, on the left, but not the one on the right. You might have situations like that. So if you're in the interior of your conjugacy class, you're structurally stable. If you're on the boundary, you're not. This is another way, equivalent way, to think about structural stability. So is your map in the interior of its conjugacy class? And to define that, you need a topology on the space of system to talk about being in the interior of your conjugacy class. You need a topology that tells you what it means to be in the interior of your conjugacy class of maps. Okay? And we will see several examples now. Okay, I think maybe this is a moment to take a very quick two-minute break and have a glass of water. Okay. So now we are going to remain in a one-dimensional setting because it's a simpler setting to illustrate some concepts. But we're going to take the step beyond linear maps to nonlinear maps where you don't have such a simple and obvious parameter space of families. So we're going to now consider interval, study interval different morphisms. So we suppose i is a compact interval. So generally you can just think of i equals 0, 1, for example. But it really doesn't matter. Everything will... Sorry? Yes, yes. So it doesn't really matter. We will study the dynamics of maps on compact intervals. Okay? So if you have a map on one compact interval, just up to the scaling is the same as any map on any other compact interval. It doesn't make a difference. And we take f from i to i, a different morphism, c1, different morphism. So this is interval i, interval i. This is just a map from i to i. Remember, different morphism in particular needs to be a bijection. In general, the different morphism will look something like this. The map will look something like this. So just a few observations about interval different morphisms. First of all, notice that the end points are mapped to the end points. So end and points. Also, since it's a different morphism, by the same reason we said before, we have that f' is not equal to 0 for all x and i, because otherwise the inverse would have infinite derivative and that cannot be. It's not part of... It's in the definition of c1, different morphism that that cannot happen. And so this means that either f' of x is greater than 0 for all x and i, or f' of x is less than 0 for all x and i. So you can only really have one of these two cases. In this case we call it orientation preserving, and in this case we call the map orientation reversing. If a map is orientation reversing, then the second iterate of that map is orientation preserving. So in some sense the orientation reversing maps are kind of special and there's a very simple classification of these, and I'm going to leave this as an exercise. So the orientation reversing dynamics of orientation reversing maps is really simple. There's not much that can happen. We leave this as an exercise. Whereas for orientation preserving there's a much richer structure. So from now on we're going to study always the orientation preserving case. All the iterates are always orientation preserving, so it's a kind of a closed class of system. What do we want to say about orientation preserving maps? If p is a fixed point, if f of p is equal to p, we say p is a hyperbolic fixed point. If f prime of p is not equal to 1, which is completely consistent with the linear case. We do not include the minus one here because we're assuming that we're orientation preserving the derivative is just 0. The derivative is positive. So if all fixed points of f are hyperbolic, then we say that f is hyperbolic. So what is the dynamics of interval diffeomorphism? So we start with the following proposition. f i to i, orientation preserving, c1 interval diffeomorphism. For all the limit sets omega x and alpha x, the alpha and omega limit sets are fixed points for f. Shall I prove it or leave it as an exercise? I will sketch the proof. It's a fairly simple exercise actually. But the conclusions are non-trivial. The conclusions are fundamental to the theory of interval diffeomorphism and show also that interval diffeomorphism have a particularly simple kind of dynamics. So here we're starting with simple systems where not much can happen. And then we will go and we will see that the systems with this very complicated omega limits. So basically what this says is that the only thing that a point can do if you choose a point x and you iterate it, it must converge to a fixed point, both in forward time and in backward time. That's what that says. So it cannot do strange things. It must converge to a fixed point. And that basically follows from the fact that this map is monotone, monotone, monotonically increasing. And so the sequence of points along the orbit is monotone increasing or monotone decreasing. And a monotone increasing sequence must converge to some point and the only thing that really needs to check is that that point is a fixed point. So we'll just sketch briefly the proof here, but it's very simple. So let x in I, if x is a fixed point, then what? Then there's nothing to prove. Then it's obvious because for a fixed point, the omega limit set of a fixed point is itself and the alpha limit set is also itself if it's an invertible map. So if f of x happens to be already a fixed point, then this is clear. Then alpha limit of x equals the fixed point itself, which is equal to omega limit and nothing to prove. Well, and that's the proof. Okay, that's fine. So suppose, so if f of x is different from x, then either f of x is greater than x or f of x is less than x. Those are the only two possibilities. If f of x is greater than x, then f of x minus x is greater than 0. By the way, what do these possibilities correspond to geometrically? It's always useful, especially in these one-dimensional maps, where you can gain a lot of intuition from looking at the graph of a map. So every different morphism of the interval, orientation preserving basically looks like this. The derivative is positive. It has to map end points to end points, which means that the end point, if this is the interval a, b, or 0, 1, it must map a to a and b to b. In the orientation reversing case, a would map to b and b would map to a. So it would go like this. It still has to map end points to end points. In the orientation preserving cases like this, a maps to a, b maps to b, the derivative has to be bigger than 0 everywhere, and so it has to do something like this. It can be like this, or it doesn't have to intersect the diagonal, of course. It could be just something like this. It could also be something like this. In fact, basically, these are the three kinds that we will consider. In these two cases, these are the only two fixed points, right? Because remember that a fixed point, okay, crucial observation. When you look at the graph of an interval map, a point is fixed if and only if the graph intersects the diagonal. This is clear, right? Because the diagonal is the line where y equals x, so it's a fixed point precisely means that f of x equals x if the graph intersects the diagonal. So you can immediately see from the graph where the fixed points are. Okay? Here it intersects the diagonal and here it intersects the diagonal. These are fixed points. It could be that these are the only two fixed points. For example, if you have this situation, or this situation, or it could be that you have several fixed points. Okay? It could also be that you have an infinite number of fixed points. What is an example of an interval different morphs with an infinite number of fixed points? Identity. Identity. Not of course. The identity is the best counter example in so many situations. Okay? Always try with the identity first. Whenever you're looking for some counter example or something, try the identity. Okay? Because otherwise you start constructing very complicated construction where the identity works. Okay? The identity works. Or you can have just something that is the identity in some piece. The identity means that it exactly coincides with the diagonal. That gives you an infinite number of fixed points. So you can try to construct something that oscillates lots, or you can just construct something that goes along the diagonal for a certain, you know, certain piece and that gives you an infinite number of fixed points. That is the identity even just on some piece. Okay? So in this case is the map hyperbolic. What is the derivative of these fixed points? One. One. So the map is not hyperbolic. These fixed points are not hyperbolic and the map is not hyperbolic. Okay? So I'm just anticipating one of the reasons to introduce this definition of hyperbolist in this case just as in the linear case is to exclude certain very unpleasant situations like this one. In fact, we will see that if this map, if all the fixed points are hyperbolic they can only be fixed points. Because if the fixed point is hyperbolic then it must cross and it cannot be accumulated by other fixed points. So it needs to be just a finite number. Okay, but that was just a little parenthesis. This is just to give some idea. So what does it mean here that f of x is bigger than x in the graph? For example, let's look at just this, this, okay, look at this graph here. Okay? For this point. This is x and this is f of x. Is f of x bigger than x or smaller than x? Bigger than x. How can you tell? Because the graph is above the diagonal. Right? So you can immediately tell for each point if you look at the graph whether f of x is less than x or bigger than x. So if instead you were looking at the pink curve let's take this point x here and then f of x is here and because it's below the diagonal this means that f of x is less than x. Okay? So here you can see exactly the situation that we described here that either x is already a fixed point or if it's not a fixed point clearly it must be either below the diagonal, the graph must be either below the diagonal above the diagonal which corresponds to those two cases that we saw before. Okay? So we suppose that in fact the two cases are exactly the same. So just for definiteness suppose we take this case what does this mean? It means that f of x minus x is greater than 0. So for example it could be let's just stick to one map maybe so this is the diagonal let's suppose the map looks something like this so let's suppose we take a point x here where the graph is above the diagonal so what's going to happen to the next iteration can we say something about what happens to the next iteration well where is this point here in your dynamics, in your dynamical space so this is the dynamical space a, b this is where initial condition x is 0 and this here is x1 the image where will x2 be what's the image of x1 excuse me that's right, that's right and how do we see is that a graphical representation so we're going to do it in two ways we're going to write down formally what we know but I think it's also very useful to understand how to look at it in the picture you want to know find out where x1 is here because when you know where x1 is here then you can look at its image by looking at the graph so how do you carry x1 from the vertical axis to the horizontal axis you just reflect it in the diagonal so geometrically you look at x1 and then you take a horizontal line all the way to the diagonal and then you take it down like this and this will be x1 and then you can look at the graph and you say ok what's the image of x1 but the image of x1 is here so x2 will be somewhere here but having x2 on the vertical axis is not much good for us what we want to know is where x2 on the horizontal axis so we can do the same thing we take x2 and I don't need what's the point of me starting from this point going all the way to the left and then going back all the way to the right so just from here I can just go directly down to the diagonal and what's the point of me now coming all the way down and then going all the way up again I can just start from this point and I can go up again so let me just draw a picture of what we're doing I'm just taking a little parenthesis to show you how you can analyze this graphically so let me zoom in to this region here the map looks like this and what am I doing I start somewhere under here and I end up with this point here which is this point here and then what I did is I said this is x1 now you come back and you reflect in the diagonal and then you go down reflecting in the diagonal means finding this point here and then I go down to x1 but then what do I use x1 for the next image which will be this point on the graph here which is this point here and now this corresponds to x2 so I take x2 and then I go back and I reflect it in the diagonal which means I move to the diagonal here then in principle I go down and I find x3 down here and then I look for the next image I come back up and go across to the graph again so really the itsets of the map you can just kind of graphically draw a picture like this which shows you that this point is moving from here to here to here wherever these images are from here to here and it is converging to this end point so this is a very useful graphical analysis it does not prove anything because it is just a picture but it helps you to get a feeling for what is going on so because the graph is always above the diagonal then the image, the next image is always bigger than the point that came before okay and so you have a monotone increasing sequence okay so repeating this argument so let's write f of let's write this as x1 minus x0 greater than 0 repeating this argument we get x2 greater than x1 okay and so the orbit xn n in n which is equal to n of x0 sorry n in n is monotone increasing is monotone in forward time so what can you say about a monotone increasing sequence of points on the real line this is a sequence of points this is monotone increasing it's also bounded that's right it's bounded before it converges that's why we're interested here we're in a compact interval before this is an increasing sequence so every monotone increasing sequence xn which is bounded converges to some p in i it's monotone increasing and bounded so xn converges to p which is what we're seeing in this picture because you're just increasing increasing increasing there's no thing else that you can do except converge either you go to infinity or you converge to something what is left we need to show that this point p is a fixed point we need to show that it doesn't converge to something since we know that this is the only fixed point we need to show that it actually reaches the fixed point because it's monotone increasing but maybe it converges to some other point here that comes before the fixed point okay we just need to check that we need to show that it converges and it comes all the way here so we need to show that f of p equals p sorry? why by construction? it's that and why might that not be the case so your observations are correct so it's almost trivial but it's not completely we're using so you're saying it converges to p and now you need to show that the fact that it converges to p implies that f of p equals p is the upper bound and so in this particular case p is the end point of the interval but how do we know it does not stop here somewhere before that f of x but we don't know that f of p is f of xn is a subsequence of xn and so by the continuity of f okay this is what implicitly you were using this fact but this is a crucial ingredient here okay you need to use the continuity of f exactly so how do we write this out so the continuity of f plays the crucial role here so what we have is that f of xn the continuity of f we have that f of xn converges to f of p just by continuity okay but f of xn has the same limit as f of xn but f of xn equals xn plus 1 and so the sequences and so the limit of as n tends to infinity of xn is the same as the limit as n tends to infinity of xn plus 1 which is the same as the limit as 10 tends to infinity of f of xn okay and the limit here is p and the limit here is f of p it's just continuity okay so the integration is completely correct but the formula depends on the continuity of the map f okay so this allows us to study the conjugacy classes of these maps because this is of course our goal we want to classify all these interval diffeumophism and classification means understanding when are two interval diffeumophism's conjugates so what do you think let's see let's start with the simplest cases so conjugacy classes so first of all just heuristically of course what this proposition says is that the fixed point play a crucial role because diffeumophism has a certain number of fixed points every other point ends up in one of these fixed points so the fixed points determine the long term asymptotic dynamics of every orbit because every orbit in backward time and in forward time ends up in a fixed point so the fixed point play a crucial role in classifying these systems and we will see this more specifically so let's start by looking at two maps that have that both have just two fixed points so let f i to i and g c1 orientation preserving diffeumophism suppose both f and g have only two fixed points so what does it mean that I have two fixed points because we saw that the end points are always fixed then it means that one this and the other one maybe is also like this notice that you do not need to be defined on the same interval so this is going to be conjugate topologically conjugate differentially conjugate do they look conjugate to you could they be conjugate topologically conjugate makes sense that they're topologically conjugate why would you say how would you convince me without a formal proof the intervals are homomorphic as intervals but now the conjugate needs to conjugate the dynamics as well right it's not enough for the spaces to be homomorphic the homomorphism needs to conjugate the dynamics so we need a homomorphism that maps h composed with f equals g composed with h so we need to map orbits to orbits so you would clearly naturally map these two fixed points to the corresponding fixed points and then can you have a homomorphism that maps every orbit here to every orbit here that's right so what are the alphan omega limits of points in here and which one is it in this case why do you think who can be more sure yeah it's obvious you don't need to think it's obvious by the argument we just did you take a point x here its image is above the diagonal so it increases right this image is above the diagonal so it increases all the points are moving to the right and therefore they converge to this end point here and the alpha limit is this end point here and the same for this right all these points are moving to here and all the points are moving to here okay so the gut feeling is that this should be topologically conjugate because you think okay you know it's not exactly the same I mean you think that you should be able to take all the points these similar things a little bit like in the linear case when everything was going to zero you think they should be topologically conjugate and this is correct okay we will construct the conjugate here what about if this case here is is doing this are these still topologically conjugate why not why do you keep thinking instead of being sure yes and so what does not preserve what does not preserve the omega limits this homomorphism so you are saying correctly that if this homomorphism maps x to a point y then it needs to map the omega limit of x to the omega limit of y right so why does it not do that when you say well the omega limit of x is this point and the omega limit of y is this point so why can h not map this point to that point sorry this point to this point can it but can you have a homomorphism that maps this interval to this interval that maps this point to this point and this point to that point that flips the interval of course you can have a homomorphism it's an orientation of reversing homomorphism because it switches the order but it's not a problem it's just switch the order that's still a homomorphism it's not part of the definition of the conjugacy there's no particular requirements on orientation in general between two topological spaces you don't even have the definition the notion of orientation so it's just a homomorphism between these two topological spaces and there's no reason in this case why it cannot switch them around in which case it should be we need to check we need to prove this but there's no reason a priori why we couldn't imagine that these two are also topologically conjugate although we need to prove we need to check that but okay so keep this in mind so we don't have much time left so I don't think I will be able to finish the proof here okay so let me postpone the construction to the next lecture but let me first say that we're going to use the same construction as we did for linear cases we're going to use fundamental domains right so it would be a very good exercise if you want to try to do it at home it's very easy in the same way because every point here is moving from here to here for every point the alpha limit is this and the omega limit is this so you should be able to find which is a fundamental domain exactly in the same way as the linear case so any point in backward time will go here and it should spend just one iterate inside this fundamental domain and then another fundamental domain here in the same way and then you map by homomorphism the two fundamental domains and you extend it exactly like we did for linear case and we construct this topological cone so we check that it's what we get is a homomorphism so I will do some or at least I will sketch some part of the construction but it is exactly the same as we did for the linear case except for you don't have an explicit form when you need to explain why this is a fundamental domain it's not just multiplication by A or B you need a slightly more sophisticated argument to show that that's a fundamental domain that's the only difference excuse me sorry I didn't hear you yes it does work the definition is the same it's just to show that well the argument is almost exactly the same just to show that you cannot jump so here you take some X tilde here you take F of X tilde this is your fundamental domain closed on one side open on the other side and everything is exactly the same just to show that you cannot jump and that if you jump you land only once basically exactly the same when we did it for the linear case we used a little bit the fact that it was exactly multiplication by A but it's almost exactly the same argument so since we just have a couple of minutes let me say first what about C1 conjugates assuming we can prove that these are topological conjugates what about C1 conjugates are they C1 conjugates exactly so certainly if the derivative at these two fixed point is different then they cannot be C1 conjugate so for this to add there's not work exact in particular in this case it does not work so this is another example of how strong the C1 conjugates is so two maps even if they look very similar even if the so if this map is like this okay even if it's even if the two maps coincide almost at every point but at the fixed point they're a little bit different the derivatives are different the maps cannot be C1 conjugates okay just to show how strong it is so we shall see that in the space we still need to put a topology on the space of interval the film of this but instinctively you can see that if we will show that all the maps that look like this basically topologically conjugate then you will have a reasonable large conjugacy class which contains a lot of maps and it is also reasonable in the sense that these maps all have pretty much the same behavior all the points mapped one of these two fixed points on the other hand differentiable conjugacy you can still have two maps that are different but in the same differentiable conjugacy class so this is different from the linear case so in these two cases we will not prove it but you can prove that if they are if they do have the same derivative then you can construct a differentiable conjugacy between here so you can have maps that are different but in the same differentiable conjugacy class but you can also have two maps that just differ a little bit in the different differentiable conjugacy class so again the topological conjugacy class seems like a more natural and robust definition of equivalence okay so in the next lecture we will do this construction we will also look at maps with several periodic points okay although that again will be almost the same and then we will define the topology on the space of this interval the thermomorphism that allows us to address the notion of structural stability okay thank you


Tensions were high tonight at the last public meeting to review an environmental review of a proposed Enbridge energy oil pipeline. As Mal Meyer reports from Bemidji, activists from all over the state came in and at one point by busload to have their voices heard. The group jobs for Minnesotans kicked off the afternoon with the gathering of community members to express their support of the pipeline. We would like to call on the Minnesota Public Utilities Commission to approve this project. They argued that while they also support renewable energy, there will still be a demand for oil. Physics prove that it cannot replace the huge energy demand that our society has. And Enbridge would like to have some of these workers to help them replace the more than 50-year-old line. I think it's just important to say that it's time, it's time to do this, it's time to move forward. According to the draft environmental impact statement, Enbridge's preferred route would need a maximum of 4,200 workers within an eight-month period. But this would not affect county-level unemployment or median household income levels. I think there's a huge number of liability issues here for the state of Minnesota. Some activists are concerned that the impact of jobs would not outweigh the potential catastrophic effects of an oil spill. We oppose a new energy corridor in the Lake Country because of the long-term risks. And they're ready to fight back if something does happen. The water protectors that are throughout northern Minnesota and throughout this country are looking at what is happening here. The Department of Commerce is trying to weigh the many benefits and risks through the environmental impact statement. The comments presented through the 22 meetings will help them clarify or edit any sections. The line should be twice as tall, exactly twice as tall, we'll get it there. Any comments submitted to the Department of Commerce by July 10th will be considered for the final draft. In Bemidji, I'm Melmyer, Lakeland News. Once the final draft is completed, Minnesota Public Utilities Commission will decide if the project is in the state's best interest. If you've enjoyed this segment of Lakeland News, please consider making a tax-deductible contribution to Lakeland Public Television.


Welcome to Toffy TV. It is the Everton News Daily. Will the pitches surrounding who's going to be the next Everton manager is no clearer today. In fact, it feels a little bit more complicated as it has made so that it seems like Everton aren't going to take more time over the decision of who's going to take over from Raffa Ben Eater's as the manager of Everton Football Club. It seems that they have given it a little bit more consideration and don't feel like a decision has to be made straight away, which with a transfer window closing next Monday doesn't seem completely right. Vita Pereira, who obviously yesterday seemed like was a strong favourite to take over the job. It seems like there's going to be further discussions with him about whether he's going to come in as the manager. Frank Lampard, I feel like has a made in the last 24 hours as a bit of a favourite amongst fans. It seems like he's the best of a bad bunch. He is set for a second interview. I've noticed on social media a lot of Everton fans now are backing him as the man they want to see taken over as the manager considering the options out there. So it's going to be interesting to see what happens with him. Wayne Rooney, it seems like has not been approached and certainly has not had an interview so it doesn't look like he is the man Everton wants to take the club forward. Roberto Martinez, his name has come up again but it looks like the Belgium FA have completely blocked the idea of him becoming the next Everton manager unless of course he was the design but it doesn't seem like Everton are going after that anymore. A couple of new names Rudi Garcia and Covatch who had been linked previously have been linked again in the last 24 hours maybe putting themselves back on the table for the job or certainly Everton maybe going back and considering them and reexamining all the evidence to see why they should be the Everton manager. Whether they get interviews or not, we'll have to wait and see but it's no clearer today but Pereire Frank-Lampard and Wayne Rooney even though he hasn't been interviewed or the three you are seeing as the major contenders now so we'll see what happens tomorrow but the club giving themselves a little bit of breathing space taking a little bit of time and who knows you know if they're giving themselves a bit of time that might mean Duncan Ferguson stays in charge for the next few games. Duncan Ferguson has spoken today, spoken to evanfc.com about the situation. He said the new manager will take charge of the greatest club in the world. I mean that is Duncan Ferguson for you. He said we need to keep everyone galvanised and keep the spirit and fighting mentality there and try our best to get the next result. I thought our shaping in Staston Villa was quite good. Every player on the pitch tried their absolute best and that is all you can ask for as a manager. You just want the players to work as hard as possibly can and I'm sure you saw that but we need to work on trying to get the result that's over the next couple of weeks. We are the greatest club in the world. We love it so much don't we? That is why the fans are so passionate, why they are local people and love the club. This is an attractive club for somebody but the first thing we need to do is get results and he's absolutely right. Obviously I haven't got a couple of weeks off now till the next game against Brentford and the FA Cup. A couple of weeks until the next Premier League game against Newcastle away and then we'll play Leeds at home. That week will be huge so the next couple of weeks are really big with only really many players away on international duty. Obviously the core is going to knock but he's the only one who's really on international duty so Duncan Fakes will have the group for the next couple of weeks and we'll see what happens and who is in charge for the next game we play. One of those players he's got is Nathan Patterson who played his first competitive action in an evidence sheet last night. He played for the under 23, played an hour and had an assist in the goal by Lewis Dobbin. Obviously another member of the first team squad. Jared Branthwaite also played an hour in the game. He done well, he looked up for it, he looked competitive as I said he created a goal and for me it's about the next couple of weeks. If he can get up to that level that they want him to be at then I think we should start him against Brentford. I think it solves the problem. I think he's got the athleticism and he's got the determination to get into this side and make that right back spot his own. We have to get a settled team for these last 18 Premier League games and have to make it work rather than chopping and changing. I just don't think John Jones, Kenny's got the quality to play in that position. I just don't think James Coleman's got the legs to play in that position so this lab was brought in for a reason. Let's get him in the side and get him amongst it as soon as possible and hopefully he can make a difference. That started last night with him playing the under-23 game so really good to see him put on a royal blue shirt and hopefully it's the start of a very good career at Everton Football Club. There you go, keep an eye on our socials for anything that happens. We are at Toffy TV, EFC on Facebook on Twitter and on Instagram of course if anything happens we'll keep you up to date on it there. If anything breaks, keep an eye on Toffy TV YouTube channel and for any breaking news we're always ready to go so give this video a like, subscribe if you haven't already. 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