 Psemo to assisted. Ako je bilo in izvah nekaj oto se potrebno sponujte, dobro ide ntakaj. Naspočijo počke vseme, tako ene malosti počke vse San 알no na našigu vse tega nožena, jih najmojeva odnžaj v tem, in včetno nekaj to v žezve in zato, in zato, najmaj, je izvah bila in posleda. Nelad najčetni. Kaj sem zelo, nezavljala, in odnene včešljem vzadovati spomaljnoh hamiltonjfo, zelo, nekaj je vene vzadvati spomaljnoh hamiltonjfo, vzadvati spomaljnoh hamiltonjfo, na izvršenju genuči nekaj nekaj. Zdaj smo polačili vzadve. Veselič sem odnet, kaj sem odnet vzadnjalo vzadve hamiltonjfo, in divinjenje vsega. Zda tudi, da jih vam ne možem postariti na zemljenje, imamo prejzipčenje, bo v진je spesjelj drage hradi in uvaj so, če so na večj stranu. Zdaj, sem tudi spojljo vsega definitje, imamo zelo za svojjo neu prevju. Zato smo bilo, da, kako so imaš obježenja hrvi genus in ta torez v dejerni kajštih, imamo površenje. Zdaj smo izgledali z hamiltonijanja sedelja. So, hamiltonija, hamiltonijanja sedelja. Daj bila zvomovina vstavljaj pospravljno, in povrčo je zdačne fjišk, da je do tomh pa spremljenje, način je, da je tezvega do stačnega. V tom spremljenju enduring je ostavljenje, do vrče in bila. Ness, kako se je dodelite, The key feature is that in this locally Hamiltonian parametrization because I want high, just started. We started 15 minutes late, so you didn't miss anything. Key feature is that it takes infinite time, infinite time along the separatrix to reach the saddle. Let's call it P. So this saddle point you will never reach it in a finite amount ta vzgledaj, kjer sem vzgleda na separatričnih, sem vzgleda in izgleda inzgledaj, in za to, da me najbolj, nekaj, kako sem zelo vzgleda na separatričnih, vzgleda, da je vzgleda, nekaj je vzgleda na separatričnih. Jih se zelo zelo, je zelo tudi zelo zelo, pa je zelo pri kratu. A več, ga izgledajemo, imamo to v tej formali, ko je začal, je začal, je začal. To je prevoj, a načal. I to je odlično deseleracije, nekaj je vse zelo, nekaj je začal, odrde. We create a key phenomenon начli proti prra adjust. So deceleration rates create different deceleration rates create shearing and this is the key word of today and it's the key word of many parabolic flows. Zato vse ne zelo. Zelo nezelo povsem, da so mi zelo je povsem, da je povsem vse. Tukaj sem moračite na praviček. Jaz bom mi se izglede in ustešel, da je transversal na različku. Zelo nezelo je, da sem izgleda, da je transversal na različku. in vsi udeljda. Vsi udeljda. Vsjah, način je, vtečnja od vrst, pojeko so dolge za dušnjenje s sedem. Poječnje v vrste na dolge prič. Voječnja je so dolge, bo mladite, da me zelenje, što je očen. Zato, ovo je skončilo, da mu grač je našličil. Dozaj se našličil? Zelo skončil je vzvečil. P je bila, bila, če blježi pri p. V veliko stvari, v veliko stvari. Zvuk je nekaj nekaj napravil, nekaj nečo se vseče, in izgleda, da se, kaj je to razkrično vseče, razkrične, nekaj nekaj nekaj, nekaj nekaj nekaj nekaj, nekaj nekaj nekaj nekaj, nekaj objevaj ta umoženja, bi jih da je dejça zelo i zelo. In je прima, ko mi je kriši, odradi, uniformne, vzpolivnje, ljude, nekaj objevaj za vlasi, in pozanj, objavaj za vlasi, in početaj za svetil. V nekaj, ki tjela z tjela, nače je vse uvršen tjela v tjela, nače je tjela vse uvršen tjela. Kaj je vse uvršen, nekaj se nekaj izvah, nekaj se nekaj izvah, nekaj pa pa vse izvah nekaj se nekaj izvah nekaj nekaj izvah nekaj, nekaj očetna. In, ko dobrovam, to se vsegače zelo vsegače. Če kaj ta vsegače? Zelo, da je zelo, če kaj to bila, če to bila vsegače z tvojnji vsegače z 50 jezetnih vsegače. To je izgleda Ergodizij, ki je odlog, ki je ležite danes vse. Znise tudi na vzve. Vspeši vzve k equidistribučnosti. Spojna. Tudi je taj tudi vzve. Vzve je vzve. Kaj ti vzve, pošlo vzve, na equidistribučnosti. Zato da tega trajaktora vzve je vzve tudi na vzve, potem pošli vzve. Kako? Zelo, ideja je, da je tudi tudi tudi tudi tudi tudi tudi tudi tudi A in je, da je komposit in tudi arg, tudi tudi tudi tudi tudi jasno. In da je, da je, da je tudi tudi tudi A . . . . . . . . . . . . . . spet Talk, in budeš lahko vzskakval. Torko je zašljučak, ki se bo, ki se prišel, je vse, ko se je vse, ki se s visim glasbo, zelo, nekaj, nemeč ne se vse. A je, vse je, ko se je vse, ki se z visim glasbo, ima nekaj, ali nekač nekaj, je tako vašljene vse in neči blizve vse, in nič, ki se razdaj, si je izroda do napas. Tako in nekaj, ga je našljička , našli, da je pravi. Zelo. bo naprejniti po vse na vse srečen prilag. Toto je vse. Nimam vse na vse. Srečen prilag je svata. Ko je po vse, po vse, po vse, po vse, po vse, po vse. Nič je vse, bo to skone. Ne vse nič ima? Števa vse. Nekaj, jo, ko je vse. Ko je vse, ko je vse. Nekaj, ko je vse. Nekaj, ko je vse. prevu. If you want the spoiler, so this is the geometric picture that I want to explain today and make precise. But I wanted to give it at the beginning. So you already have and if you're left with something out of today's lecture, this is your motto in parabolic dynamical systems very often. If there is a mixing, it happens through shearing. So this is a phenomenon, which we will sit today for Locality Hamiltonian. sve vsega loka, kaj je kaj pa tega obenazala, ali je to vse obikvitov, in se da jim se barj, in v hyperbolici, tudi je možno izgleda by Markus za barj, ne bo sedlje, kaj ne zdavlja in zrpusti prezum, za konlutator v tjerizu geodezikov in borj, kar možite v težko hore za hodnje nekaj, skaj se vse vse znači. Vse skupaj je nekaj obizs in na vseznači vsezne glasba, boste, ki bom tudi počesil vseznači nekaj glasba na obizs na drugim pljah, in začal je začal in načal je začal vsezne glasba. Vsezneč, ki je začal, začal je začal na nilj vseznači, začal je začal in počes, in nekaj je v terzo para baliki. To je vse čezne. Vse, da bomo počičati od začine, način, da imamo te rigoruse. Samo, da bomo najnemo vse in in in in in in intervalične mapy. Čest, pa in intervalične mapy. so se načal nekaj je odmah početil in bilo prezidentno, nekaj je, ki to ne je zelo, neko. Ok, nekaj ne se ne zelo. Zelo sem zelo, da se nekaj nekaj, pa naj več. Ok, zelo, da jim bo, da mislim, da sem početil, Tako, počekam, da je transversal, tako, sigma, kaj je transversal. In, da bomo izgledati, da je sigma vs. sigma, puanca rej map. Puanca rej map? Ne, ne, ne, ne. Puanca rej first return map. doesn't in so asa dynamic assistem really go back to end of eighty one hundred eighty then becomes one of the first people who the father if you want of the man on the system stand this is the key idea he had to study flows of surfaces long time ago that if I have a flow it might be useful to pass true and met to discretize your continuous time tudi v k probability. In pravično je, da so krateljne dome, počutva, in počutva na tradjektori, če se občutva odkrati tjega. S° o vrstало. Ještje to je počutva. Ja li počutva? Daj, legoj možemo ampak ušlošiti tronoviti odse k dva-djahr. t, t, z arg, do vse, ki povodite razgaz i zelo vse zelo z vrste zelo zelo, da so sem bilo zelo, da, da, da je zelo zelo in sigma lega dotku t, t ki splensityjči glasbo in zelo t, da 50i dotki t, je eno v sigma, kot kaj to opravil. Se načne, ne znam, če vzutaj, obstajte, ki so navet, zato vzal je, da zelo zelo, nekaj ne bom zelo, to je ljuda, kak je tako druga kratka, ah, zelo je spola. Se blizgeš, kako bo, kaj je čas druga kratka z večstvornega počteva, in to je, da je tako počteva, ljuda. Bama je tko, kaj nekaj zelo, nekaj, kako bo, kako je, neske, nekaj, Zdaj sem betač, da sem pripravila. Povej smo izbojene, kjer je linijer. Prejdejo, sem održavljala izbunje obržavljenja, obržavljenja oktogon, kako je opoziti večo izvamašljala in linijer. Profesivnem v obržavljenju drženja, modulov ni gluči. V mojej piste, kaj je? Ne ne bo, da je vse zelo, da je dobro. Zato sem zelo toga seksu, vzelo da je diagonal. To je sigma. To je vse vse lup, da je vse zelo, da smo videli, vse nekaj nekaj nekaj nekaj. Zato, da je toga seksu, in zelo, da vse zelo, da je t. Zelo površenem x na te diagonal. Vselo, da je, da je, da je, da je, da je, Poživajte ploče. Zelo sem vse. B je zelo b, tako sem gleda vzelo vzelo in zelo sem gleda vzelo. Tako je tko, tko je T-O-X. To je izvena razvaja, ko sem vse na kroz seštju. Zelo sem gleda, da vse nožje, kako početimo, kaj pa početimo, bo vse nožje, bo vse nekaj, bo se početimo in pa vzelo vzelo. Tko vse početimo, bo vse početimo. And you can also see that the return map discontinues, so if I, on two sides of a separatrices, this part will take one's history, the other side will do another game, right? So the Poincaré map is discontinues, so all the parpol interval maps here, all the red interval under the c side will follow c, here will follow c and come back on this red interval. So here below, I drew the Poincaré map starting an image. So each, there are four intervals, one under each side, and when I do follow the flow, they come back translated. Ok, so it's a this piecewise translation, it's a discontinues map. And, yeah, this picture I think it's very enlightening, and just even easier picture if you prefer, just linear flow on the torus, you can do it in the linear flow on the torus. In this case, you have two intervals. So here the cross session I chose is 0,1. So sigma is 0,1, maybe close. And these points will just, this 0,1 is bottom and top, right? So these points are immediately back when they are in the top. And they are shifted by an amount which you can compute is, the shift is cotangent of theta by trigonometry. And what about this remaining interval? This one hits the vertical side, the vertical side is glued to the other side, and when I come back, I'm there. So it's basically an exchange of two intervals. And this is an artificial discontinuity, because if you think of it on the circle, once this is a loop, this is nothing else than the rotation. So this is nothing else that x plus alpha mod 1. Ok, so it's a map where I rotate, I add alpha, modulo 1. Ok, so if I open this on 0,1, I see it as discontinues. Ok, good. Ok, so in the case of the torus linear flop, one kind of map is a rotation, but in the higher genus case, I really need discontinues maps. So let me recall you, recall you because you know, but so definition t from i to i, so i, say, is 0,1, or an interval is an IT, interval change transformation, standard IT. First, if there exists i1 dot dot dot id, and you can think of them as open or semi open. So sometimes today I would like to think of them as open, but sometimes you can think of them as one side closed, one side open intervals, such that. I'm looking at the picture above. So i, I can write it as union from 1 to d of this ai. This is this joint union. Maybe if I don't want, if I want my intervals to be open, mod 0. So apart from finitely many points, my intervals fill the interval. So think here. This would be ia, ib, ic, id. The union is the interval. And i is also equal to the union from 1 to d of t of i. So there images, and again this joint. And again mod 0. So the intervals after applying the IT are reshuffle to give me back 0,1. And am I done? No. If I want a standard IT, I want also to say that the map and t from ii to t of ii on each of this interval t is a translation. So I have intervals. So if you want in particular the length of td of ii is equal to the length of i. And maybe sometimes we will call it lambda i. So they are really a rigid translation on each interval. So the map is 1 to 1, apart possibly from finitely many endpoints and acts as a rigid translation on each interval. So it's a piecewise isometry. And if I drop the last request, so you can also get generalized IT. Normally when you say IT, interval exchange transformation, you just mean this translation. But generalized IT, you just have the same, same, but t from ii to t of ii, you want it to be basically injective 1 to 1, 1 to 1 defer. So you basically want allow for distortion. So each interval is mapped to its image in orientation preserving. So basically you just can distort this image. But still you want that your map is piecewise defer and the images give you back the full interval. So this one is a piecewise isometry. This one is a piecewise defer. 1 to 1 piecewise defer. So we already saw through an example that standard IT will arise by looking at Poincaré maps of the linear flow. If I do now a, let's do a lemma, 50, ah, let me do it a bit more careful, 50 locally Hamiltonian. Ah, it's OK. No, I don't need to have my minimality yet. Do I want it? Maybe I'll put it later. OK, that's fine. And sigma transversal and T from sigma to sigma Poincaré map is the generalized IT. And one can choose coordinates, can choose suitable coordinates, coordinates in your charts, such that T is a standard IT. So let's catch the proof. OK, proof. So where do that discontinuities come from? Well, we already know our flow must have seddles, somewhere. Let me draw the picture with two simple seddles. There are some incoming and some outgoing. So let consider basically how the discontinuities happen on the Poincaré map, they really correspond to points which map into the separatrix, and then are split. So what you have to do is to pull back, pull back all separatrices until if they, I don't know how to plot this here, and pull back the incoming separatrices until they hit the sigma. So consider, so basically remove points in i, in sigma, which are pullbacks of incoming separatrices. I hope you understand what I mean if I don't like it too precisely, but I think it's clear, so you remove points which lie on a separatrix and are swallowed by the saddle. Those points will never come back, so the Poincaré map is not defined there. I also want to remove pullback of the endpoints of the segments. So imagine that I have a Poincaré map, at some point I come back and I hit part of my curve, hits the segment and part goes out. This will also create a discontinuity in the Poincaré map. So maybe I'll do it in a different color. So you can also take your endpoints and pull them back, and they might give you some supplementary discontinuities. So this one again I pull it back. I don't know if it comes from here. No, come from here, something like this. So you remove all these points, which give you issue, and then the claim is that T on complementary intervals. So I remove the finitely many points. If I am in a connected component, so if I am in an interval bounded by two adjacent of these points, then I claim that the Poincaré map T is continuous. So basically two points, which are nearby, stay far from any singularity and any other reason to split up. So when they come back, they will come back. If they are sufficiently close, they will come back close. So this essentially gives you generalized 90. I said continuous is actually also maybe differentiable. So if I assume differentiable in my definition, maybe let's say smooth and smooth. This is the sketch of the first part, but I also said that you can choose coordinates where I actually see a standard IT. So far I just got a piece of a smooth map. So if I want to make it a standard IT, then I have to change my coordinates. So first let me remark that if 50 preserves a smooth area form, a smooth new area form, so fully supported without atomic, without atoms, then this implies that also the Poincaré map preserves a smooth measure, new. This is kind of new wheel measure. So it's a general fact in dynamical system. If I have a flow that preserves a measure, you can kind of imagine of kind of building the measure new by restriction. So somehow you can take a small flow box and send one, the component orthogonal to the flow to zero and you get a measure on the cross session. So it's like a Fubini kind of. And this will be preserved. So actually in our case, let me recall you that we have very precise property. We have the Hamiltonian flow is such that the contraction on the generating vector field of the smooth area form is a smooth closed one form. This is what we did yesterday. So actually new is given by eta, by integrating eta along the section. And basically what I want to do is to choose coordinates such that eta becomes Lebesgue. So if I have a smooth measure with no atoms on my interval, I can always reparameterize my interval so that the measure is actually Lebesgue. So I'm not going to write more, but this is just the idea. So in the coordinates where my smooth invariant measure is Lebesgue, then my map preserves Lebesgue locally, so it must be a standard IT. So that's how we reduce to IT. And good. And now we need to... So now we discovered that, OK, Poincaré sessions are well understood maps. And indeed we know a lot of properties of interval exchange maps. Maybe I will recall you later. And some of these properties automatically go to the flow. So maybe I will say this later, but the key thing is that to study mixing is crucially important. The Poincaré map is not sufficient. It's important the return time. It's important how long it takes me to come back to this section. So what I want to do next is to give you a way to recover an explicit description of the flow from the Poincaré map and the return time. So again this is a standard construction in ergodic theory. So this is the special flow representation. See this? Someone? Not few, so good to define. OK, so what is the motivation for this construction? Recover a flow to data from Poincaré map plus return time. So if I only give you a Poincaré map and tells you it's the Poincaré map of the flow, then you don't know what the flow is. But if I give you the Poincaré map and I tell you how long it takes me to come back, then I claim that you can recover enough information of the flow to be able to do all the ergodic theory you want and study. And this is we are going to build this explicit model of the flow which is what we will be working from now on. OK, so first construction. I have a transformation from I to I, say I is 0, 1, and I give myself a function which is positive, maybe even greater than some constant greater than 0, and maybe it will be integrable with respect to the invariant measure of the flow, but I'll write it later. So let me just plot the picture here. So I have t, and I have a function. For now I will draw it smooth. In a second we'll have to draw it with discontinuities. OK, so this is the Poincaré map and this function will be the return time function. And I want to build a flow from a discrete map. So before we took a section, now we kind of have to lift the map to a flow. So what do I do? Let's call it XF. This is basically, I'm drawing the picture in R2, but of course you could have taken any space in the base. So you don't need any measurable space. So here XF would be the area under the graph of F. So SF is the set of points XY in R2 such that X is in I and zero less than one, less than maybe F of X. So this is the region under the graph of F. And this would be my phase space where the flow will act. And now I want to basically build a flow whose Poincaré map is my given t. I'm trying to build the flow whose Poincaré map is t. And the flow is very simple definition. I will draw it first. So what I want to do is for every point in my space I want to flow vertically upward with unit speed until I hit the top. And then what do I do? I want to basically glue up to the base through the transformation t. So if this was X when I get to the top I have to come back to t of X. And then I keep moving up and when I'm back again I'm back at t square of X. And so on, right? So if I build a flow like this by construction the Poincaré map to the base is t. I'm skewing the vertical flow to the Poincaré map. So sorry. If you have never seen this does it make sense? Is it clear? So I'll write some definition. So I call it 50 and maybe let's call it 50 of t,f because it depends on t and f. A special flow need I call it special sometimes some suspension but special flow over t under the roof roof f and first I'll give you informal definition and then we'll write a formula under the roof f is unit speed vertical flow after t. The point of the form x f of x so this is the point on the top s f of x is the point on the top and this one I want to glue it identify it with t of x,0 so this point x f of x is identified to t of x,0 sorry this is I will tell you how it relates to the local chemical flow wait a second so this is for now I will give you a precise statement in a second so just be patient so let me before let me just make some easy remarks so remark is that f t preserves measure mu f t preserves mu cross lebag so im translating with unit speed in the vertical direction so lebag measure is invariant on each fiber on each trajectory so measure preserving up or down and this is indeed mu is indeed kind of the bag that you will measure for this product measure so it's a restriction and okay and I also want a formula that we will use later today so better to write a formula formula so to write a formula formula for 50 yes so I need one more definition so I need so when I'm flowing say I start by the base how much I'm flowing up I'm flowing f of x then I will continue flowing for f of t of x then I will flow for f at t square of x so the amount height of each fiber is the value of the function along the orbit so I need to introduce Birkov sums so definition the Birkov sum so s n of f of x this will be a sum along the orbit or Birkov for angodic sum and this is my notation is equal to the sum from zero so k from zero to n minus one of the function along the flow along the orbit of x so I will only write the formula for a base point but you can just shift it easily and get the formula for any other point so say that I start at x is zero in the base flow and I want to compute flow for time t so where do I get well I need to know how many iterates of the Poincare map I do in time t so I need to count in time t how many full fibers do I cover so define what I will call nt of x and this is the discrete number of iterations in time t let me write it first discrete number of iterations of t in time flow time t this will be the max n in n such that s at f of x is equal so if you think that this Birko sums it tells me the length of the fiber this is how many full fibers I can fit up to time t so then fit t of x is zero so where am I so I'm gonna flow for time t that means that I can fit n of t up and down so I go up and come back n of t times and where will I be I will be above the nt iterate of t and at which height I will be at the reminder height what is left after I removed all this full fiber so the formula is I will be above t to the nt of x of x this is the nt of x iterate of t and at which height will I be t minus s and t of x of f sorry, I spent some time to write this formula because later we will talk about shearing and this formula will be useful to compute shearing so I wanted to have it written up ok makes sense so this Birko sums are counting how many discrete iterates so I relate continuous time with number of discrete iterates and then I have a reminder which gives me how much do I still have to flow and where I will be and ok now I can I can answer your question how this relates to the original flow so and I was a little bit sloppy when I talked about one kind of map why they are defined so morally every time you have an invariant measure you have finite invariant measure you have one carrier recurrence so almost every point comes back to where you are so I wasn't worried so much about defining but now I want to also talk about minimal so if I want to represent say locally Hamiltonian flow through a suspension flow I will need minimality so that all orbits intersect my section or all but finally many so let me maybe let me plot for one second on the side let me plot Arnold flow and recall you that Arnold flow is what I have after I remove the trap it's the flow on the complement of the trap this locally Hamiltonian flow on t2 t2 in the complement is the Arnold flow right is bad right is not nothing outside and imagine so this part I don't care about so I'm interested in what is called the minimal component typically will be minimal so let me say definition s prime inside s is a minimal component of some locally Hamiltonian flow of t so s prime is a subsurface possibly with boundary with boundary possibly with boundary could be everything if the flow is minimal could be really an honest subsurface with boundary in this example it would be a torus with a hole and and the boundary of s prime is union of trajectories of vt usually it would be separatrices and set the loops usually e.g. set the loops separatrices fixed points so in my picture the boundary is the boundary of the hole is a union of a fixed point and I set the loop which is a trajectory of the flow and so I don't want basically my flow to escape through the boundary and the key to call it minimal so every every infinite trajectory of vt on s prime so not in this picture but in the higher genus picture I can still have saddles so if I surface as higher genus there will be fixed points and some trajectories will die into the fixed point so when I say infinite I mean basically not separatrices maybe infinite is wrong actually because it is also infinite on the separatrix but let me maybe say here I mean not separatrix which has infinite lengths I don't know how to say it on s is dense is dense in s prime so this what means minimal so that all trajectories are dense this is what minimal means in this context so you exclude the separatrices and the other trajectories fill your densely your subsurface and then ok and now I can connect the two where are we ok so lemma which is a general fact in some sense it's a general fact of suspension flows so if I have maybe the lemma I could have stated it later ok we start stating it and then we will add to this lemma in a second so s prime minimal component sigma cross session in this case because the flow is minimal the cross session will hit every by infinite orbit will come back to the cross session so I will hit all orbits on my flow and then 50 and s prime is isomorphic to the special flow to the special flow over t from sigma to sigma one car a map this is a general fact I'm not saying anything it's a general fact when you have a one car a session and you want to recover the flow is isomorphic to the special flow under r from sigma to r plus is the first return time function which we will compute now so the lemma maybe will have follow up in a second ok but this is general ergodic theory nonsense so I'm not claiming that my flow isomorphic to this picture and isomorphic let me also write metrically isomorably and I don't want to define isomorphism but this is isomorphism of dynamical systems in measure theory so you want one to one map which is measurable which maps the trajectories of one flow to the trajectories of the other flow and the invariant measure to the invariant measure so everything which has to be which you want in ergodic theory the dynamics and the invariant measure should be mapped into each other by the isomorphism so from the point of view of ergodic theory these two flows are the same so all properties of one will be properties of the other from the point of view of ergodic theory so we will in some sense forget about the local Hamiltonian flow as we defined it yesterday from now on and from now on our local Hamiltonian flow will be a special flow over an IT on each minimal component we are not done because we need to describe this root function so I need to tell you how this root function looks like and then we can work on the special flow but so I will do one more thing and then we will have a 5 minute break but I want to do one, the first example I want to do Arnold and then maybe tell you the general form so where do I go, let's go here maybe should I leave you the special flow maybe then let's go here so here we have so example and remark R has singularities so if if I am on, so the picture of my, I have an IT so near separatrices blows up to infinity so what do I mean so we know that we have some points which never return we know that the separatrices will not come back so for those the return time is infinite so the claim is that my function will blow up so actually some of the singularities could be given by the endpoints but all the singularities which come for all the discontinuities of the IT which come from separatrices which will give me an infinite return time and the return time is smooth so it needs to blow up so the picture is not what I lied when I drew this picture I should draw a roof with peaks so the roof is not defined at separatrices so which type this is an exercise so an exercise which so how do you know how do you compute this return time so you need to do the following so first of all if I am outside or if I don't cross a neighborhood of the singularity then the return time is smooth so I don't have to understand what happens locally near a saddle and I'm assuming that from yesterday I'm assuming that I have a simple Hamiltonian saddle and normal form if you want you can put in coordinates so that this is the Hamiltonian saddle of the Hamiltonian say so I can you can have to do a little computation so you have to plot I don't know a little neighborhood say your saddle you put it in 0,0 and you find charts so the this saddle comes from trajectories of the Hamiltonian flow with that Hamiltonian near a saddle and maybe you can put omega equal to some function dx dgy and and now your first of all so h is constant along trajectories 50 orbits so the orbits orbits are of the form y equal constant over x they are pieces of little hyperbolas, that's my picture and now you have to compute how long it takes you to exit this for example little square so you have in explicit form so yesterday this locally Hamiltonian flow it's Hamiltonian, you can write the vector field and you want to integrate so you want to find the return time in this piece of hyperbola so I'm not gonna do it and if you want so Davide has written in his paper actually he explains this reduction of minimal components to special flows over certain roofs in all details and very nice, so I can recommend Davide it's on anal ripo encara and there is a little computation which also Davide doesn't do which you can find in a paper of front check and myself this is just one reference so must anal append it so Arnold sorry Arnold gets to here he says the form is this so this is an exercise but if you want to read the details so written you can try so what is the statement that after you do this computation what you get that return time blows up let me first write it in words and then in formulas logaritmically and maybe I need to change report so I claim that this peaks this singularity have the following form they are basically absolute value of a log i near xi near nxi which is singularity so you want to say that for example for x greater than xi your return time looks like log xi plus the i is for the i singularity and plus is for the right side times the log of the distance x minus xi in absolute value that's a log singularity that's this graph and ok and maybe let me remark that ci plus depends on the saddle so in my normal form if you put the saddle in the normal form so in the normal form then somehow it's the area form which depends on the saddle so in normal form this v will depend on the saddle and then it depends this depends on v00 something like this v at the origin so ok in any case you can it's only determined by the saddle and should I ok maybe we'll do Arnold and then I will have a quick break what is key let's do the example of the Arnold flow and maybe now I will plot it like this so yesterday we also plotted it like in the torus and I want to put in your picture put the separatrix on the origin I can choose the origin along the separatrix so remember this is topologically like a linear flow and here I have my trap which I removed and now consider as a hole in the torus and 50 Arnold flow isomorphic 2 so we already did the exercise and if I take this as a section 1 km map is going to be in suitable coordinates a rotation, rigid rotation so here t is equal r alpha so r alpha of x is x plus alpha mod 1 and you can think of it as a 2 i t, as an i t of 2 intervals so isomorphic to the special flow over r alpha under the roof so what is the roof singularity here the singularity is at the origin there is only one singularity in the origin so the picture should be like this my function blows up at the 2 end points of the origin at 0 and at 1 which is the other side and it's of the form constant times let's call it c0 log x in absolute value this is the side of the origin and plus let me write c1 for a second and this is log of 1-x this is how it blows up at 1 ok and maybe I'll remove the definition of the rotation so then the same that the origin which is also 1 doesn't come back so in c1 I explore the logarithmic this is the return time the return time is logarithmic so how are c0 and c1 related so ah sorry plus maybe you have some g of x which is some other smooth part which comes from outside of the saddle so this is kind of a local behavior at 0 and the rest will be smooth so so this is what we are saying we are representing the Arnold flow as a special flow over a rotation under a roof with logarithmic singularities what was the key remark of Arnold in this paper where he conjectured mixing so the key remark is what's the relation between these two constants so I just said it's only on the saddle so if I have a saddle and I go to the right or to the left I still have the same type of singularity log times the distance times the same constant but here something a little special happens here when I go to the right I enter a neighborhood of the saddle once when I go to the left I actually enter a neighborhood of the saddle twice so this is the key phenomenon that the trap creates so Arnold key remark key remark is that in this case let's get it right c1 so when I go from the one side I do twice the same thing than 0 so c0 is twice c1 is twice as c0 so there is some kind of asymmetry and this was Arnold intuition for conjecturing mixing and we will start from this in 5 minutes and I will tell you how yes, I will tell you finally the precise statements of the theorems I was some of the theorems I was mentioning yesterday and start explaining the mixing mechanism for, okay so far we have special for our presentation we did it for the Arnold flow after 5 minutes we will write the general picture for higher genus flows minimal components and underline Arnold remarked that saddle loops, these traps introduce asymmetry in the return time singularities 5 minutes so we start in 11.30 so we are building our special flow representation of our locally Hamiltonian flow on a minimal component so we look on a subsurface where the flow is minimal and we built essentially everything we need we already proved that the Poincaré map is a IT in suitable coordinates and we prove that the return time function will blow up logaritmically when there are saddles so first maybe an aside that I was just asked a question and I meant to say it and thanks that you remind me to say it so I'm focusing on simple saddles because we said this is the non-generic this is the generic case in this locally Hamiltonian word but of course you may want to know also what happens when you have multisaddles and this is when you have a multisaddle and also you have another picture which is the picture of stopping point that stopping point so you can take a linear flow and introduce a fake fake singularity so a point which is fixed so this point it takes you infinite time to reach and you have to define locally what this flow will look like but in this case this give rise to power singularities power singularities so i.e. the root function will look like as x tends to xi will look like 1 over x constant ci x minus xi to the alpha where we need to put an absolute value probably where alpha is some power between 0 and 1 so it blows up like a power 1 over x to the alpha and this is again the case which was studied by Kočergin in the 70s Kočergin in the 70s studies suspension flow special flows over rotation and also over it with power singularities so I will focus on the logarithmic singularities but maybe I will tell you also the results in this case and there is a lot of new developments on finer chaotic properties like the spectrum of these Kočergin flows that some people in the audience Kani Gaskin, Fayad have been working on so maybe the last lecture I will mention also some things in this direction so where are we so maybe I should continue the lemma we had there because in some sense it's a continuation so maybe no I'll put one more thing here so we just saw that it creates two singularities which are like we have the same constant in the logarithmic block up but if I have set the loop then I have a picture like ci and the 2 ci so there is this type of picture simple saddle on both sides I have a logarithm with the same constant set the loop this asymmetry so putting everything together now I can tell you what the roof is in the general higher genus case and let me write lemma continuation so we have s prime of some genus greater than one two and more, more interesting and so there are two cases maybe let me write a so if phi t has actually no, ok, has has, ah, no, let me do the symmetric first s, the first case is called s for symmetric has only simple saddles in s prime so if on your minimal component you only have simple saddles the roof in the special flow is of the following form r of x is equal to the sum let me not put on what the sum of la of ci plus log of x minus xi so here I want to look at the singularity from the right so first of all I need to put an absolute value in the log and then maybe let me put this positive part so I only want to look at how you approach it from the right so the notation here is that x plus is x, if x is greater than or equal to zero and zero otherwise positive part just a notation and then plus the sum of ci minus log of x minus xi plus sorry the other side will be xi minus x so this is the distance from the left and it has a constant ci minus in front plus g of x which is most wait a second ok maybe continue here nice to continue on the same board so this is the continuation of the previous lemma because I'm specifying the form of the roof function so the roof function will have logarithmic singularities and where xi are discontinuities of t so it's important that this singularities of the roof coincide with the discontinuity of the it so where the it splits up it's because I hit the separatrix and it's also where the roof is infinite ok so they coincide it's important technically and what is really important of this case is simple settles no sorry s is for symmetric and what is important that in this case each settle produces an equal right and left constant so it's very important that the sum of the constant to the right and the sum of the constant to the left is equal so what I'm saying is that singularity will have the same constant on both sides it might happen that you don't actually see the same constant on the same side but for every constant on one side somewhere else that will be a similar constant equal on the other side so this condition is called symmetry so this is condition so roofs with this property like this these are called roofs with symmetric have symmetric logaritmic singularities ok and the notation I will use not to write everything every time is f belongs to the class of sim log of t so this means that it has symmetric logaritmic singularities at a subset of the discontinuities of t ok so does it make sense so that's a definition, sim log so it again what is f? where? f f belongs this is a class this is a notation for the class of roof functions of that type with this symmetric condition and discontinuities at the discontinuities of t it's a notation, I'm introducing that's not your question it's just that before it was between r I guess it's the same thing f is r ah, f is r, absolutely absolutely you're right thank you I'm also a bit confused about the x plus so where is the plus supposed to be in the function r of x where? here this plus over there in the log so this is just to say that I want to have this is the behavior from the right so if I don't put the plus then I'm getting a c plus also when I approach first of all the log is only defined for a positive argument it's inside the argument of the log yes the log computed at x minus x i plus I can put an extra parenthesis it's the log of, no it's not log plus but it's a variable so it's like this just to say that to make it positive and only from the right so I don't interfere with the left annoying notation it's not so and the crucially different case is case a a is there for asymmetric so if vt has at least one saddle loop it will be on the boundary of this subsurface here we have a little bit of technical conditions so I would like to say typically and I will tell you what this means in a second so allow me a little bit then r of x has the form star so let me not rewrite it again so r of x has the same form but the constants to the right don't add up to the constants so this is our nulled case so think of our nulled where you have c to c and c so this is the nulled case where you have c to c and notation in this case is that r belongs to asymmetric log of t and these are asymmetric log singularities yes exactly so that's exactly why I said typically so you would like to say that once if you have only one trap even one trap you could be so unlucky that somehow I don't know typically means here is that constants for example each saddle so each saddle will produce a constant and you want these constants to be for example rationally independent so there cannot be linear relation that can sell them out otherwise you could get symmetry but for a very rare circumstance so that's exactly a technical point and okay and why do I pay so much attention about symmetry or asymmetry because symmetry or asymmetry is crucial for mixing so now that we have this carefully defined roofs we can state the essentially the theorems which give yesterday picture of mixing in this language okay so maybe I need another more notation okay so so let me take t is an it so if I have an it maybe I will remark that there are two data so if I have this rearrangements of intervals I have a permutation which describes how the intervals are rearranged so this is the permutation which gives the order of t EI and you have lengths lambda I is the length of this interval exchange by t so I will write when I write almost every it you can take as a definition if you want almost every it that I will assume that this is a technical point I need my permutation to be irreducible and don't worry so much this is a combinatorial condition on the permutation so if you have a subgroup of indices 1k which is map to itself k has to be equal to everything so otherwise there is some trivial subset of your permutation which is invariant and you can reduce it and almost every it so I'm assuming this implicitly even if I don't say it and this means almost every the bag almost every lambda 1, lambda d almost every choice of the lengths of the intervals and maybe again recall I know that many people here do know interval exchange so let me recall you though because we use it that almost every it in the sense is minimal which means that all orbits are dense for example as the lengths are rationally independent you can prove minimality and this is due to in the 70s almost every it is ergodic also uniquely ergodic and this is the first breakthrough of tekmular dynamics independently proven by measure in which in the 80 82 probably and and on the other hand no it is mixing so they are not mixing and this is cut-talk in the 80s and I mentioned this result yesterday but translation flows are never mixing and translation flows and ITs are the same paper proves this result for translation flows and ITs on the other hand ITs are weekly mixing and Giovanni Forni and Artur Avila prove this ten years ago maybe now more 15 years ago 2007 more than ten years ago so I am not writing with mixing because we didn't define it but of course it's another piece of the history so all of these are classical results and what about mixing so in some sense these two minimality and ergodicity maybe I will write if you have a 50 special flow over p 50 is minimal if and only if t is minimal and 50 is ergodic if and only if t is ergodic so these properties don't change for the suspension flow just a reality check does this sound I'm not sure if it's familiar to everybody but if you want your orbits for the special flow say to be dense nothing interesting happens when you span the fibers so as long as you are dense in the base you will be dense in the phase space and the ergodicity has to do with invariant sets so if I have an invariant set for the flow it has to be fully foliated by fibers so it will project to an invariant set on the base and conversely so if there are no invariant sets of trivial measure the base there will be no invariant sets of trivial measure for the suspension so these properties do not depend on the roof but mixing is a sensitive then ergodic property which crucially depends on the roof so for example I can have a base which is as chaotic as I wish but if I put a constant roof then in the flow say a rectangle what will happen? Nothing a rectangle will flow as a rectangle so no spreading no phenomenon will happen so mixing depends on the choice of the roof so similarly weak mixing so sorry for but I know that not everybody works in dynamics so even this thing is better to say them and now I can state so theorem theorem symmetric, theorem s and this was put this is in my 11, 12 in my Anna's paper but I should also mention Sheklov for 5 i t and what should I mention for genus 2 and I should mention kočergin for when t is a rotation so this is the symmetric case so for almost every i t and any f in the class of symmetric log of t so for a full measure set of interval is change transformations if the roof has symmetric log singularities 50 is also not mixing maybe I will write weak mixing but weak mixing and ergodic and minimal and so this corresponds to what yesterday I described as the set 50 in the set u was it u1 or u2? probably u1 there was a set of flows which have only simple saddles so those fall into this category so through the special floor representation you can see them like this and then you have to prove that this full measure set of i t will give you full measure set of locally Hamiltonian flow in the suitable measure class and as I said this was known by kočergin in the case of rotation and then there is a special case which was dealt with before my general result and then there is a complementary theorem a which I proved for one singularity and then David extended to the case I will write to the more general case I will write now and Sinai and Hanin proved where the case where t is a rotation so this we have all references and this second theorem is for almost every i t and any f with asymmetric log so this is same picture but with asymmetry fit t and fit t is always sorry, I didn't say it but fit t is always the suspension over t under the roof f write it to implicit for any so you have t f and fit t fit t is mixing and maybe I will add ok, so this on the other hand correspond to the picture of fit t on s prime in u2 so if you have this open sets u2 prime so there is a dense open set in the space of locally Hamiltonian flow with set loops the dense open sets correspond to the constants not cancelling out so open and dense come from you don't want your constant to be you can perturb them to make them not cancel out for strange reasons but in that open set each minimal component of your flow will have such a representation and be mixing and here you can improve and you can have quantitatively mixing, so super inomial estimates also in ravoti and if you know what maybe I will say it again at the very end in the last lecture you can also do mixing of all orders higher order mixing and this is a paper of Aga Kani Gowski and myself for IT and Fayad and Kani Gowski for rotations so I will comment on this in the very last lecture because it's kind of a recent development of all these techniques and it's really exploiting shearing in a very deep and more quantitative way ok so now finally I have precise statements so how are we doing we have some little bit of no 5 minutes where I don't know I started at 15 minutes late so we have another 20 minutes is that all good good so I want to give you the sketch of the mixing ideas in this language and tell you what we are gonna do on for real in some sense on in two weeks time so first key comment let me make I wrote almost every IT here and I wrote almost every IT here so almost every IT maybe I'll make this comment almost every IT means it's a full measure full measure full measure diafantine condition so there is a quite explicit characterization of what do you need from the interval exchange to have mixing or absence of mixing and this is what I will really do carefully on the Tuesday two weeks from now so this will involve I will give you whatever the basic minimum I want to know it will involve let me just write some multidimensional continued fractions and if you know what that means it's rosivich induction so I will define this and whatever I need I want to get to this diafantine condition in the second week so it will be a key topic and this is somehow okay key point for today I just want to stay on the soft side and giving you the heuristic picture of shearing and mixing so we did it at the very beginning on the surface I told you Hamiltonian saddles introduce shearing and mixing happens via shearing if there is shearing but we didn't have yet the tools to study shearing now we have all this formalism setup in order to understand shearing so first of all it's time to okay it's time to do a toy model example okay this will be next week and the next week there will also be plus real plus proofs I will try to do the proofs these two theorems mixing and absence of mixing at least give you the key outline but today we will start with some building up some intuition and toy model say they just have f which is log x and just one side in singularity and so what happens if I take a small interval so j j is a small horizontal interval this is my curve transverse to the flow in this special flow picture transverse to the vertical flow and such why small because when I flow it say up to some time t doesn't hit the singularity so the size of j would be somehow of order 1 over t so in times 1 over t I would maybe not hit and then what do I do I want to flow it and please look at the look at the blackboard this is my baby example but I think if you've never seen it it's useful so you flow up nothing happens until you hit the roof what happens when I'm here actually this point is already gone somewhere else it's already gone it's already moved with t and but this point moved so what is happening is that this curve is reappeering but the shape here is given by the graph of f at that part imagine a little bit moving first point appears and it's already flowing up then the other points appear so what I'm gaining is a little bit of slope in the shape of the graph of f and then if I keep moving at some point this one fully reappears and then it hits again goes with this square again and this part now I'm adding more this is sloped in a unique direction so what I'm doing every time I'm picking up the graph of f and maybe I go very close at some point here so you can see the shearing happening you can see my horizontal transversal is evolving according to bulk of sums of the function bulk of sum, what I'm doing here I'm taking a piece of f here I'm taking a piece of f composed with t then I'm taking a piece of f composed with t squared and so on so now it's where it may be useful to have this formula of bit of suspension flaw that we did at the beginning so 50 of j is made by graphs this formula of t minus s n n f so this was the formula which was giving me the evolution remember we did the suspension flaw so you know if I flow maybe let me write t this is just a formula so when I flow a point I get here so for little segments where n of t is constant I'm just seeing the graph of this function the graph of the Birkhoff sums and this is justification for this picture because you see I'm taking f adding f composed with t adding f composed t squared so you see the Birkhoff sum is described in the evolution in the Birkhoff ok and say that I want to know the slope I want I'm trying to prove that this curves shear so my intuition and my picture tell me that this curves are shearing so shearing shearing is described or slope or slope of graphs is what is shearing I have a graph I just want to know what's the derivative of the direction so I just take d dx t minus let's keep it constant so I'm taking x where and t is constant so I call it n and what is this derivative t doesn't matter it's a translate and I have to differentiate the Birkhoff sum so this is the derivative of my so normally if you differentiate something like this you have a chain rule so you need to differentiate t but remember that t is an isometry my interval exchange has derivative 1 it's a translation so almost everywhere so there is no chain rule and if you forgot the chain rule you didn't make a mistake so I just have to differentiate f so what do I get is minus the Birkhoff sums of x of the derivative along the orbit so I differentiate each I just get the derivative here no chain and I get the Birkhoff sum of the derivative so this is key quantity so you have a concrete way to study shearing so shearing of my curves in this picture is given by the gross of the Birkhoff sums of the derivatives of the roof so now all this just setting up essentially the tools to quantify shearing and see whether there is a lot global shearing and we also saw in this picture that the local amiltonian saddle is producing shearing and this shearing is related to the logarithmic singularities which have where the slope picks up some some slope ok so now if you who knows the Birkhoff ergodic theorem ok almost everybody so let me tell you we know that t is ergodic so we almost every t intervalic change is ergodic from region mesur and so if my function if I have a function which is in L1 we know that everything about Birkhoff sums we know that this Birkhoff sums converge to the integral of g for almost every x when I divide it by n right so we know that the Birkhoff sums will grow linearly as the integral times n but if the life was so easy there was nothing interesting to prove so the issue is that if f has long singularities what about the derivative the derivative what is the derivative of log 1 over x so has 1 over x type singularities so it blows up like 1 over the distance and 1 over x is not in L1 so you cannot apply the Birkhoff ergodic theorem so when I teach to undergraduate students remember to check the assumptions before applying the ergodic theorem here you cannot and indeed it's not true that the Birkhoff sums will grow will grow but I can integrable function so maybe it is time to I don't know if it is time or not I will not make the formal statement but I will make the informal statement so Monday Tuesday in two weeks from now so we will show we will show that under suitable diophantine conditions on the interval exchange we will be able basically to prove some kind of Birkhoff theorem for non integrable function in this special setup so we need to have some precise notion of equidistribution for the interval exchange which will be coming from the diophantine condition and by hand study the behavior of Birkhoff sums for this type of functions and we will prove that let me write informally sn if f is asymmetry and this is very crucial if s prime of x will actually not grow like n but it will grow like constant and this constant will be the asymmetry constant times n times log n for most points I will not want the statement now for most points so most points will grow more than linearly ok so they have this extra log term why do you need asymmetry so in this picture in this picture I had only one side so everything was easy everything was shearing in the direction opposite to the slope if you have two singularities warning the symmetric log if there are the two singularities have the same power sometimes I gain in one sense sometimes I lose in the other sense so there could be cancellations between the shearing and this is exactly what we will prove typically so if f is a symmetric log there are cancellations occurs and you have to be very careful to make this constellation happen which reminds me that there is a part I forgot to tell you in this symmetric case for almost every IT we will have cancellations so we will not have shearing and this will be the crucial way to disprove mixing but let me tell you that there is result by Tchaika and Alex Wright John Tchaika and Alex Wright and they actually built in this class of symmetric there exist an IT of maybe 5 maybe the surface of 5 intervals so I forgot there is a special IT which is ergodic and uniquely ergodic but just barely and for which orbits on the base are not well equidistributed and they spend much more time on one side than the other so even if the roof is symmetric you can still gain shearing and they have a mixing mixing example so almost every is not mixing in the asymmetric case but exceptionally you can still generate mixing from a typical basis I want to maybe finish the picture here because then we will do so the picture here is then again I will make formal this statement but morally this non L1 type of singularity of the derivative give you gross of Birk of Samso the derivative did I erase it? No it's not there so let's wrap it up so if f is asymmetric then essentially what you can prove is that when you take a curve of size 1 over t say and then you can prove maybe I will draw the asymmetric picture so you have your j and you choose it so that it is not broken in time t by hitting the singularity and what do you see? you see that there is slope so slope grows like t slope is order of t log t this comes from the n log n so you are proving that this little curve will kind of the slope of this evolution will grow like t log t and this is still realistic but so then this means that there is a shear the shear I mean how much the two endpoints are far apart the shear will be the derivative is t log t the size is 1 over t the shear will be of order log t so you will have so your horizontal fixed size curve flowing will become a curve of length log t which is more and more vertical I lost everybody so let me plot it so 50 of j will look something like this will look like the more it shears the more it has time to wrap around my surface so this is in this picture what we represented in at the very beginning on the surface so short transversal arcs shear and shadow trajectory of the flow and then we will start here next time but we will see this mixing via shearing so say that you have a small rectangle A you can kind of do some fubini in the horizontal and cover each horizontal line into many arcs throw away those that are too small or throw away those which are not well distributed you can throw some small measure but most of these arcs individually will shear and shadow a trajectory so most of these small arcs will become a long trajectory will shadow a long trajectory of the flow and the flow is equidistributed so trajectories of the flow spend the right amount of time in this rectangle B so you can reduce mixing to equidistribution of the flow by i t so that's the way you prove mixing so maybe I will say just a little bit more but in some sense I will also do the opposite is also true this is the only way to get mixing in this setup and this is how we will disprove mixing so if the base is sufficiently rigid and i t's are sufficiently rigid in a sense which we will explain in weeks actually the only way to get mixing is having shearing so if the base has some rigidity and there is no shearing there is no mixing and that's how we will prove the absence of mixing result so really you should think this mixing via shearing in this contest of entropy zero dynamics is very common and very basic mechanism for shearing and really mixing via shearing and I should say it's not it wasn't invented by me it was already used by Marcos for the horocycle flow by Sinai and Hanin and Kočergin for the previous cases of these results and Basam Fayad has a nice mixing time changes of linear flows that also use shearing in this elliptic dynamics rare examples so then what is really kind of to do is to deal with interval changes in these day-off and then conditions which produce the shearing but it's a good thing to remember mixing via shearing very important feature in many parabolic dynamical systems hope to see you back in two weeks