So at this point the students are digging a hole for their disposal shot and basically to maintain contain a lot of that fragmentation being produced by the by the


It's the one and only, the real God of Motherfucka. Actually, not the bitch, it's the robotic guy. And I'm here at Boss Talk 101. Make sure y'all check out my mother's fan interview, cause I ate it up. Nothing different. Thanks. Yeah, we on Boss Talk 101. Who you like, what's your favorite artist or whatever, what you like there? Well, you know, my favorite artist is Cardi B.  Oh, we gonna talk about that. Oh, yeah. Did she? What did she say? Cardi B, stop the press. What happened? How did you end up linking with her? What did she just, she just hit you up? Yes, she hit me up. What she said?  Like, I've been trying to get it. I ain't gonna tell my old lady, I'm trying to get Cardi B to hit me up a long time ago. You're upset now. But that's serious.  It's like zero seconds. So you got some more time to wait. So what? I've been waiting a long time. Boy, tell us about it. It was actually out of nowhere. She followed me. I actually posted a video. She followed you? Yeah, she's just following me. Damn, man. I can't, I gotta wait. She very posted me for like two years. Like, I've been doing like, every time she used to jump a song, I'd do a video. I'd probably make a freestyle within 24 hours, like, already done, but in the car. Have you ever been just going, I think I was, she'd be reposting it, posting it on Twitter, posting it on Instagram. I'd be like, oh my God. What did you say when she, like, what did you do when she followed you? Well, I make reaction videos for every time, so there's a reaction from my first one. And I went crazy. I was looking at the phone. Can you cry? I dropped the phone. Yeah, I cry every time. I always be crying. Cause I do these stuff. I be getting too happy. We are emotional. I be having anxiety attacks. Like, when she hit me up, I went on a hour walk. I walked an hour down the street. She hit you up? What did she say?   I said, I don't know if none of y'all's watching, but y'all know one person that will. You watch. Have you spoken to her on the phone? No, not yet. It's coming, it's coming, it's coming. Are you gonna be able to keep your cool? No, not at all. I don't, I don't know. Like, I'm gonna like, I have my date.     Shout out to Offset, man. Shout out to my father. Offset, man, Cardi B, man.  She know I'm her stuff.  They do the walkin' and they talkin' and they talk that walk. Yes, they do. Hey, man, that's a, man, that girl's dope, bro. Ever since she came out, I've been real big on Cardi B, man. Like, that's my dog, man. Even though she don't know it. That's my dog. That's how I used to see it, I used to be like, bro, she don't know it yet, but I'm her son. Yeah, man.  No, she just was talented. I love the way. That's cool that she reached out to you, though. I wonder, like, did she, I know she did. She like, I know she went through her DMs and seen everything that I've been seeing for the past six years. It's crazy. I literally seen her voice messages talking to her and she don't respond, but you would think that she responded by how I'm having the conversation and DM, like, it's so many of them throughout this year. Wow.  Wow. This motherfucker, it's crazy. Nah, it's just your hard work paid off, bro. That's all it is. Y'all don't give up. Y'all think they don't see it, but y'all don't give up, they see it. They be watching it. You ready? Mm-hmm. Check it, man. Got a K01, boss talk 101. Let's go. Bitch, I said what I said, I won't take it back, hope I'm gonna fight all my hoes coming out the back. Love them sticks and my head on the neck, move wrong, bitch, leave your body up in the back.   We see them chin hair, show me, bitch, we clock you. Ay, where's that? G-d-d-d, where's that? Ay, where's that? What a money nigga, where's that? Ay, ay, where's that? Where's that? Yeah, where's that? Where the money nigga, where's that? Ay, ay, my Gucci boots killed him. Come on, nigga, let's see it. Hey! Hey! My boy, hey, my boy Prince Sutter down. Oh boy, Prince didn't want him to go, no, boy. But we went in and it went down. We're about to go on, you see. Yeah, we started to keep going, nigga. We started to keep going, nigga.  He ready to go. That was dope, man, you did that, man. I like the way how you's gonna bring that in though.


There's a lot of people that won't even see this because the DCU is getting rebranded. It's getting rebooted basically in 2025 with the new Superman. James Gunn's taking things in a new direction.  Well, the point is to watch a good solid movie with characters that you liked from the last movie. And I think that that's a win. I don't need to have 13 more Shazam movies. I don't even need one more Shazam movie. This was a good beginning, middle, and end picture. And I think that's a way healthier way to look at this stuff. We shouldn't be going in waiting for the end credits to see what's next. You should go in watching a film that entertains you.


The three-year application of information and communication technology for educational development, spearheaded and financed by the Taiwan government, in collaboration with the Ministry of Education, is slowly drawing its curtains, as the last three schools embarked to receive smart classrooms under the project officially cut the ribbons for their new educational resource room. The Mekud Secondary, Labry Boys and Canaries Primary Schools are the latest recipients of multimedia classrooms, outfitted with state-of-the-art interactive technology to facilitate teaching and learning. At the handing over ceremony held at the Canaries Primary School, Ambassador of Taiwan to St. Lucia, his Excellency Peter Shen, expressed his gratitude to the people and government of St. Lucia for embracing their efforts to support bringing cutting-edge technology into the educational experience of students. Today shows the completion of our smart classroom program, and it's also opened a new door for cutting-edge education experience, and I am pleased to say a total of eight multimedia smart classrooms have been set up in St. Lucia this year. District Education Officer for District 8, Joel Charles Charlemagne, thanked the ambassador and the technical team of the Taiwanese Embassy for choosing to be part of reforming St. Lucia's education system by providing the tools and the capacity needed to meet global trends and standards. We urgently need to rise to and keep up with the pace of the new tech era or else we are doomed for failure. So we thank you for your interest in the progress and reform of our education sector to be more 21st century relevant and compliant. It is without contention that an educated and skilled populace is tantamount for a thriving economy. The Sufra Comprehensive Secondary School, the Sufra Primary School, and now the Canaries Primary School have benefited from your generosity, your outreach, your assistance to schools in St. Lucia. I am very certain that this room will be put to the youth intended. It will enrich teaching practices and make the learning environment more fun and interesting for the students. The Mekud Secondary and Labry Boys Primary Schools bring the number of schools with multimedia classrooms to eight. The parliamentary representatives for Mekud North and Labry both highlighted how Taiwan's intervention puts St. Lucia on a positive trajectory for the continued and complete transition to embracing technology within the classroom. I cannot get tired of saying thank you to you ambassador. We saw what COVID did, COVID really exposed the fragility and some of the deficiencies in the traditional classroom setting. This was a long way as we say in terms of incorporating the technology into the classroom. It was a long way to bridge in that gap in terms of allowing children access to class even if they are not physically present in the classroom. The smart classroom with a computer, interactive, whiteboard and projector has changed the process of learning, make it more exciting, that full technology it transports students to another place. There would still be need for teachers because teachers must accompany the students on that journey of discovery and learning. So there are lots of things that it does, it saves time, all those notes. Visuals are very powerful and it keeps students engaged and the teacher can now experiment and have the flexibility with a lot of different aspects. Over the past three years, the Taiwanese Embassy ICT for Educational Development project has provided over 25 schools on the island with technological equipment and the smart classrooms while simultaneously building teacher capacity in the use of the new technologies to enhance instruction. From the Communications Unit of the Ministry of Education, Sustainable Development, Innovation, Science Technology and Vocational Training, I am Daniel Dubois reporting.


Sometimes winning is staying in there when that's the last thing you want to do, when every part of you wants to run and get away from the nightmare that you're going through, that you showed up today.


Good afternoon, ladies and gentlemen. Thank you so much for joining us for today's AUKUS Defense Minister's joint press briefing It is my pleasure to introduce US Secretary of Defense Lloyd Austin the third Australia Deputy Prime Minister and Minister of Defense Richard Marles and the United Kingdom Secretary of State for Defense Grant Shaps each of the ministers will deliver opening remarks And then we'll have time to take a few questions Please note that I will moderate those questions and call on journalists with that Secretary Austin the floor is yours, sir. Thanks Patrick. Now. Good afternoon everyone. Let me start by thanking Deputy Prime Minister Marles and Secretary Shaps for coming to California for our second AUKUS Defense Ministerial meeting and It was definitely worth the trip We had a very productive and wide-ranging session and today Just underscore that AUKUS is a once-in-a-generation Opportunity that will promote peace and security throughout the Indo-Pacific The department remains deeply focused on the Indo-Pacific and AUKUS underscores that fact For more than two years our Defense Forces Industries and scientific communities have led this historic endeavor In March our three democracies boldly launched the optimal pathway for Australia to acquire conventionally armed nuclear-powered submarines Today we highlighted the important progress that we've made to see that plan through I'm very proud of the sailors from our three countries who are seamlessly training together across our trilateral partnership This year six officers from the Royal Australian Navy graduated from US nuclear power school and More are on track to graduate in early 2024 Now these proud officers and sailors will be the first to operate Australia's conventionally armed nuclear-powered submarines As part of the optimal pathway, we're also working to increase the frequency of US SSN port visits to Australia This directly supports President Biden's commitment to ensure that Australia acquires this new submarine capability At the earliest possible date While meeting the highest standard of nuclear non-proliferation The ministers and I also reviewed our major accomplishments within the crucial second pillar of AUKUS Through pillar two we have leveraged our respective defense innovation and industry sectors To develop and deliver Advanced capabilities so that our war fighters can hear See and act with decisive advantage We're innovating with cutting-edge tech in several areas including artificial intelligence electronic warfare and quantum technology And by implementing strong standards of technology protection We are reducing barriers to sharing information and technology And we're streamlining our processes to deliver these capabilities and lead our partnership forward for the next generation Today we endorse several new efforts under pillar two of AUKUS Let me highlight two of these efforts The first is a maritime autonomous experimentation and exercise Beginning next year our three countries will conduct a series of integrated trilateral experiments and exercises They will enhance capability development improve our interoperability and Rapidly accelerate the sophistication and scale of autonomous maritime systems That we can deploy and operate together Second is an AUKUS innovation challenge series That will mean that companies from across our three countries Can compete for a common innovation challenge prize That will help our war fighters improve interoperability gain decision advantage and strengthen our deterrence Now these challenges will begin in early 2024 with a prize challenge focusing on electronic warfare Again and again AUKUS proves that we are stronger together And every day we move closer to our shared vision of a free and open Indo-Pacific So Deputy Prime Minister Morales and Secretary Shops I'm truly grateful for your leadership and your friendship and for everything that you're doing for our shared security And I look forward to strengthening our partnership even further into building a more secure future together Richard Well, thank you Lloyd. Thank you for hosting us here today And it is great to see you and it is great to see you Grant here in California and we have had a very productive day in this trilateral AUKUS Defense Ministers meeting and as we reflect today on What has happened in the last 12 months since we met as a group in Washington DC in December of last year It has been a truly momentous year where there has been an enormous amount of progress particularly in respect of Australia acquiring a nuclear-powered submarine capability with the help of the United States and the United Kingdom under pillar one of AUKUS in March of this year we saw the announcement of the pathway by which this would be achieved Which was a breakthrough moment in terms of Australia's defense capability and since then we've seen the standing up of the Australian Submarine Agency we have seen the commencement of infrastructure work We have seen Australians undertake training both Submariners and defense industry workers Here in the United States as as Lloyd mentioned the nuclear power school, but also in the United Kingdom We have seen Importantly the legislative work proceed to enable this to happen in both of our countries We've seen the frequency as we promised back in March of Visits of the United States nuclear powered submarines happen to Australia Indeed in the last 12 months we've seen the USS Mississippi the USS Asheville and the USS North Carolina Visit our country and we look forward in the future to Seeing more of those visits including visits of the stu class submarines from the United Kingdom But as we reflect on that year past we are also making very important announcements today about the future in respect of pillar one Next year we'll see the most significant maintenance occur in Australia of the United States nuclear powered submarine Which we plan to happen in the third quarter of next year Which will crystallize the development that needs to be occurring in terms of both infrastructure and skills within our workforce and our Submariner base So we are really pleased about the progress in respect of all of that and it is An enormous amount of work which has occurred over the course of the last 12 months But significantly in terms of today's meeting. It is actually pillar two Which has taken Center stage And indeed I think today's meeting will be regarded as a critical moment in the history of pillar two of orcas And that is the sharing and development of advanced technologies between our three countries. We're putting in place The architecture which will enable that to happen through the international joint requirement oversight council, which will give a Joint capacity to look at the technologies with which we are pursuing Our armaments directors will be working together with the tabletop exercise Next year and as lord mentioned Our innovation organizations the defense innovation unit Of both the united states and the united kingdom along with the australian strategic capabilities accelerator Working together on joint challenges Which we'll see initially that happen in the space of electronic warfare And again the legislative environment which is being worked on In our country is creating a seamless defense industrial base between the united states Australia and the united kingdom, which is so essential to seeing orcas pillar two Become operational But in terms of that architecture today, we are backing it up with practical first steps Where pratt web where specific technologies are being worked on And we are announcing a significant number of those today, which include work on quantum clock, which includes the Deep space advanced radar capability in our three countries and includes resilient artificial intelligence Which in turn will give rise to resilient precision targeting All of these are practical steps forward Which come on the back of an architecture, which has been established today in relation to pillar two And I think when we look back at the significance of today's meeting We will see this as the critical meeting Which was a watershed in the progress of pillar two of orcas Across all of our work pillar one and pillar two across the friendships that exist between the three of us personally What is really clear is that Orcas represents a powerful Combination of countries working together, which is sending a really important message to the world I think all of us have remarked on the fact that Together when we are meeting when we are discussing these technologies when we are taking these steps down the path Each of our countries is much stronger Thank you very much Lloyd Richard It's 80 years since the us navy shot down an enemy dive bomber of cape hunter on the south coast of guadalcanal Nothing particularly unusual. It was the second world war Going on at the time. But what was remarkable about that particular navy destroy Was that the fighter The actual Missile hit would hit it without a direct hit it missed in effect So this was actually a piece of fatal artillery That had destroyed That particular item but the shell itself was revolutionary because The 15 centimeter fuse was capable of detonating when it was close to the target So in a sense it missed But it also hit The tiny proximity fuse Not only helped win that particular battle, but it also helped turn the tide of the war And this miniature miracle was only possible because of a different type of fusion and that was You guessed it between australia the uk and the united states It's actually developed by a brilliant australian scientist called william buteman Who was in london developing it and the fuse was then transferred to the united states Where it turned into this tremendous game changer Today in a much more dangerous world with russia waging war in ukraine With hammers reaching havoc in the middle east China undermining the freedom of navigation in the indovacific. We've never had a greater need for More innovation to be more pioneering, which is why over the past few years orcus Has been fusing together our Transnational brain power and here at the defense innovation unit We've discussed the results a raft of game changing new orcus projects Together our nations will be launching and recovering undersea vehicles from torpedo torpedo tubes On current submarines will be enabling us to deliver more sophisticated strike intelligent surveillance and reconnaissance will be using quantum technologies just mentioned to support global positioning enhancing the ability of our Undersea capabilities including on our future ssn orcus submarines So they can stay silent and undetected And sharing in not only all of this not only with multiple undersea systems But with our p8a maritime patrol aircraft strengthening our capacity to zero in on potential dangers I think the inventor of that artillery shell dr. Buteman Would have been proud of what we've been talking about today the man they dubbed at the time mr Searchlight radar would have certainly appreciated another landmark agreement that we are signing today because In the coming years our nations will be creating a global radar network Bringing together three ground base stations one in each of our countries radars more sensitive more accurate More powerful and agile than anything that has gone before giving us the ability to see beyond the clouds and to detect Identifying track in space up to 22,000 miles away operational by 2029 that's this new initiative Alongside orcus will not just help us to protect our communications and our navigation satellites from deadly threats of tomorrow But will boost all three of our economies Back in 1943 our brightest minds proved that geography was in fact no barrier To our innovation as long as our values and our Ingenuity was in close proximity And 80 years on inspired by the creators of that miraculous Invention our alliances once again fusing our intelligence together sparking new capabilities From sea to space and igniting the ideas that will change the course of our history So Lloyd Richard, that's why I've enjoyed our conversation so much this afternoon We said we were going to Ensure that pillar two was meaningful that it would change the way that we work together And today we've set that course and delighted to be working with you both. Thank you Thank you very much gentlemen. Our first question. We'll go to Associated Press Alita Belldor To the two ministers You've both talked a bit about the Various types of technology that you agreed on today The radars the deep space And What specifically your countries are looking at in terms of what you think One of the top priorities is for improving your military capabilities through this What would be something that you were really Hoping to key in on what would aspect of it and Mr. Secretary for you if you want to answer that question also But a separate question for you Today The as you know the the truce In israel ended israel Do you believe israel has been taking the american warnings about civilian casualties? Seriously, are you or are you Concerned that even hours after the truce ended there were about 175 deaths already in gaza I'll start if you All of us how this started October 7th Hamas launched a very brutal attack Into israel Took the lives of over 1200 israelis Took over 200 hostages Which included Small children and holocaust survivors and and so We have said a number of times that We will continue to support israel's right to defend itself Now i'll let israel speak for all operations What i will tell you lita is that You know i'm i talked to minister galant on a new york daily basis as you know At each time i talked to him i remind him of the Necessity to make sure that we're that they are protecting Innocent civilians and creating pathways and corridors for civilians to To move out of the out of the battle space And they need to make sure that they're doing things to ensure civilian safety So that's that conversation is ongoing On a near daily basis We're going to continue to work with Israel and egypt and in cutter On effort efforts to reimplement the pause and i think the pause during the pauses You know we've gotten some very meaningful things done in terms of a number of hostages out of the out of gaza Plus we've been able to introduce a meaningful amount humanitarian assistance More needs to be done for sure But it's encouraging to see that we've been able to ramp up the amount that we've We've moved into gaza One of the conditions for The pauses to continue an israeli condition was that hamas be able to provide At least 10 hostages for release each day And they've at this point failed to produce names or hostages for you know upcoming days And i would also remind you that that we've seen things like A brutal terrorist attack recently on the part of hamas in jerusalem So hamas has violated the the very conditions that that they agreed to so Again This remains dynamic. We'll continue to do everything we can to emphasize to the israelis the necessity to operate within a law of war but also Protect civilians in a battle space We'll perhaps in answer to the the first part of the question Australia is a an island nation which is geographically relatively distant from Other parts of the world and so our needs Lie in maritime capabilities, but longer longer range capabilities Capabilities which enable us to project and so Having strike capabilities are really important. And when you look at something like Resilient precision targeting Which forms part of the announcements today that is critically important When you look at the maritime autonomy measures that are contained in the announcements today, that's hugely beneficial for a country like australia I think the third area which is which is covered by the specific Technologies which have been described in the announcements is decision advantage You know decision advantage in terms of the time it takes to make a decision in the battle space But also the fidelity with which the fidelity of information that is therefore the decision maker That's obviously critically important to any defense forces. It is to ours So when we look at the technologies that we are working on as three countries They are highly relevant to the specific needs of the australian defense force And to the same question about which element of it, of course, we've had nuclear subs running for nearly six decades under the sea in the united kingdom nuclear arm as well So there are two things which are exciting. Well, there's a huge number to go after Uh Simple answer to your question Is it's hard to know which of those will end up being the most relevant part of the new You can't see the future but partly and I was reflecting our meeting Given that kessinger that's just passed away He once remarked no country can act wisely Simultaneously in every part of the globe with every moment in time And I was saying in our meeting that's why three wise heads were together assuring better than one I'm excited about all these developments, but I can't tell you which one in the long run I should be most excited about today. That is the purpose of working together Thank you our next question will go to channel nine news australia johnny Seriously, thank you very much for your dear general johnny kersley from channel nine australia From secretary's minister Thanks very much for your time today. I wanted to go to initially china's recent aggression in other city For instance, so in our attacks on australia may be others. We've seen The chinese coast guard vessels ramming into philippine navy vessels But these are the types of issues firstly, but then to be raised directly lean at a leader level Uh, it has it increased your need for urgency in formulating the august agreements, particularly around some of the aspects And the increase in the urgency in bringing that partnership together a lot faster and finally if I may August is an agreement that is designed to last you described it as generational Donald trump is running for president of the united states is a mandatory American Confident that president biden is going to win the next election And and I will remain confident of that I would tell you that What we're working on here is as you pointed out a generational capability We have common uh goals and objectives foremost among those goals and objectives is to Ensure that we maintain a free and open endo pacific Our values are are are similar And we have a long-standing working relationships with each other I continue to see Bipartisan support in both chamber of congress, uh for August and That support Success going going forward so again I'm very confident that we'll be able to work together on this very very important issue And and again, I think All three of our countries See the value in the you know long-term capability that this is going to create Joe to go to your first the first part of your question The That is My concerns about And it's important that we make this public and those representations have been made very clearly to In terms of the and what does that say about the environment that we're operating in and the importance of Moving at a pace in terms of our work. I mean the answer is It absolutely highlights the need for this arrangement and it absolutely highlights the need for speed in this arrangement And I think you can see that speed I mean when you look at where we were 12 months ago compared to where we are today It is a huge leap forward in terms of the work that we've done on Australia acquiring a nuclear powered submarine capability, which for our country represents the the biggest leap forward In military capability in more than a century And you know the fact that we've now been able to identify that pathway that I said earlier We've stood up the relevant agencies. We're seeing infrastructure work. We commenced our workforce being trained The visit's happening This is moving fast. But what does happen Well, firstly though in terms of acquiring that capability It will change the character of Australia in terms of what we are capable of doing and that does not happen overnight There is no showroom where you can go and buy a Virginia class submarine We are acquiring the capability to operate and build nuclear powered submarines in Australia It will be the biggest industrial endeavor that we have ever undertaken as a nation And it is not going to happen overnight But I'll tell you what does happen overnight The fact of our three countries standing here right now Meeting in the way that we are making the plans that we are making and walking down this path together Represents an enormous deterrent effect immediately It sends a very very powerful message to the world and a very important message to the world And as to the final part of your question We are completely confident About the American system as I might say we are about the British system and indeed our own For this reason that across the three countries There is bipartisan support for this arrangement That's actually what gives it power by virtue of that bipartisan support across our three nations We can confidently say That when the three of us are not standing here at some point in the future and there are three other individuals They will be pursuing the same objectives and in respect of the u.s. specifically You know, I've had the opportunity of being on Capitol Hill In the last few weeks To talk to lawmakers around The legislation which is currently progressing through the u.s. Congress and as the lawyer said Across the political spectrum. There is Both a commitment to the alliance between australia and the united states and support for it There is commitment to orcas and the relationship between our three countries And you know, I think there is strong support for the specifics Which is in that legislative package, but that ought to give Everyone in australia an enormous sense of confidence that this is an arrangement And a set of relationships which genuinely does enjoy support across the political spectrum in the u.s As I know it does in britain as it certainly does in australia Actually, I've been on Capitol Hill unrelated in a separate visit Last month very very support for the orcas and so every confidence in As richard says in orcas being for the very long term On your actual point navigation of the of the open seas under international law or something the united kingdom takes Incredibly seriously We regularly and consistently are in the In the pacific will continue to be so Do so We already announced plans to send our carrier strike group by the queen Elizabeth class Again following a visit previously the You know be very important In the context of orcas actually just to be absolutely clear that the those international rules are there for a reason The International maritime organization takes this particular right of free passage very very seriously Orcas is a fantastic way to make that clear to Just on that specific aspect, but to these incidents that occur in the south trying to see Even the city more broadly do these need to be raised directly? Yeah, and and and actually there are there are always diplomatic processes to do that because you know the same in the air Other circumstances with Russia for example So these these issues are always raised this professional way to Behave and act in the air in the sea When countries breach those professional professional it's a natural professional pride for most navies for one thing And But no one should be under the impression That any of us are prepared to be kind of bullied out some forces which are clearly international forces But all to sail under international law much gentlemen our final question will go to NBC news robert handa Happy to take the final question. It's been thrown open Can I can I ask you? Minister mild you say that this is moving fast Is it right that you've indicated today that an east coast base for the nuclear subs Will not be selected to the end until the end of the decade? Do we really need six years to make that decision and are you kicking the can down the road on something that's Believing We've made that position clear before today That the timing in which to determine an east coast base is at the end of this decade and We we made that clear at the time that we Articulated the pathway by which Australia would acquire and you could have powered submarine capability I mean it will be an important decision But in the sequence of decisions that needs to be made that's when that's when it comes up And you know there will be other decisions that need to be made such as where we will Locate the place that we will ultimately dispose the nuclear reactors. That's clearly going to be a devial decision But in turn that's a a decision which comes later again What so it this is a matter of laying out the pathway and following it right here and right now We are evolving an industrial capability to build nuclear powered submarines out of the Osborne naval shipyard in Adelaide We are evolving an operational capacity to operate and maintain nuclear powered submarines out of hms sterling in Rockingham south of Perth We you know the first step was about having an increased number of nuclear powered submarine visits to HMS sterling which is happening as we speak and as I articulated you know the next step in that will occur Next year as a result of the meeting that we've had today So each of these is about working through the steps of the pathway that we articulated last march and The determination of the location for an east coast base is going to be a big decision But it's a decision which is to be made in the late 2020s and we're going to work through each of the decisions as they come Gentlemen, thank you so much ladies and gentlemen. This concludes our press brief


Okay, good afternoon, everyone. So I want to start by rewriting the precise statement of Birkoff's ergodic theorem, because in the last lecture there was just a little bit of confusion about the statement. So I tried to write a kind of cleaner statement, but let me write the precise statement here, because I think it's better to have it. So the statement works in abstract measure spaces, so that's why I think it's better to write it. So we have M is a measure space, F, M to M just needs to be a measurable map and a mere invariant probability measure, so then every integrable function, mu integrable function phi, the limit exists. So this is the first part of the statement, so I'm just rewriting again Birkoff's ergodic theorem, because I did not write it in its full generality and I think it would have been a good idea to have it. So in a completely abstract measurable setting, we have not assumed ergodicity yet, just the fact that F is invariant implies that this limit exists, which is in itself highly non-trivial, right? This just means that if you look at the average value of phi along an orbit, it does converge to an average, okay? Moreover, if mu is also ergodic, then this limit is exactly, sorry, I should say for then the limit, then for every phi, the limit exists for mu almost every x, of course. If mu is also ergodic, then the limit, and this should be 1 over n, sorry, limit 1 over n, sum i equals 0 to n minus 1, phi integral F i of x is equal precisely to the integral of phi d mu for mu. So just to spend a couple of minutes on this, what is the difference between these two and why do you need the additional ergodicity assumption to get the second statement? I'm not asking for a formal reason, but just intuitively, can you see the differences? So this is the limit exists, this is saying that the limit is equal to very specific value. Why can this not be true if mu is not ergodic? Why can you not hope to have this be true if mu is not ergodic, you have some idea? That's just for any invariant measure that's true, that's the definition of invariance. So for any invariant measure, F star mu equals mu, and this is true. Why can we not have this, why do we need the ergodicity for this, any ideas? So remember what does non-ergodicity mean? So we have our measure here, non-ergodicity means that mu can be, has at least two ergodic components, has several ergodic components, right? Mu is some kind of linear combination of various measures, okay? So there is a measure mu one and a measure mu two, and some combination of these is equal to mu. Now, what does the point do? For ergodic measures, what this theorem is really saying is that the distribution of the orbit of x inside this ergodic component is uniform with respect to this measure, right? Because the average value of phi, what is this? This is the average value of the function phi. Function phi can be, for example, the characteristic function of some set or it can be a more general function. But what this is doing is taking the orbit of x and is looking at the orbit of x and taking the average value of phi along the orbit. And what is this? You can think of the integral as the average value of phi also in a different way, right? This is the average value of phi on your whole space with respect to this measure. So these are two different ways of taking the average value of phi and this theorem says they coincide. And why do they coincide? Because this orbit is very uniformly distributed in your space. And therefore, if you take the average value of phi along the orbit, it will coincide with the average value of phi in your space because the orbit goes everywhere, it spends equal amounts of time everywhere with respect to this measure. Okay, this is what it's saying. So this cannot, of course, work for if the measure is not ergodic because if you choose a point that belongs to this ergodic component, it will never land in here, right? There is no reason that this should converge to the average value of phi over the whole space. Only with respect to the ergodic component to which it belongs. Just to give you some feeling for this. So how was it that I wrote it in the last lecture? I wrote it a little bit differently. So if you remember, we have the basin of the measure mu is equal to the set of x such that 1 over m, the sum i equals 0 to n minus 1, converges to the integral of phi d mu for all phi continuous. And what I said, which is really corollary of this corollary, is that if m is a metric space because we cannot even define continuous function if it's not a metric space, right? So if m is a metric space, then mu will be mu equals 1. And what is the difference between these two statements? It's a very subtle difference. I will not give a rigorous proof, but just to mention this, I think it's useful. So in this statement, this set of points for which this holds may depend on the function phi. So the statement is that for every phi, this limit exists mu almost everywhere. It means that the set of points where it does not converge has zero measure. But this actual set of zero measure might depend on the function phi. So for each function phi, there's a different set of zero measure for which this holds. And the same thing here. What we would like really is to find a set of full measure for which this convergence holds for all those points for every function. That's what we would like. And that is what we can get if we look at the slightly stronger situation in which we're looking at the convergence with respect to continuous functions, which we can do within a metric space. And in that case, what we're saying here is that there exists a set of full measure for which this convergence holds for every continuous functions. We cannot say that there exists a set of full measure such that this convergence, such that for that set, this convergence holds for all L1 functions for every point in the set. It's a very delicate difference, but it's a difference that I thought I should mention. Okay. Yes? So why would we need to assume it's a metric space? Well, in two ways. First of all, just to define continuous functions. And second of all is because the argument that you use to show that you can take this intersection. So basically, you're taking some kind of intersection of sets of full measure. So you're taking the set of points for which this convergence, for each function, you get a set of full measure. And then you take some kind of intersection of all these sets of full measures for all the functions. And so you use the fact that the continuous function have some dense countable subset of continuous functions. And you can take the intersection of this, of the full measure set for each of this countable subset. Yes? Do we need what? That is a good question, actually. I'm not 100% sure at this moment. I have to remind myself of the actual proof. Okay. Compact? Maybe. I don't think so. Yes, yes. I don't remember exactly. I will look it up and I will let you know. That's a good question. I wasn't planning on giving you an actual proof of this following from this. So I just wanted to emphasize this very subtle point, which is that in this case, the set depends on the function. And in this case, we are able to find a set for which this convergence holds for all functions. And with continuous functions, it's easier to do that. Okay. So what we want to do now is we want to start applying this theorem. As I said last time, this is a really powerful theorem because what it says is that if you know the measure, you know this limit. Okay. So really it's saying if you know that a system has an ergodic invariant measure, you pretty much know at least the statistics of the dynamics for almost every point. You know that almost every point is distributed in space exactly according to that measure. So it's a very powerful theorem. It's a theorem from the 20s, and it's still one of the fundamental theorems of dynamics. The difficulty, therefore, is to show that a measure is ergodic. And this is highly non-trivial. And as I said, in certain cases, you don't even have an invariant measure to begin with, so you need to first find an invariant measure and then show it's ergodic. Okay. So what we're going to start with in the next couple of lectures is some systems in which we already have an invariant measure and the problem is to show it's ergodic. And I will introduce a couple of interesting techniques to prove ergodicity of certain measures. So section ergodicity and circle rotations. So what is one of the simplest settings in which we know that we have an invariant measure? So if you have... So we write the circle, unit circle of length one, right, that we've used before. And we're going to prove the following theorem. This one. B and irrational circle. So there is one measure that we know already that is invariant here. What is this measure that is invariant here? Mod 1, yeah, in the circle. What is the measure that is invariant here? What? The what? I can't hear you. Yes, which measure is invariant? What measure is invariant for this map? Do you know one measure that is invariant? Which measure is invariant? What's delta zero? No, f of zero is not zero, f of zero is alpha. Unless alpha is also equal to zero. What is an invariant measure? If alpha is equal to zero, we have the identity, but alpha is irrational. We assume that it's an irrational rotation. So if alpha, as you remember, if alpha, okay, we need to review a little bit circular rotations, right? If alpha is rational, then every point is periodic, remember? And then we have the Dirac delta on the periodic points, okay? But the average of the Dirac delta on the periodic points. But if it's irrational, then we have no periodic points. But there is a measure that is invariant for all alpha, both rational and irrational. Another measure. A natural measure. An obvious measure, Lebesgue measure. Thank you. Okay? Lebesgue measure is invariant, because if you take an interval here and you rotate it, you get an interval here that is exactly the same length. Okay? It's invariant. f minus one of an interval is the same measure. So Lebesgue measure is invariant. Both in the rational case and in the irrational case, right? Just translation. So in the rational case, is it ergodic? In the rational case, it is ergodic? Why is that? Right? Right? So if alpha is rational, is Lebesgue measure ergodic? What if alpha is zero? Is Lebesgue measure ergodic? For example, let alpha equal to zero. f of x equals x, identity map. Lebesgue measure is invariant. Okay? Is it ergodic? It's not ergodic. You're sure it's not ergodic? You can take any interval here and it is invariant, fully invariant, and the complement is also invariant, and they have measures strictly between zero and one. Okay? What if x alpha is equal to one-half, for example, or one-third? Let's say one-third. Lebesgue measure is still invariant. Is it ergodic? Why is it not ergodic? Yes? Yes. That's because that's why it's invariant. Is it ergodic? Of course we can take an interval, but that's not the definition of ergodistic. It needs to be an interval that is invariant. Come on, guys. It's not ergodic. Why? What's ergodicity? Ergodicity is the definition that if f minus one of a equals a implies that mu of a equals zero or one. So to show that it's not ergodic, all you need to find is a set that satisfies this. Okay? Not just any set with measure between zero and one, but one that satisfies this. Where's the set? Yes? So you take this interval here, a1. You take the image here, a2. You take the image here, a3. Exactly. So if you take a equals a1 union a2 union a3, then it satisfies this condition, right? Because a1 maps to a2, maps to a3, maps to a1. The pre-image is exactly the same. a1 maps to a3, maps to a2, maps to a1. Because it's a rotation by one-third. Okay? So this is a set that satisfies this and does not satisfy that. You found a set that is fully invariant. It's not a connected set. It doesn't have to be connected set, of course. It's just a set of positive measure that's not fully invariant. You can see that you can do that for any rational number. You can easily do that for what we said is for every rational number, every point is periodic, and they all have exactly the same period, right? If the rational number is p over q, every point is periodic or period, q. Okay? So if every point is periodic period q, as long as you take a small enough interval and you iterate it q times, it will come back exactly on itself. If you take it sufficiently small, there will all be disjoint, all these images up to time, q and you will have exactly the same situation as you have here. Okay? Proof. This is a good exam question. So make sure you understand the argument. So if the map is rational, the big measure is not periodic. Okay? We've just proved that. I thought it would be easier to see, make a formal statement about this. So the question now is if it is an irrational rotation. So for an irrational rotation, we cannot use the same argument to prove that it is not ergodic. Because an irrational rotation, however small an interval you take, it will never come back exactly on itself because the points are not periodic. And indeed, we will show that then the big measure is ergodic. In fact, we will prove something more. We will prove, in fact, the big measure is the invariant ergodic. So what does it mean in terms of the distribution of points? The fact that it's ergodic. So we'll come back to this uniqueness point in a little while because it's very important. So by Birkhoff's theorem, corollary for Lebesgue, almost every x and every interval a, b in S1, we have that 1 over n times the sum. So we have that the frequency converges to the Lebesgue measure, right? So if we take an interval, A, B, take a point x, you iterate it and you wait and see what the frequency of visits to this interval is, then this frequency converges exactly to the Lebesgue measure, maybe. This is a simple consequence of Lebesgue measure, of Birkhoff's ergodic theorem. But in fact, we will prove in this particular case because it is the unique invariant measure, we will prove a stronger theorem. So this condition we say that x, orbit of x, orbit of x is uniformly distributed with respect to Lebesgue, okay? Just another way of talking about the convergence in Birkhoff's theorem with respect to characteristic functions. So we will prove the following theorem that says that if fS1 to S1 is an irrational circle rotation, then every orbit is... So this goes beyond Birkhoff's ergodic theorem, right? This says that this property here holds not just for almost every point, which is what you get from Birkhoff's ergodic theorem, but for every point. So the uniqueness is interesting here and we will study now some general properties to do with maps that have a unique invariant ergodic measure. So we're going to cheat a little bit. So I said that we're going to show that Lebesgue measure is ergodic, okay? So in this particular case, the way we're going to show that Lebesgue measure is ergodic is by showing that Lebesgue measure is the only invariant measure. If Lebesgue measure is the only invariant measure, why does that automatically imply that it's ergodic? Yes. Yes, also, yes. Exactly, it's only one point. So we proved before that the set of invariant measure is compact convex and non-empty. And if we prove that it's only one point and that the ergodic measures are the extremal points, they're always non-empty. Also, the set of ergodic measures is always non-empty in the case in which the space is compact and the map is continuous because they're the extremal points of the set of invariant measures. So if we show that Lebesgue measure is the only invariant measure, automatically it has to be ergodic, okay? So it is also possible to show that Lebesgue measure is ergodic by actually looking at the dynamics directly to show ergodistic. But we're going to do that for different class of maps. In this class of maps, we're going to take advantage of this property and we're going to prove that it's the only invariant measure. And that will imply that it's ergodic. And in fact, it implies additional properties such as this. So having a unique invariant measure is a very strong property of a dynamical system. So let me write this as a general definition. Definition f x to x is uniquely ergodic if it admits a unique ergodic f probability measure. So I put ergodic in parentheses because it follows from the uniqueness of the invariant measures, right? It's not part of the definition, really. It's just that if it's got a unique measure, this measure must be ergodic. So can you give me another example of uniquely ergodic system besides these circle rotations, which we're going to prove now, a simple example that we've seen many times before? What other system can you think of that has only one invariant measure? You're right, actually. The adding machine has a unique invariant measure. It's related to the circle rotation, but it's not completely obvious to see what this measure is. What about a very simple, much simpler example than that? So what kind of example? Okay, you can have a constant map that really maps everything to a single point. That's true. And why does that make it uniquely ergodic? Only the delta at that point. And do you have to map everything to that point? Or can you think of a little bit more general situation that's similar to this one? Sorry? Well, all you need is every point to converge to a unique fixed point. What maps do you know that have this property? Exactly, contractions. If you have a contraction mapping on a complete metric space and so on, everything converges to a unique fixed point. How do you know that there cannot be any other invariant measures except for the unique fixed point? So if you have some set here, you have your unique fixed point and you know that everything converges to this fixed point. So the Dirac delta in piece clearly an ergodic invariant measure because we've already said, shown that if you have a fixed point, the Dirac delta on that point is a fixed point. How do you know in this case that there's no other invariant measures? Why? It would have another fixed point, why? Yes? Right, if you had another set of positive measure or another set that was fully invariant and had positive measure it would have to be a contraction on that set. You'd have to have another fixed point. You're basically what you said is correct. Let's try just a slightly simpler way of saying this. Suppose you had another measure that was not the Dirac delta. Then there would be some other set that would have positive measure. It would have to be. So suppose you have some set that does not contain P that has positive measure for some other invariant measure. What do we know about this set? A, the fact that it has positive measure with respect to some invariant measure. Yes? Exactly. After submitted, the points would come back by Poincare recurrence theorem. Also by Birkhoff's ergodic theorem Birkhoff's ergodic theorem will show if this measure is ergodic and every measure can be decomposed into ergodic measures so you can assume it's ergodic and then this will come back infinitely many times. But even without Birkhoff's ergodic theorem that's like just by Poincare recurrence theorem you know that all points will come back. And is this true? Everything is converging to P and A does not contain P so they do not come back. So there's no recurrence. So invariant measures are absolutely connected to recurrence. As you can see from the Poincare recurrence theorem and the Birkhoff ergodic theorem that gives a much more precise quantitative statement by what we mean by recurrence, the frequency of the recurrence. So you can use this fact to understand situations like this and know that if all these points never come back there can be no invariant measure that is living in that place. And therefore in this case this map is uniquely ergodic because it has only one in a kind of trivial way. I mean it's uniquely ergodic in a simple way. But what we're going to show is that circular rotations are uniquely ergodic and that is much less trivial because circular rotations you have recurrence everywhere so it's certainly not immediately obvious that you have only one invariant measure. So the way we're going to prove it is by a general theorem. So suppose f x to x is a continuous map of a compact matrix space and suppose there exists a dense set phi of continuous functions such that so continuous functions phi such that phi in phi, there exists a constant phi bar which depends on phi such that 1 over n sum i equals 0 to n minus 1 of phi composed with f i converges to phi bar uniformly, uniquely ergodic. So there's a, I haven't finished writing the theorem there's a converse as well but let's first just think a little bit about what this says and make sure we understand the statement. So you take a continuous map of a compact matrix space and you take the space of all continuous functions on x and you suppose there exists a dense subset of continuous functions which converge uniformly to the constant function. So notice this, I could write x here but I'm not saying that for each, well I'm saying that this converges as functions. This is a family of functions and they converge uniformly to this constant function here. It's a bit like saying that for every x you converge to a constant to the same constant uniformly in x. We have uniform convergence of continuous functions. Then f is uniquely ergodic. At the moment there's no reason you should see any connection between uniform convergence and uniquely ergodistic but this is what we're going to prove. Let me state the converse also. There's a converse. Conversely suppose f is uniquely ergodic then for all phi x to r continuous there exists some constant phi bar that depends on phi such that, so let me call this condition here star such that star holds if it is the unique invariant measure then phi bar is equal to the integral of phi. So what's the difference between this and Bilkoff's ergodic theorem? So here this of course is obvious for the invariant measure. For the invariant measure we clearly have, well we don't have the uniform convergence. This is what Bilkoff's ergodic theorem says that this for every x this converges to the integral. So it's clear that if we have this convergence it must converge to exactly this for almost every point. And here we're ashamed that in fact it converges uniformly as a function which means that it converges for every point uniformly. So it's a much stronger form of convergence and this follows from the fact that it's uniquely ergodic. It turns out it's like a strengthening of Bilkoff's ergodic theorem for the specific uniquely ergodic case. Because there's nothing else it can converge to that. If there were other measures then these averages would have to converge to other things but because there's nothing else they can converge to in some sense they have to all converge to the same limit. Roughly speaking. Okay so this is the statement and then we're going to use the statement to prove that irrational circle rotations are uniquely ergodic. We're going to find a dense set of continuous functions and prove uniform convergence. And then we're going to apply this to show some strong kind of uniform distribution property for circle rotations. So first of all let me just make a small remark is that this holds the mark theorem also holds for complex valued functions observables phi x to c. Okay this is actually simple even in Bilkoff's ergodic theorem. I mean all these theorems it's simpler and it's more natural to study for real valued observables but if you look at the proofs you see there's no difference. Okay you can do everything for complex valued observables and we will use this a little bit later on. So maybe before I start the proof of this theorem let's just take a couple of minutes break and stretch your minds. Okay so let's prove this theorem now. So there's two parts the first part that says that if this happens it's uniquely ergodic and the second part which is the converse. So the first part we're going to split into two lemmas. First of all we're going to show that if you have this uniform convergence for a dense set of functions then you have the same uniform convergence for every function. And then we show that if you have uniform convergence for every function then f is uniquely ergodic. So two conceptually easy steps. And then we will prove the second part of the theorem. Okay so let's state the first lemma under the same assumptions I won't write them but under the same assumptions as before we have a continuous map of a compact metric space. Okay if star holds for all phi in phi then it holds for all. Okay so let's prove this it's basically just a kind of approximation theorem. So let's write to simplify the notation let me write this bn of x phi is equal to 1 over n sum i equals 0 to n minus 1 phi equals phi of x. Okay this is the averaging that we always work with and we can think of as a kind of bulk of averaging operator in some sense as we shall see. So if phi is in phi okay what is the assumption? Star says that there exists some phi bar which depends on phi. Okay such that this is exactly the average so this is a function such that bn of x phi converges to phi bar uniformly in the variable x uniformly in x. So now let psi, what did I use? x or m here, x I think for this space. X is continuous, arbitrary continuous function so we want to show that the same thing is true for x. So let epsilon greater than 0 and choose phi in phi such that so what does dense, what does it mean to say that this set is dense in the continuous functions? What topology? That's right. Uniform topology that's what. So if we don't specify talking about continuous functions you always talk about the C0 topology or the uniform topology. So by the density what does this mean? This means that we can approximate psi uniformly. So there exists phi such that the supremum overall x in x of psi of x minus phi of x is less than epsilon. This is what it means that we're epsilon close using the density. And what does that mean in terms of the bulk of averages of psi? So this means that at each point of the orbit of x phi and psi are epsilon close. So these errors, the difference between these two is accumulating a rate of epsilon every iteration but you're dividing by n. So the average error is still epsilon. So we have that bn x phi minus bn x psi is less than epsilon for all x and all n because it's uniform. Uniform topology, uniform convergence and so since we know that this is converging to phi bar, that means that this is also uniformly close to phi bar. So we have that the supremum overall xn of bn x psi minus phi bar is less than epsilon. And the infimum of xn of bn x psi minus phi bar is also less than epsilon. So does this mean that this bn x psi are converging to phi bar? Is this converging to phi bar? Does this mean that these are also converging to the same phi bar? Why not? Exactly. Very good. So just a remark here. This is not showing that these are converging to phi bar because the choice of phi depends on epsilon. But what it means is that this and this are close because they're both close to the same thing for a given epsilon. And so we have that the supremum of overall xn of bn x psi minus the inf of xn bn x psi is less than 2 epsilon. Since epsilon is arbitrary and this shows that they can be arbitrarily close and so they're also converging. So star holds. So a simple kind of approximation argument to show that that's why we take a dense set and this uniform convergence shows that we can extend it to every continuous function. And now we show that if it holds for every continuous function, for lemma, if star holds for all phi x to r continuous then f is uniquely ergodic. So why is this term? Why does it imply that there's only unique invariant measure? Unique ergodic invariant measure here. Well, let phi x to r continuous. So by Birkhoff's ergodic theorem we have that 1 over n integral i equals 0 to n minus 1 phi composed with fi of x converges to the integral of phi d mu for almost every... So how do we complete the argument here? What is our assumption? What does star say? Remember star? Light. And so that's light, that's light. So phi bar is equal to integral d mu and 1 over n sum of i equals 0 to n minus 1 phi composed with fi of x converges to the integral of phi d mu for all x in x. For every x it converges to this. Uniformly. So what was the final observation? If it converges for every phi, sorry, for every point, for every point of time I have it just converged to this. Can there be another invariant measure? We're trying to show that there exists only one ergodic invariant measure. So we pick one ergodic invariant measure and we get this. Now suppose I pick another ergodic invariant measure. Then what? Then I would get exactly the same thing and I would get that for every point I converge to the integral with respect to a different invariant measure. But this convergence is unique, right? The convergence is unique. You cannot converge to do different things. So that would mean that for those two invariant measures the integral with respect to every phi coincides and therefore they have to be the same invariant measure. Okay? If mu tilde is another invariant ergodic measure, probability measure, then we have that 1 over n i equals 0 n minus 1 of phi composed with fi x converges to the integral of phi d mu tilde. For all x and so mu is equal to mu tilde. So this completes the proof of the fact that if you have uniform convergence for a dense set of functions then the map is uniquely ergodic. Notice that we've used Bielkoff's ergodic theorem in the proof here. Okay, so now we want to prove the converse. Suppose that f is uniquely ergodic. Then we have the uniform convergence for all continuous functions. So now lemma suppose is uniquely. So then for all phi continuous takes this phi bar such that star holds. So again by Bielkoff, we know that we have this convergence almost everywhere, right? So we know that 1 over n sum of phi composed with fi of x converges to the integral of phi d mu from mu x. So let phi bar equals integral of phi d mu. Okay, and we want to show that we have actual uniform convergence for all x. Okay, and we show. So how are we going to show that star holds? How are we going to use the unique ergodicity assumption? What do you think? What's the strategy of proof? Yeah, we've used Bielkoff's ergodic theorem to choose our constant, right? We want to show that star holds. So we want to show that for every phi there's a phi bar. So now by Bielkoff's ergodic theorem we know what this phi bar needs to be. If this is going to be true the only possibility for phi bar is this. Because we already know that this converges for almost every point. What we need to show is that this converges uniformly to this. Okay, so how are we going to prove that it's going to be uniformly using the unique ergodicity? So basically we're going to look at the convergence for other points and we're going to show that this would be a contradiction. If they converge to anything else then you must have another invariant measure in there. If we suppose we have a point here for which this uniform convergence does not occur we can show that this must converge to some other constant that is gives rise to another invariant measure. So let's try to do that. So what do we need to show? What's the negation of uniform continuity? So suppose by contradiction that star does not hold. So what's the negation of star? Okay, this is not completely. So then the negation is that there exists a continuous function. There exists a continuous function. There exists epsilon. There exists sequences xk in x and nk going to infinity such that 1 over n sum i equals 0, 1 over nk nk minus 1 of f i composed with f i of xk minus phi bar is greater than or equal to epsilon. So how are we going to get a contradiction from this? Well, if you remember how we put the existence of invariant measures we took sequences of this kind and we showed that any limit point is invariant any accumulation point of the sequence. So we're going to do something very similar. Now define a sequence of measures nu k is equal to the measure 1 over n sum i equals 0 to n minus 1 so this should be nk nk of f i star delta in xk which is just the same thing by the definition of this is equal to 1 over nk the sum i equals 0 so this should be nk always nk minus 1 of the delta Dirac in f i of xk So then we have for every k what do we have here? We have the integral of phi with respect to this measure so what are we trying to do here? So we're trying to contradict we will show that in fact this cannot happen so any kind of sequences any choice of points and sequences and so on because of the uniform convergence this cannot happen this is actually converging uniformly to phi bar this is what we're trying to show so by contradiction we suppose we find these sequences so this is for every k this is greater than equal to epsilon so we're going to construct these measures which are specifically built by looking at these times where these averages are far away from the average that you're supposed to have and therefore we're going to get a limit of these measures which cannot be the original measure because it's far away from this limit and therefore that's going to prove that there exists another measure so let's look at the average of phi with respect to this nu k by definition this is equal to the integral of phi we respect to this measure here 1 over nk sum i equals 0 over nk minus 1 of delta fi of x this is this measure and here I can write this as I can take the sum and the average out as I've done before 1 over nk the sum i equals 0 to nk minus 1 of the integral of phi with respect to the Dirac delta measures in fi of x xk thank you and this is just equal to 1 over nk the sum i equals 0 to nk minus 1 of again remember when you integrate with respect to Dirac delta that's just the value of the function at this point so this is just phi composed with fi of xk okay so by defining the measures in this way I get that the integral of phi with respect to the nu k is exactly this and this is exactly the bulk of average that we had here so combining with this we just get okay and so we have that integral of phi d nu k minus phi bar is less than equal to epsilon for all k okay so remember that this phi bar is the integral of phi with respect to d mu right so what are we going to do here we have a sequence of probability measures here we can take a limit point okay so by the weak star compactness of the spatial probability measures there's a limit point right so m m is compact so let nu in m such that kj converges to nu some subsequence kj tending to infinity then because of this uniform bound we have then we have the integral of phi d nu minus phi bar which is the same as integral of phi d nu minus integral of phi d mu greater than equal to epsilon okay in particular so we have the nu is different from nu and how do you know there's invariant so arguing exactly as in the proof arguing proof of the Krilov Bogoljubov Bogoljubov okay Bogoljubov theorem we have the nu is invariant okay but ergodic averages is f invariant but for all ergodic components nu ergodic averages ergodic converge to converge to the integral of phi which is different from phi bar and this is a contradiction well maybe it's like more simple I could simply say that it's a contradiction because nu is f invariant and it's different from mu from this and therefore we have contradiction to the assumption of uniqueness okay let me say that actually it's a little bit simpler the contradiction is that nu is f invariant and nu is different from okay contradicting okay so today has been quite a technical quite a technical day I think we will finish here and the next lecture we will apply this result to show that the circle rotation is uniquely ergodic and we will also prove a nice little application to number theory in ergodic theory there's quite a few applications to number theory and I will give you some examples as we go along okay so I think we can finish here for today thank you