 Hear me I want to thank the organizers. Oh, actually Ali are the other organizers here. Then let me only thank Ali Now I want to thank you for showing up on this slow lazy hot Monday afternoon Yeah, so I changed my title because half an hour is too short to prove anything But I can give you an example so the goal is just to give you an example and maybe I'll explain why it's new But when I explain when it why is it new? That's right. So I want to explain why it's new, but the danger is once I explain it you'll think it's not new but Because it's not But it still is okay No, it's true. It's true. Okay, so Why okay, let me talk about old examples, but first here's the setting three manifold to a three manifold Partially hyperbolic they feel and Maybe I'll just give a quick definition. So there's a f invariant splitting Oh, I'll give half of a definition that everybody gives There's three invariant sub bundles and then In my setting everything is nice. This is all the estimates are nice This is uniformly contracting this uniformly expanding and the center is uniformly in the middle It's all the nicest definition possible Okay, so I now have to tell you what the old examples are. So here's the old examples and By old I mean one-year-old So what are the old examples? One is just on us off on the three Taurus so you have One expanding Eigen Valley when contracting one and the third one is in the middle and any on us of example on T3 Is parts to have a ball like any an ass off like that Okay, then what's the second old example? It's you taking an ass off on T2 multiplied by the identity and this is on T2 times s1 So you have an ass off everywhere and then it's it just sheets of an ass off So that's that's example two Very old and then what's and then you take time One maps on ass off flows So there are not so flows which Look like this Except EC is just a flow direction. So nothing is happening And you can just look at the time one or time two time to pie map doesn't matter and you'll get an example so you have these three classes of examples and I guess maybe I should write down this matrix here because it wouldn't be a nice talk if I didn't write this matrix down So let me write the matrix down again Okay, so now it probably might be a nice talk Okay, so there these three examples and there was this classification you want to classify all types of examples and the classification was these three are the only type of examples so or Dynamically coherent, so I should write that down So I don't think this has been defined and maybe I should Define it. What is dynamically coherent for mine? It's EC is integrable, but then you get all the other bundles are integrable also and let me not say all because Some of them are impossible So EC Okay, so if you assume this Then there's some classification that says These are the only three types of examples and it's up to our the only examples up to Something called leaf conjugacy Okay, so what is leaf conjugacy? So I have to tell you what leaf conjugacy is. Okay, so I've a draw this I Have a manifold and a map F Sorry I'm too strong for this chalk and and I rather map G What is the leaf conjugacy as a map H and then now this is not I Have to draw H. Let me draw H in a way. Let me draw this So this map looks like this So there it's not the diagram does not commute and hopefully I drew it in a way to show you it commutes up to a Straight line. So what does this mean? This means if I first do F and Then I do H There's another way to do it. I can first do H and then do G These two guys on the same center leaf So the diagram doesn't commute you might be off, but you're off by on the same center leaf The center is one-dimensional in this game So it says, you know, these are the only examples and these are very simple examples I should say simple we have no idea what this is. This is like a black box But that's still we can assume if this is simple then they're all simple examples Okay, so the object of this talk is There there's some other examples and that's what I'll talk about. So there are some other examples But before I talk about building the other examples, I have to go to older examples so these were Old examples and now older examples So what are the even older examples? So 1976 Franks and Robinson gave an example of a quasi-anosov, which I probably won't define on T3 Connect some T3 so what's quasi-anosov? It's maybe I'll define it quickly. I won't write it down It's a vector in the tangent bundle is either expanding in forward time or backward time and They do some surgery and they build this example then 1979 Is Frank's Williams example Of a non-transitive anosov flow. I will talk about that example in detail But I will not do the other one because they're the same Okay, because same type of Construction two steps to surgery That's the easy step And the second step is to check that you've done the surgery and then things don't work check if It is still Anosov All parts to have a balling or whatever. So this is the hard part. The easy part is to take something and glue something else Okay Okay, so let's see my learning. No, I'm fine on time. So the hard part is not that hard because Right around the 1970s. I don't know. I'm not a good historian There was a mania's thesis. He was just graduating so That was that was very useful and then you could just plug in money as thesis and Check that their examples were actual real examples. So let me talk about money as thesis and now it's probably old Everybody knows How to do this Okay so if a psi t is An anosov flow and you have to check two things psi t has a chain recurrence set The on the chain recurrence that You what do you have to check? You have to check that The stable center stable of an orbit trans is transverse To center unstable of an orbit gamma is Gamma is an orbit so you have a check transversality And you have to check that they're the same dimension three Center stable Has constant dimension. So this was a black box and I'll go through this Construction and I'll probably just say by mania's theorem Everything is okay And these guys have to be transverse because the bundles are transverse Okay, so now let me start Talking about the example and now you can you can interrupt me. I don't know how complicated things will get but so now this this is the best part of the talk for me because I'm going to stop writing because I cannot write And I don't think in words, but I'll draw a lot of pictures. So this is the best part of the talk for me I probably want to stop writing any words down Let's see if I can pull that off Maybe I still have to write some words down. Okay, so this is what you do And this is the same thing for the quasi-announce of map. What you do is you This is all t2 is gonna be t2 And then you flow up This is a I'm gonna first construct a map on t2 cross an interval So this is unit speed flow I'm gonna flow till I reach the top and then I'm gonna glue by a map and the map I'm gonna glue by is the DA map and then I'm gonna build like a flow on that object Now that this is a closed three manifold and what is it gonna look like? Okay, so if you don't know anything about the DA is to derive from an awesome map and The choice here is gonna be there's gonna you're gonna blow up You're gonna take it an awesome 2 1 1 and you're gonna blow up a sink and you're gonna make a point of source Sorry You're gonna make a point of source. So you'll get this kind of You'll get a standard foliation, but then you'll open it up or unzipping. It's called unzipping some some time So this is a source So you just blow it up But the way you can do this is you can keep one foliation unchanged So I think I'll use red to draw stable. So This is a stable foliation So the red is stable And The white is unstable. So this is the first talk with color chalk. That's good. Aren't you glad you stayed around? Okay, okay, and then so this is step one Now this looks horrible. This is not even close to being an awesome flow, but you're not done yet And what happens here is this guy's an attractor So this is a source this white stuff is an attractor and the white is crossed like an interval here Which I don't know how many pictures to draw. So maybe I'll draw these pictures The stables are vertical. I'll draw a few of them. The stables are vertical and that's going to be important Okay, so now this looks horrible. What are you gonna do with it? Well, I'm gonna drill out a torus in the middle So let me tell you what the torus looks like Okay, so if I'm gonna glue from here to here this guy's a source So it's gonna expand a piece over here. Now, let's see. Where am I gonna draw that piece right here? Probably so I don't want to crowd a picture, but there'll be a torus here, which I'll draw right here Torus looks like this and it's and it's important. It's got that shape Okay, so that's a torus because This piece is identified with the one on the bottom and it gets bigger because of source So you can draw a transverse torus that way and I said transverse. What is it transverse to? Well, these straight guys is transverse to So the straight guys here They all hit it transversely and the transverse curves look like this and I'll draw a more detailed picture of that and Why am I doing this because I hope I didn't erase this you have to check some transversality You have to apply money at some point or you don't have an awesome flow Okay, 15 minutes left perfect timing think I'm halfway done here Okay, and I'll also draw What the foliation looks like on this torus? So because you probably have seen this foliation before. What does the foliation look like here? Let me try to draw it. So this is the undrulled torus and what does this foliation look like? It looks like these rape components as you can see. I want to make sure it's And this matches I've drawn a few pieces here And there's one which is completely transverse with corresponds to this and in fact It's to this one in the back. So this is the other one Okay, so step one now there this looks horrible you've built out a torus you have Something messed up going on. How do you solve this problem? Well? You might as well double down if you're losing a muscle double down and that's what happens You look at it's evil twin here. What's his evil twin is you do? Oh, you look at the mirror image. Why is it evil because it's a mirror image. Oh Mirror images are evil. So what does that mean? So I look at the inverse of this map I do the inverse of this construction. So maybe I should draw some pictures here Let's see. Let's start this way and what's it's gonna look like so Now I have pictures like that And this is a sink There being a source and it's along the stable Wait, I should yeah, that was the wrong color, right? It was the wrong color. That's red And the white is straight here now. I should draw some straight pieces And then you have a corresponding torus here, which looks like this now it looks Fatter that way another way and then You have these pieces and you get another rape for lesion Which is now drawn in white but if this was If the here, this is Santa stable. This was Santa stable This guy is center unstable. I think Okay, and now how do you check money is you've drilled out a torus in the middle here Fill out towards the middle here and you're gonna glue them And if everything is transverse by money, you're done. That's all you have to check is transversality And you have an anosso flow and why is it non-transitive because you have an attractor and a repeller right here Okay, so how do you glue this piece to that piece? Here's another picture. This is a talk full of pictures How do you do that? Okay, so I will not change the white because I have it here now. Let me draw the red right over it Hoping everything is transverse and I will not crowd the pitch area. You can see everything you can make it transverse That's it. So you said to glue things So what did you do? You started with the suspension you took out a torus made sure some fullation was transferred to the torus Did the the opposite the mirror image and you glued them together and one was sent a stable one Was sent unstable and because you can glue them transverse. So you have an example Okay, so old example of an anosso flow non-transitive because you have an attractor and a repeller Okay, so what do we do now? So this is why this example is maybe not new. What's the new example? Okay, so there's a transition in the middle going from good to bad a good to evil actually going from evil to good So I drew the picture correctly. Usually I just the evil one is the one you draw second on the right-hand side That's that's the definition. I'm not making this up when following the definition. Okay So what is what is happening here now in the middle in the middle? You have a passage So this one's the attractor you have a passage from So there's a torus here, which I'll draw this way and you can imagine this fullation on it This is fullation and there's a torus right here and you have this fullation on it Okay, and then the flow up thing. I'm way. I'm drawing it is flowing this way It's flowing this way and of course. This is t2 times an interval here an interval here so there's a piece and Fullyations are very nice here that means and let me tell you how nice they are if I translate up like this I still maintain the same fullation or I translate down and if I go move up and down here I get the same fullation and the middle I have horizontal lines moving from evil to good. That's good So what do I do here? I I look at a time one map and I compose by a twist So this is a new example now so it's a very simple idea and Of course the hard part is I have to check I get something which is Parts to have a ball like that's the first thing and then I have to tell you it's something new and Then Otherwise it's not gonna happen. Okay, so I compose with the dain twist I tell you what the dain twist is it so let me tell you what the dain twist is It's a bunch of translations here. I don't do anything Here I translate by one And here I'm translating by zero and then slowly here. I have bigger and Bigger and bigger translation is still I'm translating by one So what maybe you haven't seen a dain twist in dimension three So I'll turn on the picture about dain twist in dimension two is what a dain twist is I have an annulus I keep this fixed and This is rotation by 2 pi So it looks fixed in the neighborhood over here, and I just keep rotating by More and more angle until I reach the end and I'm rotating by 2 pi Okay, so this one in the surface. It's easy to check that Often is easy to check If I do a dain twist here, I get something a non-trivial non-trivial in the mapping class Which means it's not isotopic to your old map So here if I want a new example, I have to check I get something new And then you have to check and there's a theorem that helps you it says if the two pieces I'm gluing Hyperbolic then the middle all dain twists and on trivia. So I have to check I get a new example So that and that's one thing Okay, what else I have to check I told you I don't want to write anything else down What else I have to check I have to check that I have some transversality Conditions, okay, so let me just give you a very simplistic proof, but I'm not giving proof I'll give you some reasonable argument, but you may or may not believe but okay, so what I look at it I look at the center stable here, and what am I doing? I'm composing by dain twist afterwards, right? So what happens? This is a fundamental domain the center stable domain the center stable moves off into the attractor and the Good guy, and then I compose the map so it remains the same here. I'm not doing anything to send a stable So that's unchanged here And now the set on stable I will change But we argue we don't change so much. So it's still transverse So you get something you check some bundles are transverse and then So and then you have an attractor repellent and we prove its parts to have a ball Okay Five minutes. Let me tell you why it's dynamically coherent. It's also easy Okay, dynamical coherence Which means you can integrate stuff? Okay, so we prove this very easy fact Let V be in the center And I iterate it So why is this easy? because I have an attractor here So every point no matter where you start is going to come into the attractor, right? And here I have some uniform estimates and here you're not growing you're not shrinking and In the backward iterate I Come to the evil guy And again, I have some uniform estimates. I only spent finite time outside some neighborhoods So and there I've uniformed estimates like a not so flow there. So I get these easy estimates Okay, this is all you need to prove Dynamical coherence by first few shoe and they throw some fancy words around like Lee up enough stable and then in fact that that proves It's uniquely in a group. They unique. Also. This condition is open See one open you make a small perturbation you get the same Condition because we are tractor doesn't get destroyed repeller doesn't get destroyed Dfn for all-in Yes, because Small perturbation will not kill the attractor repeller So the C1 open so what that means is this is stably dynamically coherent That's what that means. So the example and also it's not it's a new example because it's not homotopic to any anasa flow You have a question. Okay, so there this is not the only new example. There are other new examples You can you can have two sources and you can drill two holes and do stuff you can have five million holes and start gluing them together there's not the other examples by other other people that do this construction and It's still transitive afterwards And then there's other constructions that say why do one day and twist start composing day and twists? So they all the other examples, but if you want to see one you want to see Franks Williams Oh, you want to see Franks Robinson and you see Franks Williams, then you see this and there's a straight line from there It's a straight line from 1976, but today is 2016. So it took too long but I'm happy to be part of it and This conference also. Let me stop here. Thank you