I practice fashion design. That's why you look so spefi. Thank you. Are you shy? No, I'm not. But you look like such a cool guy to be friends with. Where can we get you on social media? I'm on Instagram. I'm on Facebook. I'm on Twitter. I'm on Twitter. What's up? That's my number. Just checking. Hi Kwee. What is your name? My name is Flora Linke. I'm a student at the University of Nero. You can say I'm a journalist so far, but I'm an upcoming journalist. I'm an upcoming journalist. See what I need to do. Have you both told her she looks nice? Can we just take a second and tell her she looks amazing? Thank you. Thank you. That's even better than what I asked for. I like it. Today we're talking about foolish relationships. What do you think of foolish relationships? According to me, a foolish relationship is when one partner is serious with the relationship and the other one jokes a lot. The other one puts all the commitment in it and the other one jokes with it. What would you take a foolish relationship? A foolish relationship or rather the foolish person in the relationship is so invested in the relationship that the other one says, let me use the guy as a burden here. He's not there. Some are just up to a short term relationship but the guy is really serious about it, thinking about children. You've already got a marriage. You've even named three of them. The guy doesn't even know your dad name. That's what I call him. He doesn't even know his dad name. What do you think of foolish relationships? A foolish relationship is something let me call it weird. Let me use an example. For example, if you're all 21 and start planning a honeymoon right now and yet we don't know about our future, that's like a foolish relationship. Really? Yes. You don't have any investments in your life. How should you plan for a honeymoon? When people are 35 years, don't even know what kind of honeymoon they want. You see? Can I tell you a secret? I know a friend of mine, we used to go to the same primary school together. He got married at 24 and his pride was 22 at the time. He was more focused at 24 than some guys are right now at 40. So I don't think really it has anything to do with age. Honestly. I was using that as an example but I'll say it's just a messed up relationship. I'm so interested. Why do you think it's messed up? Do you think youth and youth don't? It's okay youth and youth but let me use the term fat boys. Yes. From now on we're just going to use the first letter F and then we're going to most. You want to say something? Okay. So today it's all about you. Just use the hashtag why in the morning or man crush Monday. Just give us an example of what you think a foolish relationship is. You can get us at WhiteFive on Facebook WhiteFive channel on Twitter. My guy always wants money from me and when I don't have he threatens to boot me.  Me or him? That's a good question. Kwanzaia Mwisha is a good question. Who is the fool really here? Queen, let's start with you. So the guy asks for money from the girl. And when you don't have not when you've eh, so at the end I would be cut. Okay, so when you don't have he threatens to I think get physical with her. In my own opinion that guy is taking advantage of the girl or maybe the girl is too is in love with the guy and the guy thinks he's actually doing her a favor of being together with her. Ha, so she has to try to keep him. Ha, so he demands for everything like money that's it could be a manipulative relationship. Wow. What do you think? I think ladies like that lady there is very difficult to find. I wish I had one. Wow. Angusha CV mami. There's someone whom I take better care of you here. But I think the lady is very good but the guy is really taking advantage of it. They are going to the extent of threatening the lady to live when the lady doesn't send him the money that's being manipulative of the lady. I think if the lady has the money and the guy doesn't have he can help the guy to some extent but if he doesn't have it there's no need of threatening the lady. I think the lady is a great lady and that guy is really taking advantage she will look for someone better like me. Ha, ha, ha. Will you take advantage of your house? No, I won't. I won't take advantage of you. I will take advantage of it.  You want to ask for things. Not just for things. How would you advise this? My advice I will tell the girl to just step out and stop making commitments to what is not perfect because that is not perfect. But no one is perfect. But you won't expect if I was the girl and he was the man I wouldn't say to my relationship you are beating me up because of money. That wouldn't happen. So the good thing for it is just to walk away. You will find someone. Don't even slow clap for you. All right. I like that guys. Get with the program at WIJ-54 on Facebook at WIJ-54 channel on Twitter.  Do you think that is foolish? I don't know because if you are completely dependent on a woman you are just a sponsor. You are completely dependent. That is how we have grown.  The point of growing up was to try and make it out on our own. It's a thing now. Guys are okay with it now. The world is changing. Oh my gosh! Millenials in Tani Maliza. So it's okay? Maybe it was a situation where this guy got fired and then this girl consisting him. He has no other option. In the scenario when I did my business and came to work I said buy me things Love is not defined by money also material things and connections. Love that? You can never know! I like how are you? Bi idea kuko na huna nga, mtk ya kwa kia? Hachish angry kwa kwa kia ni, kwa kia? Refi kwa mtk ya ! Aka quèili kia, mtk ni kwa isti wiki na kwa ifwendonna isiwezwezu kwa jila kwa hini, kwa hinaji raha mtk ya kwa isiwezwezu kwa kia haji, kwa kia na agaka kwa isiweza kwa hina ni lakia kwa jikia kwa chika wiki ni kwa kia kwa huna nga kwa hii kwa hina ni kwa hina ni kwa iiti kwa hina ni kwa ipwendonna isiweza Most, uki fawatina you will find that that guy atuhio pesa kipiwa he doesn't do anything useful with it mibi anachuku atu ane ndaku liwa na maa gays, maa he takes other girls out That's an example of a man who is just there akuwapo to his wasting time for the girl Ya, ya. Ya, ya. Spend a lot more energy convincing girls to pay for them things than actually doing the job. I think they are good ladies and they are good men. Kuna, some relationships have seen that the lady steps in to help the guy when the guy likes something. And the relationship works well. The guy is not choosing the money for useless things. He extends the money well. Going apart with friends, such stuff. Apatoo, don't die. Chiza sayi. Won't we all dance? No, you nani pay ma. But when you are a lady is such a good lady capable of giving you support, like financial support with that, you shouldn't waste that money going to party. Kamuna tagaku party, just go and assault yourself and get the money. But if the lady is financially stable and is ready to help you, I think you should be more creative and look for a good way to invest the money instead of going and wasting the money. I think it should be understandable if the man doesn't ask for money every time and is not doing a project. For example, if maybe they want to start up something for the benefit of both of them, then he tells the girl, I'm thinking of getting an apartment and I have this amount of money. Maybe you could add up something. But it's not just coming every other time. It's jengi ma zengi. Na taka kwen na api na taka kwen. It doesn't make sense. Pocket money. You're crazy. Hi, Val. Mimini wini. Wiyu jamana se maheizi niyo aju ataki bibi mlevi. Wow. And it's him who taught me how to drink. Wow. He fooled me and I see him more foolish now. I want to go but not before I punish him. What's the worst thing I can do to him? I don't like punishing people. No baby, where much are you now? Okay, just leave, leave, leave. What's your advice? Ono ga nacheka. Yata kwen na jipanish mwenyewe. If you go to the... I think when someone adds you in one way or another, the best thing you should do is to forgive. Forgiveness has a great impact. As long as you forgive the person, you will also feel it. And you will have to go. The burden will go. But if you don't forgive someone mwenyewe kuwachi in one way or another, the pain will always be with you. The pain will always be with you. Kwanzao dendo umuze, the next person. Kwanzao dendo umuze, the next person. Okay, but now that we've talked about that forgiveness, papa kidogo kuwa ati. Okay guys, help me understand. Can I direct this to you? So you get into a relationship with someone. You found them maybe soba and then maybe we're Sunday away. We're not even married. We're not married. We're not married because we're not in a relationship. We're not married because we're not in a relationship. So when one of those... So now girl them and a penda and say, I really love him and I just want to spend some time with him. And this is how he spends his time. So I might as well just start drinking with him. A ya kwa na kunwa, a shanzo kunwa. Then suddenly you don't marry her because you cannot marry a drunk woman. You don't get it, do you? What are those? I will put them ati slikwin, slikwinivi. No, I will put them natural. You know, someone who doesn't do makeup at or short hair and our first lady and all this. Aha. Venakula on point and as a selfish studio yote and a joke picker. But you have a side check when you buy hair for $20,000 and you want your wife to be natural. What is all this, gentlemen? Help me understand. Let me like rephrase it. Okay, let me not rephrase it and let me say, you know, a man cannot be satisfied by one thing and siyati ni like, ni angibu kutumi yadim kama gini pig it's like you're trying to meet your perfect match. So you need to work. You need to like... Keep searching. I haven't seen people should like have side checks and side wards. No, but... Niliambia, they're called supporting agents. Yeah, they're supportive. Very supportive. Okay. So what do you want to say to your wife when she comes back? Annaweza. But... Annaweza. But in those situations, tenna. Uh-huh. Labda, this girl that he is with may be to say ni kichwangumu, very disturbing, not understanding. And then, you know, wale ma diem, very bossy and all that, the man won't be comfortable. Yes. And then Pia, if we get a bit intimate, wakuni mana, you won't even... So the man will like look for someone else who can help him better. There was this adaka phrase someone mentioned to me during the weekend at EO.  So waif sa agichfunguwa, sa if she doesn't keep herself proper, if she doesn't look good at a tembe atu because her man is to be satisfied. Okay. Ya crazy. What do you mean at the... First of all, umedi shape juya. Uchi ya kaza. Hai! Manzinin beguza kus me fajen kai, bi alafu, when they serve tembe nam tu mingineju. Why? Why are you disrespectful like that, guys? What is the problem? But there are some ladies who don't take good care of themselves. I'm a judge here. So... Suli se muna penda personaliti ya kia. Suli se mani ni perfect match. Suli se mani soulmate. You can love the personality but dem piya na faku take good care of herself. Siya di. Yame, usha wa dem, she comes to the house. Yame za wa toto, you have children. Siya na kwa she's dirty. Adi mumbainu ni chafu. She doesn't take good care of herself physically. Ame nona na plakulati. Hai! High point of not anima slikwins. Wow. Yes. You have a take against my case. I'm sure some of them even don't know how to cook. If you told them to cook for you, my friend, you'll eat something different from what you expect. Would it be a foolish relationship if you got into something and you assumed u dem mani za pika but I easy pika? It's it. I love it. I love it. Why is it foolish? Because... Okay, so why is it foolish? Kusukunze meleni oza je pika? Nachwa, kuna some girls use a very petty excuse. Like sisi kuitu. Sisi fwangi. I don't cook. Tuna piki wangwa like. Inlisku to konishi pika ako. Kibandaski? Yes. Futenda kibandaski. Kuna 24 hours. Ndakonishi. Okay. Hai Val. I'm Joki. Chaliangu. Anashindanga kini sho nani pelika this big, big moz. Hadisai nangoja but it's been two years now. Two years. It's not a foolish relationship. Atikwenda torifa zivi. Chapa-chapa labs juwap yuko. You know. Adi uudem doa. Akwa niya foolish relationship. So foolish relationship. Saki lasi kuna doa vabi atikwende. So why should the lady expect much from a guy when the guy can't do that? Nika mo boy that guy really loves the lady. Nani kip promises. Nani kip kumdanganya so that he can move the relationship further. I'm glad you said that because another gentleman told me wa stana tunapenda wangu. Like they want to tell us the truth but we get upset with the truth. We like being lied to. Do you think that's true? To say mou kwez. Let's be very honest. In that case me I'm thinking sometimes we are guys like you get in a relationship na fukia guy na fakum pelika mall na fakum pelika sedikost yourself unadua you can't afford that. Maybe the girl even isn't there for kupeliko api ama api it's just a mutual thing.  And I'm sure that guy takes other girls to your two rivers and I want to be her. We are equal social media. We are equal social media. Oh my gosh. Is it a foolish relationship when dey manashin doa kikdanganga like she can't see she's being lied to and the guy is lying very convincingly but dey manashin doa na danganga but they just continue like assuming like nothing has happened. Is that a foolish relationship? That's a foolish relationship. But there's so many insecurities in there you know you know oh my gosh all my friends have boyfriends you know that's the problem with us girls to go so soft such that unapenda umtuna friya mmukotu and I'm sure guys aizu kwa kwa namtumu just someone once told me men are only searching such a kwa ume weza he's still sad. I think if you go to if you enter relationship with that mentality don't go far even if you know even if you know that just keep the mentality that my guy only love ask me alone at least you'll move the relationship further okay oku kwa apu since dey manashin doa na nimi me peki hangu at least search elsewhere nisi jwe now I'm born of search period who are still in search now I'm serious because we're out here killing us first of all as tana kwa as tana na gumbana leave my man alone just kwa ume wengi I can't allow men to appear atasivu bide kwa gii ati tit for tat it's not at your game like that it's like oh you want to see other people okay go see other people now me the people who want to see me let me see them also it's very simple kwa tana kila mstana akuna line ya wanome you've seen young eme got pregnant and she's the tomboy of all tomboys wanome ni wengi so stop killing yourselves first of all and stop killing other girls it's going to be alright hi val you'll be here my girl ni so cute but nimi jingatu niri na tu na tu rali kwa hi ya ya ya akodaw nali pa tiaishu le so so juni fuli sit ko fuli relationship let me tell you this is funny in a way but wanome I think you should date people either at your IQ level or a bit higher for example juzi but last year there was a situation was watua thika wali chimba sujui nini tunnel wakaiba kwa bank and it would have been perfect until he uncle aliyenda ka pelikia shore pesa shore aka piga selfie aka post facebook how now and that's how they were caught you were there with a solid business plan made it lady ruined it atiyo kwa tu katupigapicha na why now such robbery scene in movies nali kwa tu iho do you agree that you need to but pia na nyim na penda kuches na akili zahos tana so she can't be too smart ona kia kia if you get a girl who can control you hi what is this controlling aspect because men are very afraid of us wio usiponi pia pesa minkafugonga baby hiya I want to quote it directly from the Bible it is written in the book of Genesis 3 16 that through this women will bear will have will experience pain while giving birth and they will always desire to they will always desire their husbands according to me the desire means the ladies will always want to control their men but the men will always be on top of them which is written in the Bible so however how much the ladies good, sweet you will always decide one point or another to be on top of the man but the guy will always be on top so it is it is something that has always been there from the beginning of time literally from the time of Adam that is there after this particular time he just wants to control men but men will always be on top there is this guy I met over the weekend he was very vocal to have conversation and he will be content so Ati he tells me in a case where let's say your lady gets physical with you she is very aggressive he is more afraid of where we kumkali and belia ma beshtizake than you as a person do you understand? if you are in a social setting boys will have a good time and watch a ball and they will say I want what maybe I will say if you are in a social setting now the next day when he comes without you to his boys he will say he is more afraid of just that among men than you as a person why now? because it is a foolish relation and then he will say I want to be on top and then I will be like in that reaction he will say I want to be on top I want to be on top so it becomes humiliating disturbing too we had a conversation about battered men like men who are being and he will say I want to be on top they just go ham you will say stop getting so angry you need to fix yourself and then the second thing is that's a foolish relationship because why are you first of all being physical why are you attacking your man and why are you allowing yourself to be attacked because it's a horrible cycle what do you think is happening here? first of all I don't know how to control how to control but at the same time I'm not saying that the lady how to deal with her boyfriend but as a girl always understand let the man be the man in that relationship yesterday we had issues in the house but outside they are a man's ego and I'm very in a Ate na za kwata jeyo outside they just behave like a person with understanding like a sophisticated person so kwa iti kwa kwa kwa kwa kwa kwa kwa kwa kwa kwa kwa kwa kwa kwa kwa kwa kwa kwa kwa kwa kwa kwa kwa kwa yuki j Indians Telikwaj team mali tuo mendo kwa kwa w extra kwa kwa kwa yuki yaal kiwa wele, but so when you insist kukaku wa pasenda as if you paid for the mafuta of a deka ya bimpa kaili wuna jam yaw pweli like guy in CND like more on a couple, like you know why I want to sit in front what's that all about? boys how do you interpret that? is that a foolish situation chip? I think I'm a potato maker now my friend there my friend in that seat and the lady's mentality that seat is preserved for her alone and sometimes that's waving foolishly that lady is foolish I think sometimes they are trying to they know girls try to incite each other let me as an example of what you have given like a girl would go and tell her gang that you know what I cannot be with my boy and then the friend comes in seat actually as the god driver so you see they go inciting each other and then we say her and we are together and I'm with him and then no there's no way there's no way there's no way there's no way there's no way there's no way there's no way there's no way my girl Anna Sema had shita fanya paka wedding night but I was sad and she didn't even want to go I know I am the fool here so number many chanoe me me atakumi piya kwa ohhhh me habibi liya I've never seen anyone sitting somewhere with soil so that's what is happening in your issue Una pika kwen tu mingi nga na kula ma say So stop being fun Jumping to another religion Go fly Hi, what do you think I'm sorry? It's not nice Nkwa niya tapia mingi ngekwa hii nge kwa 2 chesana 2 Let's play this game till the end Nkwa niya wedi nihitu tafikanda uskopa But the guy is the fool here Finally The guy is the fool here because Una mwiwaji, una eka til marriage alafu Yoself kumuna joa na pianafu ngini Why don't you confront her ni niya both that Umulizu Ah mami bida gali yangali aka onapa Gemi nezakwa chingi Oh It's all making sense now It's all to see someone in a way Iri na ways to tell Apa ni ka tata Oh my gosh I'm so sorry Kamao Let's go and be alright Let's perhaps wind this up Your last words About foolish relationships I would like to say If someone notices something strange in a relationship No matter how strange it is Just don't hurt yourself Just walk away Because there are perfect people And good people around us We just need time to connect with them If you're not dating, it's like an issue You come out not having a house Not having food So it's a need So it's high time people need to style up And know the right time will come You don't need to hurt yourself because of someone or something I want to finish with you My last remark I think when you want to enter a relationship Relationship is not a joking thing It's not for kids It's a joke Are you serious? You tend to the perspective of If you let your man go to get a job You didn't need someone to solve it And pay the bill So if your partner is not serious So walk away You can find good people somewhere Someone such is not a try to joke with It's a serious thing That's why many people get depressed And end up killing themselves So if you want to enter a relationship Please be serious It's not something to joke with It was there from the time of Kenabra And they were serious with Sarah Without 100 years 400 years without acting So who are you? So if you are entering a relationship If you are a lady But they want to be serious So be serious with your man And make him want him to be Not that you are looking up to Men should be more serious No, even ladies should be serious If you want to be serious with them The next time you take opportunity That relationship will be perfect If you want to have lunch Then you can buy dinner Say it to me I'll help you I'll help you It's 50-50 If you want to be serious Then you can buy Dempia and go to shop Or you can buy a kwa So you can buy So if you want to stop Then you can buy gas If you want to be serious Then you can buy Then my guy is an engineer He will support me If you are an engineer Then you can buy Then you look for a lady To support you in your relationship And do not enter a relationship Those ladies will enter with that perspective Okay last one I would say first Before you get in a relationship You get to know each other I love to- Ino ni o kanako. Are we fakumet- Are we in a serious relationship? Are we in a situationship? Are we in a monthly relationship? Is it a serious one that Not needed to swear on the word? I love to- Nowadays relationships this make me about money We didn't pay a lot of money, but then you can start from somewhere, finally, if you end up together, you'll be happy, palimli tokana, where you've gotten. That's very deep, okay. I think everyone has their own, their separate growth pattern. You need to know the person you're with, how they're growing. That scenario where there's a very conservative girl and maybe a bit of a not shy guy, yeah, they meet.  But yeah, na jua, growth ya ke means, ayende hanye, choke, de arudi apa. Where meanwhile na fkirem na grupa moja. No, know how your person is growing. Nde oneeka, mta kum na pata no kumbele, ama mta kum ki enda evi. Just be aware of your person. Nda kuchuna, but okay, very quickly. Now these guys are entering a relationship kutade moja me, sayma, this one is wife material, this one is girlfriend material, codes, what made. Nda wachuna, mkisa na wacha.  Tafadali, someone maybe is using you as that term and someone is using you as a wife material. But even guys don't know that those ones that are their wife materials are also girlfriend materials to someone else. So you should be serious with one person. And ladies also be serious with one guy.  Ogapa wa stana, misawa. Namchona na nenele anga. Okay, let's kuna mtuna kumga. There's a meme for Ate Walami, be like, oh I miss Kevin, oh my gosh. Mke na ako, kuna mtu tuapa na ni sumwa, na najo atu na miss, lakini saatu. So she's covered like five guys in that kuna John Tez, kuna K-4. Nga jata jamtu, I want to say I like that spade, but it's not nice girls, okay. Ate Walami on Facebook, Ate Walami on Twitter. Hashtag of the day, hashtags of the day. Monday health, man crush Monday and a white in the morning. Thank you so very much for staying with us from 7 to 10 in the morning. Kulikwana badi din kiyasu jikuna kaji sa'ii. But I think you should have an amazing day nonetheless. Should we wish them an amazing week? Just smile away.


Okay. So today we begin by completing our study of the space of invariant and ergodic measures. So again, let me recall we're talking about a compact metric space, the Borel sigma algebra. We have a map F from M to M continues. We have the space M of probability measures, of Borel probability measures on the space with the weak start topology. This is a compact set. We have M F to be the set of mu of F invariant probability measures which we proved in the last lecture is also compact and convex and non-empty if F is continuous. And then we have the space, the set of ergodic measures, mu in M such that mu is F invariant and ergodic. And today we will prove that this set is non-empty and more specifically that mu is ergodic if and only if mu is extremal point. So it is curious that mu has this kind of, if you remember the definition of ergodicity is purely dynamical. There's nothing functional analytic. But it turns out that this dynamical definition has a correspondence meaning in the functional analytic sense in the space of invariant measures. So remember that an extremal point is a point that cannot be written as a linear combination of two or the two or the different points in the space in the set. So one direction, both of the directions are non-trivial. Suppose first that mu is ergodic. Suppose mu is ergodic. So we will show that it is not an extremal point. In other words, we will show that we can write it as a linear combination of two other invariant measures. So by ergodicity, sorry, sorry, I'm sorry, I'm going to go the other way. Suppose first that mu is not ergodic, sorry. We're going to show that it is not an extremal point. That's what I, so if it's not ergodic, what's the definition of not ergodic? It means that there exists a set A and M such that F minus 1 inverse of A equals A, F minus 1 of the complement of A equals the complement. So this is actually always true. If you have a set that satisfies this, you can do a little exercise and you can see that clearly the complement satisfies this too. And the measure of A is strictly between 0 and 1. So we're going to use this set. So what we have is our set M, okay. And then we have two sets, A and A complement, which are fully invariant by the dynamics, which means that all the pre-images of A stay in A. In particular, it means also all the images of A stay in A. You can check that also, right. This is a stronger notion than being forward invariant. So A stays in A in forward and backward time, AC stays in AC in forward and backward time. They both have positive measure. And now we can define two sets, two measures. Then probability measures mu 1 and mu 2 by mu 1 of B is equal to mu of B intersect A over mu of A and mu 2 of B equals mu B intersect A complement. Understand the definition of these measures? So we're basically taking mu 1 is the measure mu concentrated on A and normalized. So of course, if you take mu 1 of all of A, it will have measure 1. Mu 2 of AC will have measure 1. So mu 1 lives completely on A and mu 2 lives completely on the complement. And you just take the proportion of A gives the measure mu 1, mu 2. So it's easy to see that mu 1 and mu 2 are probability measures. And notice that we can write mu as a linear combination. So we can write mu is equal to mu of A mu 1 plus mu A complement mu 2, which is the same as writing T mu 1 plus 1 minus T mu 2 where T is equal to the measure of A, which is strictly between 0 and 1. So what does this mean? This means that if you take a set B, the measure of B is given by the measure of A times mu 1 because mu 1 is just, so if you want to check this, measure mu of B is equal to mu of A times mu of B intersection A over AC. AC, right? So this cancels and this gives mu of B intersection A plus mu of B intersection AC, which is clearly equal to the measure of B. This is the way I should check it. Simple calculation. So I've shown that it's a linear combination of two probability measures. Am I done? Is there anything else I need to check here? So I've assumed that mu is not a gothic and I want to show that mu is not an extremal point of mu F, right? So in other words, I'm proving it in this direction. I'm saying if it's an extremal point, it's a gothic. And to prove that I assume it's not a gothic and show it's not an extremal point. So I assume it's not a gothic and I write it as a linear combination of two probability measures. Am I done? No? Well, no, no, you just need to prove for one T that it can be written as a linear combination for the T strictly between 0 and 1. This is fine. This is a linear combination. So I've shown that it's not extremal. Extremal means that if you write it like this, either T equals 0 or T equals 1, okay? So what else am I missing? Mezza. What am I missing here? What else do I need to prove for this part? Sorry? I didn't hear the question. Okay. The question is what else do I need to prove? I assume that mu is not a gothic and I want to prove that mu is not an extremal point. So I need to show that it's a linear combination of two probability measures in MF. So I construct these measures and I show that mu is a linear combination. Am I done? Yes? Yes? Are measures mu1 and mu2, they are the right measures? Sorry? They are probability measures. Is this the special probability measures? Ah, so what else do I need to show? That they are invariant. I need to show that mu1 and mu2 are invariant, okay? It's not enough to show that this is a linear combination of two measures because that is easy. Remember, MF is a subset of the space of all probability measures, right? MF is just the invariant measures, okay? Remember that we have EF is contained in MF, which is contained in M. So it could be much easier to show that this is a linear combination of two measures here, even if it's an extremal point of this, right? So we need to show that it's a linear combination of two distinct measures here. So we need to show that mu1 and mu2 are invariant. So that's not difficult. We need to show it. So it remains to show let B in M be a polar measurable set and then what does it invariant mean? So we want to show that mu of F minus 1 of B is equal to mu1 of B. So we just check it. So how do you check something like this? Well, let's start using the definition of mu1. So the definition of mu1 is that this is mu of F minus 1 of B intersected A over mu of A. Very good. Very good. So now we can write this as mu of F minus 1 of B intersection F minus 1 of A. We needed to use some property of A, otherwise we couldn't do this. So mu of A because, right? So because F minus 1 of A equals A by the construction, by the definition of the set A. And now what? So F minus 1 of B intersection F minus 1 of B is the same as F minus 1 of B intersection. So we can write this as mu of F minus 1 of B intersection A over mu of A. Mu is invariant, exactly. So mu might not be ergodic, but it's invariant by definition. So by invariant, this is equal to mu of B intersection A over mu of A. And this is just equal by definition to mu of B. So similarly for mu2, and so mu1, mu2 are in MF. And so if mu is an extremal point, so if mu is an extremal point. So we've proved the first part, first one direction of these implications. Okay, any questions? So there's a couple of non-trivial steps here. The first one is here in which we used the definition of the set A, which goes into the definition of mu1. And the second one is here where we use the fact that mu is invariant, even though it's not ergodic. Okay, so now we need to prove, yes. Yes. Yes, thank you. Yes. This is the definition of mu1 of B, yes, thank you. Okay, now we want the other direction. We assume that it's ergodic, and we show that we cannot write it as a linear combination of two invariant measures. So now suppose mu is ergodic. And suppose by contradiction that mu is not extremal. Suppose by contradiction that mu is equal, that there exists mu1 and mu2 in Mf, such that mu1 is equal to T mu is equal to T mu1 plus 1 minus T mu2 for some T between 0 and 1. Okay, so then we will show that this implies, okay, it's not exactly by contradiction, sorry. We will show that if it's of this form, then this implies mu1 equals mu2 equals mu. So kind of trivial situation. We will show that this means the only way this can happen is if mu1 equals mu and mu2 equals mu, and then this is a trivial identity. We cannot do it for different measures mu1 equal to different mu2. Okay, so how do we do this? Okay, so this is a not completely trivial argument. So notice first of all, so we will do it for mu1. We will show that mu1 equals mu, okay? The proof that mu2 equals mu is the same. And in fact, it's anyway it's an immediate consequence once we prove this. So notice mu1 is absolutely continuous with respect to mu. What does this mean? Remember what absolutely continuous means? Yeah, so this means that if mu of B equals 0 implies that mu1 of B equals 0. And this follows immediately from this, right? Because this is just a linear combination. So clearly, if the measure mu is equal to 0, then you must have mu1 equals 0, right? And sorry, if mu of A is equal to 0, then you must have both of these must be equal to 0. So remember what we said about absolutely continuous in various measures. That absolutely continuous in various measures always have a density, right? This is a theorem called the Radon-Nickerdim theorem, a measure theory. So by Radon-Nickerdim theorem, Radon-Nickerdim theorem takes this mu1 has a density with respect to mu. Let's write it as h1. Sometimes it is written like this, d mu1 d mu. Such that the measure such that for all measurable sets B, we have that mu1 of B can be written exactly as the integral of B of h1 d mu. This is the Radon-Nickerdim theorem. So this is for every set B. So the statement that mu1 is equal to mu, this is true if and only if this density is equal to 1 almost everywhere with respect to mu. So mu1 is equal to mu if and only if h1 equals 1 mu almost everywhere. Because remember again this integral is just the graph under the curve, right? And so to say that this measure that you get is the same as mu, if this is the graph, this is our set M, this is the density h, right? If this density h is 1 almost everywhere, then the area under the graph, if you take a set here of a certain size, the area under the graph will be exactly the same measure as that set when you enter it. So this is what we want to show. This is enough to show that this density is 1 almost everywhere, mu almost everywhere. h1 equals 1 mu almost everywhere, which means there's a set of full, the only points where it's not equal to 1 will have mu measures 0, which means it does not affect the integral because it's a set of mu measures 0. Okay, so define equal to the set of points where h1 of x is less than 1 and c is equal to x where h of x is greater than 1. How are we going to show that h1 is equal to 1 mu almost everywhere? We're going to show that both of these sets have mu measures 0. And that will imply that it's equal to 1 almost everywhere. So I'm going to show it for b because the proof for c is exactly the same. So how do we show it for b? So we will show the mu of b equals 0 and mu of c equals 0. And this implies what we want to show. Okay, so how are we going to show this? Well, first of all, notice that by definition mu 1 of b is equal to the integral in b of h1 d mu. Okay, this is just the expression we have there. But now I'm going to split this set b in two ways.  Okay, it's not immediately obvious where we're going now, but just check what I've done is I've split b into two disjoint parts. One is the part that is the intersection with f minus 1 of b and one is the part that is outside f minus 1 of b. A priori at the moment we don't know anything about b or f minus 1 of b or anything. We don't know the structure of any of these sets. But we know these are well defined and we know these are disjoint by definition and therefore this integral is equal to the sum of these integrals. And we do the same thing for mu f minus 1 of b. We write this as f minus 1 of b h1 d mu. And we write this, sorry, mu 1, thank you, mu 1.  Okay, so I want to write it slightly differently. This is f, it's the same here, but differently.  So I just write it slightly differently. And what do we know here? What do we know about these two measures? Mu 1 is invariant, so they're the same. Okay, so we know that these are the same. So since mu 1 is f invariant, we have that mu 1 of b is equal to mu 1 of f minus 1 of b. And so what do we know about this? That this sum is equal to this sum. But we also know that this is the same as this because this is exactly the same set here, even though I write it slightly differently. These are not the same set, but these are. But so this means, so we want to know whether these are the same sets or not. And in fact, this implies, so therefore we have that integral of b over f minus 1 of b, h1 d mu, is equal to the integral over f minus 1 of b, b, h1 d mu. So notice that the measure of these two sets, so the measure, the mu measure of b of f minus 1 of b minus b is equal to the mu measure of f minus 1 of b minus the mu measure of f minus 1 of b intersection b. Okay, just a little bit of splitting up the sets in slightly different ways. And this, now I use the invariance of mu and this is equal to mu of b minus mu of f minus 1 of b intersection b. And this is the same as mu b minus f minus 1. So the measures of these two sets are the same. They have the same measure. So we are integrating the same function on two sets that in principle are different, but they have the same measure. What is the value of h1 on each of these sets? What is the value of h1 on this set? What do I know about the value of h1 on this set? It's less than 1. What do I know about the value of h1 on this set? It's greater than 1, right? Because this set is inside b, this set is outside b. So here I'm integrating a function on a set where the function is everywhere less than 1. So how much is this integral going to be? Less than 1. It's everywhere less than 1. So this would be less than 1, less than the measure of this set, right? And what about this integral here? It would be greater than the measure of this set. But the measure of these sets is the same. So contradiction. So let me write this down. So by definition of b, we have that the integral b f minus 1 of b h1 d mu will be strictly less than the measure of b minus f minus 1 of b.  This is a contradiction. What does it contradict? Sorry? Well, that's right. What does it contradict? What is it? Exactly. Exactly. It contradicts the fact that they have non-zero measure. It would be a contradiction if they had positive measure. Therefore they have zero measure, right? But this would lead to a contradiction if the measure was positive.  And what does it mean? This means that b, this does not mean by itself that b has zero measure. It just means that the difference between these two has zero measure. The difference between these two has zero measures. So this means that f minus 1 of b equals b up to a set of zero measure. So even though it does not mean exactly that f minus b equals b, it means that the difference between these two has zero measure, which is what we just showed. So by ergodicity, so I know that when we defined ergodicity, we said that if the map is ergodic, it means that any set satisfying this has either measure zero or one. Remember, this is the definition of ergodicity. It turns out that this is simple to see that in fact this is true, even if this is true only up to set of zero measure because sets of zero measures don't really contribute anything. So by ergodicity of mu, we have that mu of b equals 1 or mu of b equals 0. So our conclusion of all this is that either b has zero measure or b has full measure by doing this calculation. So what do we get? So assume by contradiction that mu of b equals 1 and then we get 1 which is equal to mu 1 of the whole space m is equal to the integral over m of h1 d mu. And this is equal to the integral of b over b h1 d mu and this is less than mu of b which is also equal to 1. So let's make sure we understand all the terms in this equality. So we're assuming that mu of b is equal to 1. Why do we have this strict inequality here? Because on b h1 is strictly less than 1. So we have this strict inequality. Why do we have this equality here? That's right because we're assuming that mu of b is equal to 1. So integrating on b or integrating on the whole space is the same because you're integrating over set of full measure which is the only thing that counts. And then we have this just by definition of h1 mu 1 is just this integral and mu 1 is a probability measure. So the full measure is equal to 1. Full measure of the whole space is equal to 1. So mu b equals 0. Similarly, we show that mu of c is equal to 0. So h1 is equal to 1 almost everywhere which is what we needed to prove. Okay. Let's I think this is a good place to take a couple of minutes break. We've completed the part on the space of invariant and got it measures. So let's just take a couple of minutes. Okay. So this completes the first results about the structure of the invariant and got it measures within the space of probability measures. Of course, this is not the reason we're studying them. We're studying them because we're studying that dynamics, the motive, the definitions of invariant and got it because this has some dynamical implications. So this is what we're going to start now. What are the dynamical implications and why are we interested in invariant and got it measures? So probably the first interesting and significant result using the notion of invariant measure is a theorem called the Poincare recurrence theorem which was proved way back in the 1890s. So it's a really classical theorem, but it is still used very much in dynamical system. It's quite an amazing theorem in what it says about the importance of invariant measures. So as usual, so this is really very general. It doesn't even need to have continuous, right? So we just have M is some measure space is some measurable function. So by measure space, I mean a space that has some sigma algebra and a probability measure and mu is invariant with respect to F. Then for every A measurable, of course, with mu A positive measure for every set of positive measure, mu almost every x in A returns to A for some iterative, iterative, for some positive iterative. Absolutely, of course, yes. It can be arbitrarily large for different x. So all this says, this is our space M, this is our set A, and it says that almost every point after some time comes back to A. So the reason why this theorem is really remarkable, apart from the fact that actually the proof is very simple, it's just a few lines, is that it seems to have almost no assumptions and the conclusions are fairly abstract, but still they're quite remarkable because as you know, there are many systems that have no recurrence, right? Can you give me an example of a system in which this does not happen, in which you have sets without probability measures, just of some dynamical system in which points never come back? Yeah, if you have the real line and you just take x plus n and you move to infinity, or if you have just a fixed point and everything is converging to a fixed point, or there's many cases in which things don't come back, okay? But in those cases, you cannot have an invariant measure sitting on these regions of points that never come back. This is what this theorem says. So invariant measures is intrinsically related to recurrence. And recurrence is very interesting because that's where interesting dynamics happens, is when things come back all the time, like in irrational rotations or 2x mod 1. The interesting dynamics is when things come back. If everything just goes somewhere and never comes back, there's nothing interesting is happening in that region, right? So without really any assumptions, saying just the existence of an invariant measure means that things come back. This point might come back after 5 units, another point after 10, another point after the million. It could be unbounded the return time that points need to come back. So let me first make a comment that in fact a slightly stronger version of this theorem is that almost every point comes back infinitely often. The proof though is a little bit more complicated and I will skip it. There's not an immediate consequence of this because the fact that almost every point comes back does not mean that you can then apply it again because the image of all these points could have zero measure. So it could be that if you take a full measure set in A and you look at all the points that come back, where they come back, they might come back instead of zero measure of points. So they might never come back the second time. So the fact that they come back infinitely often is not a direct consequence of this. So you need a few more lines of proofers and you need to generalize the proof a little bit and you get this infinitely often. If you want you can try to do this as an exercise but I just wanted to remark that it's not an immediate consequence but it's true. So we will just show that it comes back once. So let's define a set A. Sorry, I've got my set A. So let me define A0 is equal to the set of points x in A such that fn of x does not belong to A for all n greater than or equal to 1. So what do we need to show about the set A0? Yes. Because this is the set of points that never comes back. So if we show the complement of this is the points that do come back. So if we show that this has measure 0 its complement inside A will have measure 1. We'll have full measure in A. Almost every point in A will come back. So we will show that the measure of A0 is equal to 0 and then we will have proved that theorem. So it is sufficient to show the mu of A0. For all n greater than or equal to 0 let an equals f minus n of A0. So they didn't hear you. Yes? Yes. Almost every point of A. So mu almost every x in A. So the set full measure set inside A. So whatever the measure of A is like full measure set of A. If mu A is positive, yes. For some positive iterate we look at the future. Yes, non-zero in particular, yes. So what is this? This is just the pre-image. So these are just the points that will land in A0. Now what are we going to do with this? Well, we claim that all these pre-images are disjoint. A n intersection A m is empty for all n m greater than or equal to 0 and n different from m. And why is that? How do we show that they are disjoint? Well, suppose by contradiction, suppose there exists some x in A n intersection A m. Then we would have f n of x would belong to f n of A n intersection A m. If x belongs to A n intersection m, I can just take the nth iterate. This means that f n x obviously belongs to the nth iterate of A n intersection A m. And I can write this by definition of A n and A m. This is f n of f minus n of A n intersection f minus n of A 0. And of course f n of f minus n of A 0 is just equal to A 0. And this is just f of n minus n A 0. So we suppose, I can suppose if these are different, I suppose that n is bigger than n. Just without loss of generality. So this means this is some positive iterate of A 0. Intersection A 0 contains a point so it's not empty. So A 0 intersection f n minus n of A 0 is not empty. And what's the problem with this? Exactly. This contradicts the definition of A 0. Contradicting. So what have we just proved? What have we just proved? Yeah, with this contradiction, what have I just shown? What has this proved? Disjointness. So we've shown this. So what are we going to do with this disjointness now? If they're disjoint, we can take the union of them. So all the sets of the form A n and greater than or equal to 0 are pairwise disjoint. So what do you want to do as soon as you get a set of sets that is pairwise disjoint? We take the measure of all of them and take the union. So let's take the union A n and let's take the measure of the union. And because they're pairwise disjoint, this is just equal to the sum of the measures. So each A n, this is equal to the sum of the measure of f minus n of A 0. And what is this measure equal to? Exactly. The measure mu is invariant. So this is equal to the sum n greater than or equal to 0 of mu of A 0. And how much is this measure? Exactly. So this is either 0 or infinity. If mu of A 0 is 0, this is 0. If mu of A 0 is positive, this is infinity. And what's the problem with it being infinity? It cannot be infinity because those are disjoint and this is equal to this and this is of course less than equal to 1. Because this is some set in our space and this measures less than equal to 1. So mu of A 0 equals 0. And this completes the proof of the Poincare recurrence theorem. So this is the first result about invariant measures, using invariant measures. And this is a basic result but it's still used very much. There are many situations in mathematics where you use the Poincare recurrence theorem to show that points come back close to each other. Because remember that you can take a very small set of positive measures. It means what you know is that if you take a really small set as long as it's positive measure, you know that points must come back to this set all the time. Okay, this is invariance. What about ergodicity? So that's enough proofs for today. But before we finish, let me state the fundamental theorem which we will not actually prove. But it is very important. So about 20 or 30 years, yes? Excuse me? Yes. A n? The last sentence? Greater than equal to 1. Okay, I'm sorry, maybe that's true. Why do you want n greater than equal to 1? A 0 is also disjoint. Everything should work with A 0 also. I'm sorry, I shouldn't have erased first, but I don't think it makes much difference. Okay, so there is a theorem that strengthens Poincare recurrence in a dramatic way. Much, much stronger and much, much more interesting. And this is where the notion of ergodicity comes in. So we want a much more precise description in terms of the invariant measure. So let me remind you that if mu is an m, so now we still have a general, so okay, we still can work very generally in measure spaces. Measure space, f, m to m, measurable, okay, mu. So for any probability measure, not even in the very measure, remember we define the basin of attraction of this measure is equal to the set of points x such that 1 over n sum phi composed with f i of x i equals 0 to n minus 1 converges to the interval of phi d mu for all phi. So remember we discussed the first couple of lectures. We discussed the fact that often that we're interested in whether this sequence converges or not. Okay, and if it converges, so this is the same as taking the sum of the Dirac deltas. Okay, the convergence in the weak start-upology to mu of the sum of the Dirac deltas in the iterates of x. So we're interested in whether this converges and to what measure it converges. Here we're given a measure and we want to know which points actually this time average is converged to the space average of phi because what this says is that in some sense the measure mu describes in space the weighting of the measure mu describes this orbit of the point x. Okay, so in principle there's no particular reason for this b mu to be non-empty. So the theorem we have is the following. Okay, so sorry to define the theorem the way I want to define it I will assume that this is actually a metric space. So it's just a little bit stronger. There's a technical reason for which this needs to be a metric space. So theorem, I will define it a kind of short version of this theorem. This is the so-called Birkhoff's Ergodic theorem. Birkhoff's Ergodic theorem and it was proved in the 1920s and it says let mu be an half invariant ergodic probability measure then the mu measure of b mu is equal to 1. So I try to emphasize at the beginning how this convergence was not something that you could take for granted and in general there's no reason why these time averages should converge to some measure. What this theorem says is that if you have an ergodic measure then almost every point with respect to the measure satisfies this convergence. So almost every point satisfies this. So once you have an f-invariant ergodic measures there exists a set of full measure whose dynamics is precisely described by that measure. So they all have in some way as we shall see only in some way the same dynamics, statistically the same dynamics because this is a limit in which you say that asymptotically the distribution of this orbit corresponds to this measure but it doesn't mean that these orbits stay together and do it all at the same time. You might have all these orbits they might do completely different things they might be in different places at the same time but on average they all do the same thing. Just like again to use the example of tossing a coin if I toss a coin and you toss a coin the heads and tails will come up at completely different moments for the two but statistically when we both toss them for a million times almost half of them will be heads and almost half of them will be tails. Exactly the same thing. So the dynamics of the orbit could be completely different but the distribution in space asymptotically is the same for mu almost every point. So in fact just to get a better feeling let me say a direct corollary of this. Corollary is that under the same assumptions for all measurable A in M the limit 1 over N of the proportion J between so what does this mean? You take a set A, you take an arbitrary point X and you look at how often so you take an N what does this mean? This is a set, a set of what? This is a set of indexes J for which FJ of X is in A and you take the number the cardinality of the set so you just count how many times you fall in A. If I'm tossing a coin I count how many times I get heads for example because I'm taking N iterations the maximum cardinality is N and the minimum cardinality is 0 of course and then I divide by N so what I have here is a number between 0 and 1 because I have always something between 0 and N here divide by N I get something between 0 and 1 so as N increases I get a different amount of time so I look at the iterate sometimes it falls in A then it comes out and then it falls in and then it comes out and then it falls in and I'm looking at the amount of time that it spends inside the set A asymptotically the average amount of time that it spends inside the set A so what this says is that for every measurable A this is true and I need to add one more thing for mu almost every X not for every X of course so the proof is just an immediate consequence of this so how do I prove this? what I know is that almost every point for almost every point I have this convergence this is Birkhoff's ergodic theory and how does this convergence relate to that? it's very easy change phi to the characteristic function exactly so let phi equals the characteristic function of A then by Birkhoff's ergodic theorem for almost every X we have that one we have that this 1 over N cardinality in 1 j that fj x belongs to A is exactly equal to 1 over N sum from so I indexed them slightly differently here so I'm sorry maybe I should have just written 0 to N minus 1 to make it more consistent but it's exactly the same thing of course 0 to N minus 1 is equal to the characteristic function of A sum from I equals 0 to N minus 1 composed with f i of X so this is just exactly a different way of writing this so here I'm just counting I look at f i of X this characteristic function is either 0 or 1 depending on whether f i of X is in A so if it's in A this is 1 so all I'm doing here is exactly counting the number of it as where I belong to A and dividing by N and by Birkhoff's ergodic theorem this converges exactly to the integral of the characteristic function of A d mu and this characteristic function of A is equal so in fact most of the time this is really the way you think about Birkhoff's ergodic theorem it's more general because this convergence happens for all continuous functions in fact so this is what the price I pay for not stating Birkhoff's ergodic theorem properly is that of course this characteristic function is not a continuous function but Birkhoff's ergodic theorem says that actually this convergence happens for all measurable functions so I'm sorry I wanted to state a kind of more compact version of Birkhoff's ergodic theorem so let me just leave it at that so let me tell you that Birkhoff's ergodic theorem actually says that for almost every point this convergence happens for every measurable function so it's more general than what I wrote okay I thought that that would be sufficient for what we want to do but of course I forgot that I want to use this application and this is not a continuous function so it works it doesn't work okay so for any measurable F yes there's no it's a completely measure theoretic theorem it's a theorem that works in measure spaces for measurable maps with invariant measures and that's why it actually works for measure L1 functions not just not necessarily continuous functions okay so there was some reason I wanted to state it like that because it was cleaner to state so the key point is that here is that this limit does not depend on the point X of course right this is this is one of the key features of this result is that in general the orbit what I was saying before I'm just going to repeat it if you start with a point X or a point Y they might have very different dynamics they might go in very different places at very different times but if you look at the statistics of the orbit in other words you only look at something like this the frequency of time which you're going to A then almost every point has the same frequency of visits to A you can in some sense think of this as the probability of A and this is exactly where the connection between probability that's why we call it the probability measure right so if you just don't know if you wait lots of times and then you pick one iterate the probability that my point is in A the probability is exactly the probability of A that's the probability that your point is in A so it's a probabilistic description of the dynamics it says we maybe cannot describe exactly where you are at every point but we know that if you wait for some time the measure mu describes the probability of your points being in that particular region of space and that's how it is okay so I think we will finish here for today in principle I could prove because of the gothic theorem but because we don't have that many lectures I think I will skip it's not even that difficult I will give you some notes with the proof it would probably take one lecture or maybe one and a half lectures to prove it but I would prefer to go and look at some examples so what we will do in the next lectures of the course we will look at some examples in which we will use the gothic theorem to describe the dynamics so once you have the gothic theorem what you want is you take a dynamical system and you want to find an invariant measure and if you can find an invariant measure that is also gothic and then that measure tells you something about the dynamics because it tells you the statistics of all the points so the problem is two-fold take a dynamical system find an invariant measure and show that this measure is gothic or find an invariant and gothic measure, probability measure okay so we will start with some systems in which Lebesgue measure is invariant and the problem will be to prove ergodicity so we already have the invariant measure and then we will look at situations in which you actually don't even know what the invariant measure is and then show that it's gothic


Hey Gems welcome back to my channel so this is just a quick video to update everyone on what's going on with gem solistic, what I am offering now, where you can find me next, and everything that's happening. So gem solistic will be partake in in the 2023 agriculture fair at the Agriculture Fair Grounds on the island of St. Croix. It's a three-day event taking place on February 18th, 19th, and 20th. So if you are on St. Croix or will be on St. Croix be sure to look for gem solistic boots and come by, stop by, see what we're gonna have, ask questions, let's just connect. At the Agriculture Fair I will be having more herbal teas available. I'm talking about herbal teas for wound wellness. I am now diving more into the wound wellness and holistic lifestyle aspect of gem solistic and I am so excited. I've been taking my time to kind of start putting that part of the business out there because I did start as a skincare product line. However the reason why I named my business gem solistic is because I am all about health from a holistic standpoint meaning looking at every aspect. Not just what I put on my body but what I put in my body. I am pretty much, gem solistic is pretty much a reflection of me and how I live my life. So anything, so anything that you see I pushed through gem solistic is because it's a part of my lifestyle and I don't only use natural products on my skin. I take what I put in my body very seriously as well and I take wound wellness even more serious because our wounds is the portal to life in this physical realm. So I just want to help women heal their wounds and live their best and most healthy lives as they possibly can. So at the agriculture fair I will have teas for fibroids, picos, endometriosis, hormonal imbalance, fertility. Even for the men I'll have teas for prostate health and I'm not sure what word to use to help your stamina. As well as some other teas but those will be the main ones. I'll also have more holistic tools available other than just candles. I'll have sage and palo santo.  It's also on my website but now I'm really promoting it and putting myself out there to offer these services to people. So be sure to stop by my boot during the agriculture fair ground. During the agriculture fair this year taking place February 18th 19th and 20th. So this year I'm also going to be implementing scheduling on my website for those of you who've been contacting me to book consultations. You'll be able to do that on my website this year. I'm trying to implement it before February but I'm only one person and there's only so much I can do. So if I can it's going to be implemented before February before the agriculture fair so that I can lead people to my website when they ask me but if not it will be implemented by the end of March. You'll be now be able to go on my website look at the services I offer either book a free intake consultation or book an actual consultation where you pay right then and there and book it. So that is what's new with gem solistic. We are just diving deeper into wound wellness and living a holistic lifestyle and that means focusing on mind body soul and spirit. Okay stay tuned if you don't already make sure to follow me on social media at gem solistic on instagram and facebook or my personal instagram at ja underscore and press underscore. The links are in the description box below. If you want to see more behind the scenes footage of my life be sure to check out my patreon at ja and press. The link is in the description box below as well and that is all for this video. I hope you all enjoyed it. To all of my melanated women out there protect your womb by any means necessary and to everyone watching protect your peace by any means necessary. I wish you all love light and prosperity and you always see me in my next video. Peace.


This is Mr. Garry Bosman, now brother of Solomon. He's from London, Solomon accepted Islam some 25 years ago in Dubai, where he went for a research surgeon in the Middle East. This story is incredible. We've picked this story to feature on the show today for two apparent reasons. First, his transition from black and difficult Christianity to a dedicated life of Islam is authentically incredible. Second, what he expected of Islam, asked a rigorous and dictatorial religion, turned out to be very false. I'm going to say, okay, when did you last read the Bible? And you say, oh, I don't remember reading it at all, you know, or maybe I did it at school. So, you know, the Lord's Prayer, and maybe you went to church for somebody's wedding or a funeral. And that's about all you know. And that's what most Christians are in the UK, the majority, so-called Christians. Assalamualaikum, ladies and gentlemen. Welcome to another episode of the Open Minded Thinker Show. Please like, share this video. Don't forget to subscribe to the channel to help us grow. Like a lot of regular English guys, you might say, I would have said if there was a box to tick on a form, what's your religion? You would tick that box Christian. And then someone would say, okay, when did you last read the Bible? And you'd say, oh, I don't remember reading it at all, you know, or maybe I did it at school. So, you know, the Lord's Prayer, and maybe you went to church for somebody's wedding or a funeral. And that's about all you know. And that's what most Christians are in the UK, the majority, so-called Christians. So I was one of those. I arrived in Bahrain in the first of June, 1984, which was also the first of Ramadan. So one of my first experiences was to see people fasting. But the first thing I really noticed was to hear the call to prayer coming from the nearby mosque. I heard this voice singing out over the streets. And I heard another one. It was like in a cannon you could hear stereophonic sound as they were picking up. And I asked somebody, what's that guy saying? Actually, the guy I asked didn't know. But after some couple of tries, I found somebody that told me what the guy is saying. I said, oh, that's interesting. That's a very straightforward thing to say. It's not something sort of weird and poetic. It's just a straightforward, you know, come to the prayer. You know, that's what it's saying. There are two points that are worth highlighting here. One is the fact that he described his parents as Christians but not dedicated Christians. According to him, there were Christians that adopted secularism and only participated in what is today called the ceremonial faith. But what people don't seem to understand is a clear reason why there is a decline of Christianity and the rise of the so-called lukewarm Christians in the United Kingdom and, indeed, every inch of Europe. Now, essentially, in a minority in this country, not because we've been overrun by Muslims before anybody leaps to that conclusion, but simply because people are... Apart from the changing attitude towards religion, open-mindedness and tolerance among a younger generation, the decline of Christianity has thus far been propelled in the wings of the increasing rate of scandals and controversies that have knocked the Church in recent years. The Church has been criticised for it handling of cases of sexual abuse and this has led to a massive loss of trust and confidence among many people. I was working with Muslims. They were fasting in that first month and they could see I was interested, so they started to give me answers to simple questions. So you pray. How often do you pray? What do you say when you pray? Oh, okay. I can't say I was overly excited about that. I just thought, that's interesting. It's something different. So I started asking them and that was like 7 p.m. and 8 p.m. came and went and so did 9, 10, 11 and 12. At 12 o'clock, let's go rest out at the restaurant and we stood in the street. We stood in the street for another half an hour and I kept asking my questions and it was somewhere between midnight and 1 a.m. We're standing in the street here in Gahood in Dubai and I guess it went quiet and one of the guys said to me, so have we answered all your questions? And I said, yeah, I guess you have. The power of any conviction lies in asking questions and receiving genuine answers. There are people who think the vast in philosophy of religion or the theology of Christian exegesis, they are wrong. The truth remains that Islam can be explicated by genuine philosophical methodology and deep theological explanation. The guy featured in Brother Solomon's journey were armed with knowledge and by what the result is now explained Islam very well to the curious brother, obviously the one. Tuesday night, I think, and on that Friday, I embraced Islam in Moscow and Dubai. One of my biggest concerns around the time that I embraced Islam immediately before and still immediately afterwards was that I felt I might be getting into something that was too prescriptive, that I might find that this new way of life was going to be like a straight jacket and it was going to tell me too much, restrict me too much, give me no options, no choice, lose freedom of choice. That turned out to be true and not true at the same time in a very happy way because, yes, it does tell you what to do but it tells you in the way that your father tells you what to do if he's a good father. He's giving you the best advice. You can still choose what you take of it and how you do it.  You're still a human being but now you know you're being given good advice and that's very comforting, isn't it? Lessons abound in this incredible story. First, Muslims must direct their dower to those so-called Christians currently sitting on the fence. Those who are undecided they form the bulk of those who are spiritually hungry and remain starved of the truth, fruits of truth. Second, open-mindedness has propelled the growth of Islam among British people. Britain is an open society where everyone, anyone has a right to think the way he or she wants and act accordingly. This freedom, which is enormously lacking in Muslim-majority nations gives people the choice to find the truth. While I don't blame the Muslims for tightening their society, it will be safe for one to stay here that the Western world has shown that the Christianity that preached was so much a hoax. The Muslim world has continued to keep the faith untainted thus driving the religiosity and traditional truth of godliness around the world. It looks Islam in the future will be the bastion of truth and the only place one can see sane worship of God after the jowl devastates Christianity and Judaism by establishing false replicas May Islam succeed. Until next time. Assalamualaikum


Yeah, thank you for the introduction. So this talk will be about AS-like ciphers. And AS-like constructions have been a very popular design choice. And a wide range of primitives have been constructed, their block ciphers, hash functions, permutations, all which use ideas originally proposed in the design of AS. And also for all the competitions we had so in step three, we had a lot of AS-like designs. And probably also with the lightweight competition we will have a lot of AS-like designs. So what we mean with AS-like designs, we consider here the following building blocks. So we have a round function, and it consists of these three sub-functions. So the first one is just the application of an S-box. Then the permutation is applied, which changes the position of the elements in the state. And finally, matrix is multiplied on each column. And a lot of the designs fall into this category. So about the S-boxes, so there has been a lot of research on how to construct S-boxes. I think we also have a session later this week on S-boxes. And it's the only nonlinear component, so it's a very important building block here. But there has been a lot of research on this, and it will not be the focus of this talk. Then similar for the mixed columns operation, there has been a lot of research on how to design these matrices, what properties we want, how to get them efficiently. Also, the next two talks in the session will be about this. And this talk will not be about this. What we are going to talk about is this middle layer. So how do you change the position of these cells? So arguably one of the most important criteria for block cipher but also other primitives is resistance against differential and linear cryptanalysis. And each one of these building blocks plays quite a crucial role in constructing a primitive secure against these attacks. So the S-box directly influences the probability of a difference going through it. The matrix multiplication, it directly gives you a low about on the number of active S-boxes in a single round. But the relation of how you do this permutation is a bit less clear what happens in general. And it still heavily influences the number of active S-boxes you get in total. So the typical goal for every designer is, yeah, we want to have as high as possible number of active S-boxes over many rounds. So as an example, A-S has a mixed column with branch number five. And this gives a quite simple constraint on what can be achieved in one round. So branch number five here means just if we start with a single difference, it goes through the S-box and shift rows. And then it becomes a full active column. And this is kind of the constraint we have here. So the sum of the input differences and the output differences has to be bigger than five in the case of mixed columns. And this allows quite nice arguments on the minimum number of active S-boxes you get over more rounds. However, for other choices of mixed columns and shift rows, it becomes a bit less clear. So there are a lot of other designs following this strategy. So for instance, Midori, we heard about it before. It uses a branch number four. So OK, we would have this constraint. They have to be at least four if we sum up input plus output. But actually, there are transitions in there which are not possible. In the case of Midori, you can never go from two to three active through mixed columns. And the picture below, this very complex picture, shows you what happens through the skinny mixed columns. So skinny mixed columns is extremely lightweight. It only has a branch number of two. But not all transitions are possible. So actually, if you look at this, you will see there are only two transitions here which go from one to one and reach this minimal bound. So if you now want to analyze the size of more rounds, it becomes quite complex in the case of those designs. There are some results known on this. So for case of AS, yeah, we know if we have a square state and the matrix is MDS, then shift rows will be optimal. And there's also been some additional work on this where they look at also other shapes of states and other properties of this mixed column. So but the problem that we solve in this paper is now if you're given an n times n state and we fix the mixed columns, what is actually optimal choice for permutation with respect to differential and linear cryptanalysis? So let's look at an example. So we want to choose this permutation now. So let's assume we have a four times four state. Then we already would get like 16 factorial possible choices for this permutation. So this is quite a big number, but maybe still feasible to handle. But the issue is we need to kind of evaluate each design now if we fix the mixed column, which permutation we kind of want to know what are the bounds on the active S boxes. And this might be quite out of not be feasible for us to run for all the permutations in this case. So what we have to do is we kind of have to limit the search space. So the first observation we made here, so if you have a permutation p and a permutation theta. So if theta commutes with this matrix m used in the mixed columns, then the cipher where you use the permutation p and the cipher where you use the permutation theta inverse p theta have the same properties. And this can be quite easily shown here. So down here we have the round function. And if we change the position of the theta inverse with the S box, so we just swap them, we then have to apply this permutation also in the key. And now if the theta commutes with the matrix m, we can also swap the position of the mixed layer and the permutation here. And then we can see, OK, we have theta and then apply theta inverse, so it just becomes the identity and we get the same cipher basically. And we use this to define an equivalent relation. So we say we have two permutations, p and p prime, and they are m equivalent if such a theta exists with these properties. And the nice conclusion from this is all permutations which are m equivalent will have the same number of active S boxes. There's some drawback because we don't actually know how to test m equivalence between permutations. So we have to go a few more steps and see how we can achieve this. So what we did is we make the job a bit easier. We first look at the weaker equivalence relation, which we call weak m equivalence. The only difference here is theta has now a structure. So we say theta is composed of two permutations, phi and pi. So pi only permutes the columns with each other and phi only permutes elements inside each column independently. So if theta is of this form, then two permutations are weak m equivalent. We need one more tool, so which we term you the structure matrix. So we have here, it's probably easier to understand an example. So we have an example of a generic permutation. So on the left you see the elements of the state. On the right, how they get permuted. And this would be the structure matrix. And the structure matrix shows you from which column how many elements get moved to a different column. So first column, zero elements get moved to the first column. One element gets moved to the second column, zero to the third, and three to the fourth. Another example, in this case, the second column, two go to the first column, one to the second, one to the third, and zero to the last column. Here they are distributed to all different columns, so you get all ones. And here again you can see how it works basically. So what we know showing our paper is that there is a efficient algorithm which can enumerate all the permutations up to m equivalent. And I only give you the basic idea, so if you want to have the details on the algorithm, it's best to look it up in the paper. But the idea is we can enumerate all these permutations up to this week m equivalence for a given structure matrix. And there are not too many valid structure matrices, actually. So if you consider four times four state, there are only around 10,000. And some of them also have some symmetries. And you can further reduce this number. What we can do now is we can just pick one of the smallest representatives of each of this equivalence class and then carry out the crypt analysis on those. So this still only gives us this weak m equivalence. So it's kind of the obvious question then when does this weak m equivalence actually imply m equivalence. And there's a nice property on m which can be used to show this. So if we take this matrix m and use it as a adjacency matrix for a graph, we can test this graph for strong connectivity. And if the graph is strongly connected, then weak m equivalence will imply m equivalence. In the paper, we now use this approach to look at two ciphers. So the first one is Midori with a lightweight energy efficient cipher. And it's probably the most obvious target because it introduced using this arbitrary permutation. So often you want the most structured permutation, so you have some implementation advantages. But also in hardware, it doesn't matter because it's often, if you do a round-based implementation, just rewiring. So our question was, is the permutation used in Midori optimal? So first, yeah, this mixed column is used in Midori, as I said before, as a branch number four. And a good thing is if we look at the graph of it, it is strongly connected.  So it took a few days then to, on a normal PC, to find all these permutations up to m equivalence. It turns out there are around 2 to the 22 classes. So we then pick for each of these equivalence classes one candidate and try to determine the number of active s boxes. So first, we tried this there with mixed integer linear programming, but it turned out to be a bit too slow for running it on all these 2 to the 22 permutations. So we then decided to implement some custom branch and bound algorithm to do it faster. But you can also find here. So this is basically the summary of all our computation. So I guess for full effect, you should look into the paper and zoom in. But just to show you the general picture, so if you can spot that like a red line going through all these other blue and green lines, so the red line is the original permutation, the blue line are all permutations which are at least optimal in one of the rounds and the green are then another random sample of permutations. So the main conclusion from this is, well, the original permutation is actually optimal up to 12 rounds, except for nine rounds. It seems an outlier here. But for any higher number of rounds, it is never optimal. So when you look at the picture, you will see there are a lot of blue permutations on top of the red permutation. But it's also mostly outside the number of rounds of Midori, so maybe it doesn't matter too much in this case. It's also quite interesting there's not a single permutation which really stands out here. So a lot of the permutations, they are better for some rounds. So it really depends on what parameters use, which permutation actually is the optimal one here. We also show a nice proof in the paper, which gives you an easy to check condition on your permutation. So if your permutation, p, p squared and p cubed, if they all have the structure, all one structure matrix, then you'll get at least 28 active S-boxes over six rounds. In the second example, we looked at is the skinny block cipher, which we also had a few times already. So it uses AS shift rows and a very sparse mixed column, so you only get a branch number of two. And it's, again, an interesting choice here because the mixed column seems very simple, and the shift rows is a bit structured. So maybe if you use a more generic permutation, you could improve the bounds. So we applied again this algorithm to find all the permutations, all the equivalence classes. But in the case of skinny, it turned out there are quite a lot more, so it's almost two to the 40 equivalence classes here. It already took 24 CPU days to find all these equivalence classes. So doing this crypt analysis, counting the active S-boxes for all these two to the 40, then we probably would have submitted the paper only around now. But we then decided to filter it a bit further. So we reduced the permutations and only picked those permutations which have at least as good diffusion as the original one. There's still around three million permutations left, and it took roughly 3,000 CPU days to test all of them with the Matsui algorithm. We again get, like, this picture here. So red line is, again, the original skinny permutation. Blue are all the permutations which are optimal in at least one of the rounds. And green is a random sample of 10,000 permutations. And we can see here quite clearly that actually the skinny permutation, even though it's quite structured, it holds up quite well, and it doesn't make too much difference if you loosen this restriction to only operate on the rows. So to summarize, I think our paper gives a quite nice theoretical understanding of how these permutations behave, how they influence the number of active S boxes. And it's especially, I think, a useful design for any future designers if you want to make a new design. You should maybe look at this, and it's often actually feasible to test the full design space. So if you have a four times four state, there's a very good chance you can actually find the best permutation for your design. This concludes my talk, so if there are any questions, feel free to ask. Thank you. Thank you very much. Any questions for Stefan? There's one here in the front. Yeah. So assuming you fix some arbitrary key schedule or even don't put some, could you do the same kind of thing to evaluate security for a related key model? Yeah, we didn't include the related key. So the problem here is really probably that your search is going to be much more expensive on the active S box count. So this might be quite tricky than to do. Exhaust is searched. Yeah, you still have to do for each of the equivalence classes here the test on the active S boxes. So if you can do that efficiently, then it might be possible. OK, thank you. More questions? So I have a question. Would it make sense for the best choices to then look for characteristics or even differentials to find out whether this really also active S box is a good indication for the best attacks? Is that feasible or? So that's, I think, quite difficult to evaluate for the AS-like ciphers because you have this strong alignment to really say, this permutation is better than the others in this differential setting. So I think that might be, at the moment, I don't think we have such good tools to evaluate these properties for AS-like ciphers. And it might not be feasible to do this for all the permutations. But only for the best ones, you find. For the best ones, you can, of course, do that, yeah. And what about other attacks, like structural attacks? Do they depend on the permutation or is this orthogonal? It depends on the structural attack, of course. So for some, it's probably a good. So the nice thing is also you can pick the permutation which is best in this equivalence class, then you could also generate all this permutation in this equivalence class, and then check those again as a second filter against the structural attacks. Also for if you want to do efficient implementations, you could also do that. You just test one in the equivalence class and then test other attacks and implementation properties. OK, if there's no more questions, please join me in thanking Stefan for his talk. Thank you very much.


As capitalism goes from crisis to crisis, bosses have gone on the offensive against paying conditions across the UK, but workers are responding to these attacks with increasingly militant fightbacks of their own. A strong example of this is the workers of St Mungo's Housing and Homeless Charity who have recently forced into indefinite strike action against their employers. We headed down to the picket lines and rallies to find out more about their struggle, asking the question, how did these events develop? Our pay dispute has been going on for 18 months. I think it's important to say that this is not about pay, this is about working conditions and this is ultimately about the safety of our clients. We've been asking for a 10% pay rise, which at this point is quite out of date. We've had inflation that's been over 12% and in real terms we've actually lost 30% as frontline workers over the last 10 years while senior management has increased their salaries by over 385% in the same time period. So you have this massive pay inequality in the organisation. I think we're trying to address that. We're trying to negotiate. We've come up against a real diligence for senior management who have constantly said it's financially unviable despite the fact they clearly find the money for senior managers. We said enough is enough, we're going on strike. We announced some definite action recently, just because senior management still refused to come to the table. Things that they might be changing a bit now. But yeah, we also said well as long as we're not getting a decent pay off of we're going to keep going. After 18 months of struggle and a decade of worsening conditions the workers finally decided that enough was enough. For me for example when I entered so I get 25 days annual leave but before that was 28, before we would have two night workers in my service now people are alone working at night. Also really really important, our teams have been like stripped apart so we would have specialist workers, we would have mental health workers, we would have substance use workers, we would have activity workers and none of that exists anymore. We cannot keep duty workers, they're the ones who run the building. The building has been falling apart. It's well known, it's a really old building, it's an old workhouse and we've had leaks in the kitchen that have lasted for months because we go through duty workers so quickly. These leaks and other issues, smashed windows, broken doors they take months to fix sometimes and it's really frustrating because the duty worker that's the lowest paid role in our service so we can't keep hold of them. A lot of them have left in sort of anger, some of them have moved over to other jobs because they just need more money to get by. It was only after senior management's refusal to negotiate and their belligerent tactics attempt to break the strikes that forced the workers to take the drastic measure to taking definite action and start making the radical demands to open up the books. I work for an organisation that pays their CEO more than the Prime Minister and senior management has increased their salaries significantly so obviously the money is there, it's just distributed in a very specific way and if I look at the developments over the last years that means just a road in frontline services and we're supposed to be a frontline organisation so I think it's absolutely crucial that we step up at workers at this point this is not about pay, this is about saving what we can of these frontline services. So from the start of the negotiation I think one of the key demands has been that some mongos show their accounts the last accounts we can see, say there's 22 million in reserves if what we're being told by senior managers is that there's only 11 million, 10 million left something serious has happened and we can't see any explanation of what's happened we can't move forward with negotiations unless they're going to be open so we have to force their hand because whether or not there's the money there to pay us we need to know if there's not something very serious has happened and we need to know as well so it's just as important that we keep demanding to see those accounts. Our reps had a meeting with our CEO two days ago and had to leave after 20 minutes because continuously from the beginning throughout there were verbally abused this is not a way to have negotiations. You know they're going to sort of try and divide us and you get this a lot in the charity sector I suppose they'll try and sort of use you know people sort of own goodwill against them so they'll say you know your clients are going to suffer because of this it's going to cause a lot of disruption and we've always been really clear that you know what causes clients disruption in the long term is the high tone over it's the fact that you know every sort of client has experienced all the instability of their key worker having to leave because they're kind of full of their bills but you know they're not getting enough pay their work loads are too high and so clients have been incredibly supportive clients know we're striking they understand that we're not just striking for us but we're striking for them as well because we want to be able to sort of stay and do the work they're all passionate about we want to be able to help our clients and so we've seen an amazing amount of support in the big line in stark contrast to what senior management have tried to sort of say is that we don't care about our clients and it's just completely nonsense. It's clear that this is not purely a dispute about pay but about bigger trend of conditions for all frontline workers suffering while bosses find more and more money for themselves It's all part of the bigger trend of sort of you know resources just being and susceptible to kind of the central bureaucracy and away from frontline services yeah it's for the heart of the organisation is what we've always said it's for pay but it's for make sure it sort of remains you know what it was set up for to help sort of homeless people not sort of to you know to pack the pockets of senior managers It's really really important to emphasise that if we want the services to continue public services to be provided workers need to be paid properly a lot of people say they can't afford to do this job they love this job but they can't afford to do it because they can't pay their bills and I want this to change and I know we can change it As bosses continue to belligerently protect their profits during periods of economic crisis and with no solidarity for workers being found across the pillars of the political establishment more and more layers of the working class will be dragged into these disputes but the workers of St. Munger's are showing the way forward strong organisation, militant demands and definite action and resolve


Everyone, Basel Chapman, we're looking at the 11am Eastern Time Tiger Financial News Network update down to 160 at $35,981, making that pattern. We've been talking about it as a potential age pattern, lowercase age pattern. If it takes out $35,915 the next day or two, that's a big negative. It says that the daily chart, not the weekly, but the daily chart has gone from a cell signal to a cell mode. We're looking at the S&P actually doing quite a bit better. It does down 0.44%. S&P is only down 0.14%, holding near the all-time higher of $47,18.50 at $46,94. Yesterday it caught within four points. This is still good action. We're going to be watching NVIDIA today because look at the QQQ. QQQ is trying to rally nicely. It's up $0.84 to $3.98, just as fractured under the $400.99 all-time high. That means that the SMHs, the semiconductors when NVIDIA comes out with its earnings today, tomorrow's action is going to be so important all the way around. We are looking at something like a Qualcomm in the same sector doing fabulous all-time high today as we look talking at $186.33. What about the IWM, where the Russell 2000 acting quite weak as the small caps are not finding favor at this time? We've got gold. Gold is up, let's see, it's up $10, holding quite nicely at the higher level. You're looking at the dollar probably pulling back a little bit now. Yes, just down three ticks at $95.89 made a new recovery high today of $96.25. In my show of the target technicians our moment ago, I didn't actually get to the TLT bonds, but bonds right now, $144.81 down again. That means yields are going higher. You've got to be real careful and you've got crude oil, crude oil is trading down now two points, making that H-pattern as well. This is now a very mixed market, very selective. You've got to be careful. Watch the VIX index. The VIX index right now is up just a little bit, $0.26 at $1663. If at any point today the VIX manages to get into the 17s, be careful. The market will have another self, but if it holds here at $1664 or actually pulls back to the $1650s or lower, that will allow the market to have a pretty decent balance. Okay, we'll have a wonderful day. We're about to hand you over to, I believe this is Tommy Jr's show, a repeat of the show, and that is the market kickoff, fabulous show. You also had Teddy, which he was talking about the currencies, et cetera. Really interesting. Great program all the way through today. Stay tuned to Tia for that. Have a wonderful day.


A planned fish-off on Lake Bemidji was cut short today after the Minnesota DNR intervened. Reporter Sherell Moore is live in studio tonight to explain what happened. Sherell. Thank you Dennis. A group called the 1855 Treaty Authority had planned the fish-off for the day before the Minnesota State Fishing Opener as a demonstration to exercise their treaty rights. Two men headed out onto Lake Bemidji and placed a fishing net in the water. The net was then confiscated by the DNR and the men were issued citations. The treaty authority says they are disappointed with the outcome of today's events and say they plan to take the citations to court. Unfortunately, the game wardens came confiscated the nets, you know, not acknowledging or recognizing the treaty rights. That's the message we like to say today is we have a right. These are our inherent rights as tribal members is to hunt gather and fish on a seeded territory land, to feed the families, to feed the children and the elders. And this wasn't the first time for the fish-off before the opener. A similar demonstration also happened eight years ago in 2010. Dennis, back to you. All right, thanks Sherell for that live report tonight from our studio. The DNR issued a statement to Lake Lannous about today's incident on Lake Bemidji saying the DNR's position is that off-reservation harvest rights do not exist in the 1855 treaty area and that state laws will be enforced.