SPEAKER 1: So our first speaker this morning is Anton Zorich, who will tell us about the Lyapunov exponent from the Hodge bundle and diffusion in periodic billards.
ANTON ZORICH: Thanks, everybody, for coming in the middle of the night.
And thanks to the organizers. I realized that this conference resembles rather ICM than a standard conference. And I realize what an effort requires an organization of such a conference. So thank you very much. I appreciate. And special thanks to Karen.
I decided to present measured foliations and Teichmüller dynamics on a particular problem. So the plan of the talk is to present first to formulate the problem and to tell the answer. Then I will tell the technology. And I apologize in advance. Plenty of colleagues know this technology much better than me. And I learned it from them. So I will be telling some elementary-- extremely elementary stuff. Forgive me.
Then we'll arrive to some more than technology. And at the end of the talk I hope to present the solution of the problem-- not only the answer but the solution-- using all of this technology. And so here are first examples of measured foliations.
So they are everywhere, in particularly the solid state physics. So if you can see their electron transport in metals, then the particle stays on the surface of constant energy defined in the quasi momentum space on the one hand. And on the other hand, it has to stay in the plane orthogonal to magnetic field where in the quasi momentum space and not enter in the coordinate space.
So the particle really stays in the plane. And from a mathematical point of view we have just a Z3 periodic surface in R3. We cut it. We slice it with parallel planes. We get lines. And this lines are electron trajectories. And the question what are the trajectories.
And if you quotient everything-- if you quotient you R3 by their translation-- by their integer lattice, you get a compact surface in a torus-- in a 3 dimensional torus and a measured foliation on the surface. So this measure foliation is exactly the object we're interested in.
Actually, this problem is my first problem in mathematics. It was suggested to me by Sergei Novikov in '82. And I was 20-- young and ignorant. And I even managed to make some progress on this problem, proving that for open and dense set of directions over the plain, minimum components of the corresponding foliation are just tori with holes. So the trajectories for open and dense set of directions go along straight lines.
But of course, very, very fast I got stuck. And at this time I thought I'm the only person in the world who is working with measured foliations. So I just abandoned the subject for 10 years and came back only 10 years later. And one of the main mathematical regrets of my life is to ignore for this 10 years that actually, by the end of the '70s Bill Thurston has already created beautiful theory which I was not aware of.
So however, it's-- I suggest another problem actually also suggested by physicists in the beginning of 20th century by Ehrenfest and Ehrenfest so-called wind-tree model to sort of present measured foliations. And this problem-- the initial problem is stated in terms of a billiard. So you have a billiard in the plane with periodic obstacles-- periodic rectangular aligned periodic rectangular obstacles. You send a billiard trajectory. It travels in your plane, reflects. And the question is how it propagates in the plane.
So this rate of propagation is called diffusion rate. And here is an old theorem of [FRENCH]. It says that the diffusion rate is the length of piece of trajectory power 2/3. So here's one definition of the diffusion rate. Another one is as follows. You just let your trajectory travel full time T. You'll get the curve in the plane. Take a diameter of this curve. Take the log of diameter and divide by logarithm of time so that if the limit exists or this ratio, it's the diffusion rate.
So as a matter of fact if you make a random walk in the plane, you have square root of time. Here you have t to power 2/3. So your guy is not completely drunk but slightly dizzy.
So a typical feature of these kind of problems is that changing the height and the width of the obstacle will get different billiards. But it does not change the diffusion rate. So from the point of view of this theorem these guys are also sort of generic rectangles. So you have your lined walls with very narrow holes-- still the same t power 2/3.
Or you can just make a charcoal plate, leave very narrow corridors- no matter-- t power 2/3. Well, of course, you have [INAUDIBLE] logarithms. I cannot control the coefficient in the front of t power 2/3. So it's sort of rough estimate. But still it's curious that these 2/3 does not change.
One can change the shape of the obstacle. For example, one can consider wind-tree like this. And-- or for obstacles-- so if you plant your wind-tree with different spices of trees, you can change the rate, which here in this particular case depends only on the number of their right angles. And it tends to 0 as you can see. So double factorial means that it's the product of all even numbers. And this is the product of all odd numbers up to 2m and 2m plus 1.
So this is the way to suppress the noise in the room. You can plant plenty of trees like these which might be a really tiny but perfectly aligned. And since their diffusion rate tends to 0 you'll suppress the noise.
Also the same as in the previous problem-- the diffusion rate does not depend on the particular shape of the obstacle but only on the number of corners. So we can make a carpet-- complicated carpet like this. If m is the same, it would be the same diffusion rate.
AUDIENCE: [INAUDIBLE] question.
ANTON ZORICH: Yes.
AUDIENCE: If you rescale, can you build a continuum with it?
ANTON ZORICH: I have to think about that. So now I want to abandon billiards and arrive as fast as possible to surface foliations.
ANTON ZORICH: Yeah, sure.
AUDIENCE: Those numbers-- 4m and 4m minus 4-- is that some theorem of geometry--
ANTON ZORICH: Oh, yeah. Yeah. This is sort of Euler characteristic. Yes, this is true, though. So to pass from billiards to surface foliations, let's start with the simplest billiard-- a rectangular one. And this is-- well, this-- the construction which I'm telling is called [INAUDIBLE] construction. And it comes actually from mathematical Olympiads for high school students.
Instead of reflecting a trajectory, one can reflect a billiard table. And the advantage is that the trajectory continues just as a straight line you see. So this angle is the same as this angle. And we can unwrap the billiard table along trajectory. And we'll get a straight line from our trajectory. I suggest to cut to-- so the initial trajectory in the rectangle goes in at every moment-- it goes in one of four directions. I suggest to assign a color to each direction and color the corresponding trajectory here.
So we have 4-- in this particular case at any point we have 4 possible patterns. Let's quotient over equivalent parts. If we-- yeah, and I forgot to tell that if we pulled this billiard table, we'd get original trajectories here. So it's sort of equivalent. So now, instead of considering this infinite unfolding, let's quotient over equivalent [? patterns. ?] And of course, we'll get a torus. We have four patterns. And our identification give a torus.
And our trajectory now becomes a winding-- irrational winding line-- oh, rational binding line on a torus. And again folding it back, we get the initial trajectory. And this is the way people study billiards and rational polygons. They unwrap a billiard to a compact surface. And trajectories are unwrapped to a measured foliation on this complex surface.
So in our wind-tree problem-- in the initial wind-tree problem when the obstacles had the rectangular shape we can take-- well, there are again 4 directions. So we take 4 copies of our billiard. We can make an additional quotient over Z2. And we get a surface which is glued from 4 tori with holes and their identifications between these holes. So we get a surface of genus 5 instead of a torus as in the previous example and a measured foliation on this surface of genus 5.
And now if we want to study the diffusion rate, one can check that what we have to do in this terminology is the following thing. Take a long piece of leaf of your measured foliation-- so straight line foliation, join the ends and count what is the intersection number of this cycle C, which depends on the length of trajectory with the cycle H, which is represented as linear combination of this basic cycle so cycle V.
And it tells you how far you went to the right or how far you went up on the initial plane. So you can immediately recognize the initial problem in terms of this language. Just you have to see it for-- I don't know-- 30 seconds observe this picture carefully. And you realize that this is exactly absolutely equivalent.
So from now on and almost up to the end of the talk my goal would be the following. I have a measured foliation on the surface. I want to study how their cycle obtained from a long piece of measured foliation behaves in homology. So this is the problem.
To study this problem, I have to present some background, which is Teichmüller dynamics. And I sort of follow ideas of Bill Thurston. The first observation, which is absolutely elementary but very deep is that surfaces can wrap around themselves in a complicated way.
So here's the first map of a surface-- the most elementary map of a surface. To itself, it's a Dehn twist. You take a torus. You cut it along a circle. And you twist it by complete turn and then glue back.
So this is one way to see this map. Another way sort of more algebraic is to consider an integer matrix with determinant 1. It maps an integer lattice to itself, which means that it maps a torus, which has a quotient of R2 over integer lattice to itself. So oops. Pardon me.
Yeah, so this is the first way to represent our surface diffeomorphism. This is another way using the lattices. And one more way is just observe this thing as follows. So if we take a fundamental domain over the initial lattice, which I'll say a square torus, it maps our fundamental domain to this parallelogram. And then we can take a pair of scissors, cut this triangle, and glue it here, because this side is identified to this side.
So by this cut and paste we get the initial square. And we see that we map our initial torus to the initial torus. Now concerning this map I can come-- so one of-- the other way to see it is as follows. Let's apply this transformation continuously-- not immediately a matrix like these. But let's put here parameter t. And let's decline our parallelogram continuously.
What do I observe here is that we get a curve in the space of tori. If we decline just a little bit, we get completely different torus in terms of flat metric. But after time 1, we get back to the initial torus. So we get the closed curve in the space of tori.
So this is the most elementary example. The next one-- let's compose two Dehn twists, one along meridian and another along parallel. Well, nothing complicated-- now the matrix-- well, it's again can be represented as before by a linear transformation.
Now, the matrix is product of the corresponding matrices. And their fundamental domain is mapped to this fundamental domain. And one can repeat the construction with cut and paste. Now it would be slightly more complicated. But we can cut and paste. And we get, again, a square torus.
And what is important, which is sort of obvious from algebraic point of view-- we just have a map-- an integer map, which maps the lattice to itself. What is slightly less obvious from this colored picture is that everything is extremely smooth. When we are cutting and gluing nothing happens. It's just two different ways to unwrap our torus into a flat [? pattern. ?]
So when I cut and paste I carefully-- so I take care that say-- you'll see this side is red. This part is red. This thing is glued to this thing. This thing is glued to this thing. So I carefully preserve the gluing rules. There is no discontinuities in my procedure.
Now, I arrive to pseudo-Anosov diffeomorphism. So this is automorphism of a surface now. The measured foliation appears. Consider eigenvalues of this matrix and corresponding eigenvectors. So there are two eigenvalues. One is more than-- is greater than 1. One is less than 1. So our linear map extends everything in direction of unstable eigenvalue and contracts everything in direction of contracting eigenvector.
So we get the surface automorphism, which preserves a pair of foliations and expands everything in one direction of one foliation and contracts everything in direction of the other foliation-- of transverse foliation. And in surface automorphism like this, it's called pseudo-Anosov diffeomorphism.
I should say that for many years I knew Anosov, and I was curious who is pseudo-Anosov.
Now I know that this pseudo-Anosov is actually Bill Thurston. Now I know.
So and one more trivial remark is one can consider this map not as a sort of immediate map but as a continuous deformation of the initials torus. So we have pair of foliations on the initial torus. And we can start the form the initial torus contracting in direction of one foliation and expanding in direction of the other foliation. When we apply it a little bit we spoil the flat metric of the initial torus. We get a completely different torus.
But our observation says that after an appropriate time t0, which is logarithm of all eigenvalues, this expansion and contraction brings us back to the initial torus. So once again we get a closed curve in the space of tori. And we'll see that this closed curve actually-- well, one can see that this closed curve is actually like a geodesic in the space of tori. So pseudo-Anosov diffeomorphisms define closed curves and actually closed geodesics in the moduli space of Riemann surfaces. So this is a really important observation.
I'm speaking about space of tori. Let me represent the space of tori. How can one parameterize tori. So there are several models which one can use to describe a flat torus. One of them is a lattice in R2. If you have a lattice in R2-- I don't want to distinguish between tori which can be obtained by a proportional rescaling. So let's rescale the torus in such a way that the shortest geodesic has length 1. So the corresponding lattice vector, the shortest latest vector, has length 1. And also I will turn the lattice in such a way that it becomes horizontal.
So this vector represents the shortest lattice vector of all lattice. This vector is turned and normalized to be placed of the standard vector. Let's take the second one. It's somewhere. But the second shortest vector of the lattice is-- sorry. By convention, let's take it in the upper half plane. It's our choice.
Well, it is supposed to be longer than the first one. So it's supposed to be outside of this unit circle. And also, it cannot before it. So this is definitely not the second shortest vector. It should be between two parallel lines. So we get a domain like this. And it's really the fundamental domain in the space of lattices. So this is the space of tori. It's a nice surface.
So one should notice that this side actually is glued to this side. This arc is glued to this arc. So we get something like this. We get a sphere with a cusp, which responds to tori which are very narrow and very long. And yet, it has two orbifold-like points which correspond to tori which have extra symmetries.
And actually, this picture represents perfectly well other moduli spaces for higher genera. They are never compact because your surfaces can degenerate. Well, I am speaking about moduli spaces of flat surfaces.
You can have arbitrary narrow and arbitrary long cylinders. So the space cannot be compact. And also, it has some orbifold-like lossy, because some flat surfaces may have extra symmetries which are not typical for all flat surfaces. So this is really a good image of what is space of flat surfaces.
To convince you that everything which I was-- yeah?
SPEAKER 1: Yes. Yes. Yes. So I just arrived to it. Yes, of course. Of course, when you are in genus higher than 1-- different-- not higher, different from 1, you cannot have a perfectly flat metric, but almost. So here's an example of almost perfectly flat surface of genus-two. Let's take an octagon, a regular octagon, and let's glue all the opposite sides.
So first I want to convince you that we get a surface of genus-two because, well, we can see it as a square from which we chopped out the corners. So if the corners wouldn't be chopped out, we would get a torus. Now when we are starting to glue it, we first get a torus with a hole because the corners are chopped out.
Then, well, I cheated a little bit. I turned their rectangular hole by 45 degrees. But when I glue this side to this side, as suggested, I get a torus with two holes like this. And now I have to identify these two holes. I get a torus with a handle. So I get a surface of genus-two. And our surface of genus-two in endowed with a flat metric which comes from the flat metric on this [? pattern. ?]
And the only particular thing is that it's really perfectly a flat metric except at the single point which comes from the vertices of the octagon. All these vertices are identified to one point. And at this one point, the total cone angle is the total angle which we have here, which is 6 pi. We have a conical singularity.
But still I really insist that this flat metric is wonderful and mimics the flat metric on the torus because the flat metric one the torus-- well, one of the special properties of this is that it has trivial holonomy. If you make a parallel transport of a vector along a closed curve, it comes back exactly in the same direction. Here it's the same story, because all our identifications are just coming from parallel transport. At no point, we twist it whatsoever.
So our flat metric has trivial holonomy. And if you take some direction at some point and claim, this is the vertical direction, you can transport it to all the other points of the torus without any contradiction which might come from closed trajectories. And from now on, I can see that this vertical direction is a part of the structure.
And by the way, another hidden structure here is complex structure because you can consider the hyperbolic metric in the same conformal class. Or you can see it differently. You can declare that your pattern lives in the plane, in the complex plane, in the plane C. Since the choice of vertical direction is part of the structure, I am not allowed anymore to turn my [? pattern. ?]
So the only thing which I can do is to parallel transport it in the plane z. And I can see that the coordinate z there, complex coordinate z on my surface. And even more, I have a holomorphic differential dz, which is well-defined on the resulting surface because all their identifications are just parallel translations, z equals z plus constant.
But I will sort of-- mostly I will hide the structure. I will try to remain with flat surfaces [INAUDIBLE]. To be honest, this is absolutely equivalent language. One can speak in terms of spaces or flat surfaces. One can speak in terms of the moduli space of pairs, complex structure plus polymorphic one-form.
Group action. Now we arrive to a key point. So this spaces or flat surfaces are extremely [INAUDIBLE] structures. And one of the most important ones is that there is a huge group acting on this space. And this huge group is a SL(2,R). Just unwrap your flat surface into a polygonal [? pattern. ?]
So here, you can see that my polygon is spatial. The sides are distributed into pairs of parallel sites of equal length. So it really represents a flat surface. I can identify this thing to this thing by parallel translation, this thing to this thing, and so on. I get a flat surface as in the previous slide, exactly with the same properties.
And also, I can put another plane. Only after parallel translation I have vertical direction, distinguished vertical direction, which is part of my structure. Now apply in your linear transformation, I prefer the ones with determinant 1 because they do not change the area of the flat surface.
You get a new polygonal [? pattern. ?] So new flat surface with a flat metric over the same type. The fact that pairs are distributed into pairs of parallel sides of equal length is preserved. So I can glue back the surface.
We have an action on the space of all flat surfaces. And part of this action is a one parameter subgroup will just shrink everything horizontally and expand everything vertically with the same ratio. And the key theorem on which based all the story is a theorem due to Howard Masur and William Veech is that the action of the group of SL(2,R) and this diagonal subgroup is ergodic with respect to the natural finite measure on each connected component on every space like this.
So I can see the spaces of flat surfaces with fixed types of conical singularities. In the example which we considered, there was a single clinical singularity with the angle 6 pi. If you go too high, in genus-two, you're going to have two possibilities. Either two conical singularities with angles 4 pi or one conical singularity with angle 6 pi.
When you go to higher genera, well, there are more combinations, but basically everything is described by partition of 2g minus two integers. It's like degrees of zeros of corresponding holomorphic one-form. And of course, our linear transformation preserves this property. And on every connected component of a space like this, the map is-- well, this action is ergodic.
And now I would like really to insist why I'm so excited about this theorem. In plain terms, this theorem says the following. Start with a randomly chosen octagon like this. Well, with this property, the pairs are distributed [INAUDIBLE]. The sides are distributed into pairs.
Then, if you shrink it horizontally and pull it vertically for an appropriate time, you can get arbitrarily close to a regular octagon, for example. Looking at this picture, it seems contradictory. Well, you never get close to a regular octagon. Right?
But I am not speaking about polygonal [? patterns. ?] I'm speaking about flat surfaces. We're allowed, as we did for the torus, we are allowed to take a pair of scissors and start cutting and pasting at any time we wish. And you can cut and paste-- well, of course, preserving their identifications-- as much as you want.
So the theorem says that if you choose the time in a smart way and if you choose your cutting and pasting in a smart way, then you can get arbitrarily close to any point, for example a regular octagon. And this is much less trivial. So this is really highly non-trivial theorem.
AUDIENCE: Is that what you've got [INAUDIBLE]. This one is periodic?
ANTON ZORICH: This surface is not yet the same.
AUDIENCE: This one is not going to be [INAUDIBLE]?
ANTON ZORICH: I'm saying almost. Arbitrarily close to. Not exactly. So if you take the surface by random, if you-- yeah, in my example. But the torus, it would be exactly periodic.
AUDIENCE: Arbitrary should be almost every--
AUDIENCE: Almost any [INAUDIBLE].
ANTON ZORICH: Oh, yeah.
AUDIENCE: What does arbitrary mean?
ANTON ZORICH: Yes, arbitrary was it wrong? This word arbitrary is at the wrong place. At least it wakes up everybody. Yeah. Wonderful. It worked. It was not on purpose.
I presented sort of 2/3 of the necessary background. Now, I want to come back to measured foliations. So remember that the object which we want to study is the following. We have a measured foliation on the surface. Now I can formulate differently. We have this flat surface and straight line foliation.
And we want to study the following thing. We take a very long piece of our straight line trajectory, close it up. We get a cycle in the first homology. I want to study, what is the behavior of this cycle on the first homology when the piece of trajectory becomes longer and longer and longer.
Let's start, again, with a torus and to study how can we choose when we close up the cycle. One of the ways is to choose a transfer segment. And as soon as our foliation hits the segment, we just join the endpoints of the piece of trajectory.
So our foliation will hit many times this transverse interval. And we get a sequence of vectors in the homologous space. And the simple theorem says that there is a well-defined limit if you normalize this cycle, which becomes longer and longer by number of iterations, then as n tends to infinity, we get a limit and this is true not only for torus.
But there is a theorem-- of Steve Kerckhoff, Howard Masur and John Smillie and I am happy that two of the authors are here-- that for any flat surface, directional flow in almost any direction is uniquely ergodic So this means that in almost any direction, we have asymptotic cycle. And this asymptotic cycle is one and the same for all trajectories in this direction. Good. This is asymptotic cycle.
I suggest a procedure which allows us to calculate this asymptotic cycle faster in the case when our foliation is spatial. Suppose that our foliation corresponds to a pseudo-Anosov map, like this foliation and direction of expanding eigenvector for this matrix. Take a closed curve and apply any closed curve, say meridian or parallel of the torus, and apply our map of this closed curve.
So these are two sample curves. And when we apply the map g in homology, the corresponding cycles, of course, they map two vectors which follow they're expanding eigenvector. Just because, forget about everything, it's elementary linear algebra. You are applying a matrix like this with one eigenvector which is clearly dominating. So basically all the vectors except the one which goes in the unstable direction, they will get stretched in direction of the top eigenvalue.
Well, the same behavior is true for these first return cycles. They get stretched along the eigenvector corresponding to top eigen number. And the way to see it-- well, it's not a proof but it's sort of a very convincing argument. Consider this first return cycles segment.
To see them, it's easier to conceive that they're [? pattern ?] of a torus which correspond to this first return map to a segment. So I take a segment, I repeat it periodically. And I see how my foliation comes back to this [? pattern. ?]
I forgot to say, and this is important. I suggest to take this transverse segment in direction of unstable foliation. So when I apply my map once, since this cycle goes in direction of unstable foliation, it just is mapped on this subsegment of itself. And these guys are stretched in this direction. So we get another sort of building exactly in the same lattice. But this building would be stretched, would be shrinked and stretched.
And the first return cycles to this longer segment are mapped to first return cycles of the shortest segment. And we have just seen what happens to the cycles when they are mapped under this transformation.
I forgot to say that when you start playing with measured foliations, you run into a map like this almost immediately. So I was not an exception. And fortunately for me, I was not too naive to think that I'm the first one who discovered such a wonderful map. So I started to ask, what is the name of this map.
And I was lucky enough to visit a conference at the [INAUDIBLE] in '92. And I ran into John Milnor and I dared to ask, do you know, by chance, this map. And he said, yeah. This is called interval change transformation. There is a person who knows just every bit of information about this. And this is Bill Veech.
And this was [INAUDIBLE] just got a brand new mathematical reviews on CD ROM. It was just top of technology. I went to the library and I printed out their math reviews for all papers containing the word interval change transformation. And I've been working in this domain ever since. So I use this occasion to thank John Milnor, who sort of directed me to the right spot.
So there is nothing special for the torus. You can do exactly the same with other surfaces. You get similar story in your first return cycles when you choose this transverse interval in the direction of unstable foliation and your flow. This is direction of your-- sorry. This is contracting foliation. This is expanding foliation. The first return cycles through a longer interval are mapped to first return cycles to shorter interval, which is a convincing argument that we can apply renormalization.
To see the behavior of these very long first return cycles to a very tiny transference interval, you can pretend that-- well, I can't. Our foliation corresponds to Anosov map. Well, then we have to use the matrix, the corresponding matrix, and their homology induced by this Anosov map. And everything is fine.
And now, let us use the theorem of Masur and Veech, saying that, actually, this flow which contracts everything horizontally and expands everything vertically is ergodic, meaning that if we follow this flow for a long time, we can pretend we have map. And there is a theorem, which is called multiplicative ergodic theorem, which says that this analogy is much stronger. It says not only that you can imagine an ergodic flow as a flow, a sort of repetitive flow along a very long closed curve.
Moreover, the holonomy, there is sort of a virtual matrix which mimics a matrix of Anosov map. We'll see it in a second. And now, to finish this story with asymptotic cycles, so if we trace for surface of high genus, we trace these points in their 2 [? g ?] dimensional space representing first return cycles, well, we know that they expand in asymptotic direction. But we're interested in more detail, so let's put a condimensional screen in our space and project everything to this condimensional screen.
We'll see that the projections of these vectors to the screen are accumulating along a line. So all these vectors live [? morally ?] around a two-dimensional plane. We know this plane. It's spanned by two top eigenvectors of corresponding pseudo-Anosov map, if we have a pseudo-Anosov map, or something analogous in [INAUDIBLE] situation.
And then we can put a screen of codimension two, repeat the procedure, and we'll have-- never use new devices. We get a picture where deviation in principle direction is described by some special power, in the next direction, by the next power, and so on.
These powers are called Lyapunov exponents. And the formal statement says that the deviation from the-- there is a flag of subspaces for-- now, I am not in a pseudo-Anosov situation. I am in almost any situation. We have a flat surface, and almost any flat surface-- vertical foliation, horizontal foliation-- you construct the segments of first return cycles, and they behave as if they mimic the situation with the pseudo-Anosov map.
And this rates of deviation from asymptotic direction, they are one and the same for almost all points of each space like this. They're called Lyapunov exponents. And they correspond Lyapunov exponents of the Hodge bundle along the Teichmüller flow. And this will be explained in a second.
I should say that I have cheated here, because the fact that there are strict inequalities everywhere here, is two deep theorems. The first strict inequality is proved by Giovanni Forni in 2002, and all the remaining strict inequalities are proved by Avila, Viana. So the theorem, as strong as it's presented here, incorporates this two very deep results. And they're highly non-trivial.
So I have to explain what are these Lyapunov exponents, finally. First, forget about all the details which were presented. Let's consider the general setting. Suppose you have a vector bundle, a flow on the base, and suppose that this flow is ergodic, and suppose that you are lucky and you have a flat connection in the vector bundle.
So you can play the following game. You take a fiber of your vector bundle and you float along the flow, and from time to time, it comes close to the initial point. You close up trajectory and you get a matrix of holonomy. And you continue. Your matrices start to multiply. And you can ask yourself whether there is some conversions, in some sense. And yes, there is.
There is a well-defined matrix of mean holonomy, which can be defined as this. So I multiply by a conjugate, and now I have symmetric matrix. I can take roots, and this limit exists. And the eigenvalues-- or rather, logarithms of eigenvalues-- are called Lyapunov exponents.
So Lyapunov exponents, they mimic eigenvalues of a pseudo-Anosov diffeomorphism. You pretend that your flow is periodic, and then you have just one pseudo-Anosov diffeomorphism along a very long trajectory. Eigenvalues, these are the guys.
Now, what is the manifold and what is the vector bundle, in our case? The manifold is our space of flat surfaces. The flow, I already presented. It's the flow which is contracting everything and expanding everything. And the vector bundle is the Hodge bundle. It associates-- so we're working with space of surfaces. We can consider their homology-- or cohomology, rather, in this case-- of our surface, and we get a vector bundle.
And we have a natural flat connection with this vector bundle because, since it's not an abstract vector space, it's cohomology, we have integer lattice cohomology. And there is the only way to deform our vector space in such way that the lattice follows the lattice. And this is the flat connection, which is called Gauss-Manin connection.
Now, one more remark. The holonomy matrices here come from diffeomorphisms of a surface, so they preserve intersection form. They correspond to symplectic matrices. They have extra symmetry of eigenvalues, and that's why the spectrum of Lyapunov exponents is symmetric with respect to change of the [? sine. ?] Remember, my Lyapunov exponents are logarithms of eigenvalues, not eigenvalues themselves.
OK. Now, in the way I presented this story, one can get the impression that-- so the only persons who gave some input in this story is Lyapunov, Hodge, well, famous German mathematician collaborators Gauss and Manin, Thurston, Maser, and Veech. And myself.
Well, this is not quite correct. And actually, this is dramatically wrong. The whole story is based on-- it's really, in the sense of sequence of good luck, coincidences, and incredibly strong results, which are obtained in the last 20 years in adjacent areas.
So the relation of the Lyapunov exponents to the deviation spectrum, and the first idea how to compute the Lyapunov exponents, it's our [? games ?] with Kontsevich in the mid '90s. As I already mentioned, the inequalities for Lyapunov exponents, that's two deep theorems of Giovanni Forni and of Artur Avila and Marcelo Viana.
Now, to apply ergodic theorem, you have to know ergodic components. So you have to know connected components of the spaces of flat surfaces. And the classification is it's our work with Kontsevich.
Now, volumes are computed by Eskin and Okounkov. And this is a highly non-trivial computation. So volumes of this space. And they are needed to get numbers.
So you don't want to have just qualitative theory. You want to have [? at least ?] 2/3 at the end of the day. And to get these 2/3, you have to know volumes.
Also, there is a whole marvelous story of counting closed geodesics on individual flat surfaces, and this is developed by Veech, and by Masur and Eskin. And one of the key ingredients is the Siegel-Veech constant, which is responsible-- it's the coefficient in the exact quadratic asymptotics for the number of closed geodesics on the individual flat surface. And this is developed by these people.
Also, to get 2/3, you have to ask [INAUDIBLE] to classify all SL(2, R) invariant manifolds in genus 2. And I am not mentioning plenty of wonderful results around. I am mentioning all those which are ingredients in this wind-tree problem mentioned at the beginning.
So there are plenty of things-- well, there are results of John Smillie and Barak Weiss on unipotent flow. There are results of Martin Moeller and the [INAUDIBLE], and Lyapunov exponents of [? cyclicals. ?] There are plenty of marvelous results here. I'm a lucky man.
So state of the art. We arrive to state of the art, and to the solution of their wind-tree problem. First, Lyapunov exponents, it's a great thing. But usually, you cannot compute them. It's something completely transcendental.
Here, in this particular case, we are so lucky, and there are so many beautiful structures that it is possible to compute-- well, not the individual ones, but the sum of Lyapunov exponents. And here's the formula. It is expressed in terms of these cone angles plus this mysterious C area. This is the Siegel-Veech constant, which comes from this theory of counting closed geodesics on individual flat surfaces.
And I cannot explain what is this, but I can give a formula for this at least which gives a sense of what is going on here. It's some polynomial expression in terms of volumes of [? strata ?] or flat surfaces which correspond to possible degenerations of flat surface normalized by the volume of the initial [? stratum. ?] Well, with weights, which is explicit combinatorial factor rational number here. It should be completed, but it's doable.
So it's something which can be-- so this formula shows that if you know the volumes, you can compute this explicit combinatorial factor. You can compute the sums of Lyapunov exponents. By the way, how one can-- yeah.
AUDIENCE: Some of the ingredients that you mentioned suggest that it might be possible to get out of the symmetric component, and you also [INAUDIBLE] about that in the exponents. Is that [INAUDIBLE]?
ANTON ZORICH: The ingredients which I show suggest that one can get other symmetric polynomials in Lyapunov exponents. Probably. I forward your question to Simon Philip that he has an idea about this. And, well, he is next door. But it's highly non-trivial because, if you play just elementary linear algebra and you consider, say, from the initial bundle, which products and so on, you have just sums of Lyapunov exponents.
AUDIENCE: Is it also believed that the individual exponents are transcendental, in some cases?
ANTON ZORICH: I'm afraid that yes. But since recently, I have a hope that one cannot compute them, but estimate them, so you found some bounds--
AUDIENCE: The coefficients [? would ?] be rational?
ANTON ZORICH: Yes. So that's why-- we are not com-- so Simon Philip tried to convince me that something like this-- well, some other functions and Lyapunov exponents-- are probably computable. If it's true, that would be wonderful. I do not quite understand his arguments.
So how one computes the volumes, by the way. One of the way to compute the volume is to compute integer points in the huge ball. And integer points, in this particular case, are flat surfaces tiles with squares.
And in a particular case, when our surf-- yeah, and I forgot to say that, up to now, I considered flat surfaces which are obtained by just from a polygon by parallel translations of the corresponding sides. One can consider slightly more general metrics where the holonomy is not completely trivial, but where you're allowed making a parallel transport of your vector flip it to the opposite direction. So it's slightly more general situation, and it also corresponds, this time not to holomorphic one-forms, but to quadratic differentials.
So in particular, one can consider [? mirror-morphic ?] quadratic differentials with, at most, simple poles on CP1. And starting from now, this is what I am interested in for a while, for five minutes.
And, well, to compute volumes, you have to compute how many surfaces with prescribed corner angles of genus, and surfaces of-- how many flat spheres with prescribed corner angles, tiles with at most, I do 1 zillion squares you can find. And, well, spheres like this are branched covers over standard spheres. So actually, the equation of computational volumes is equivalent to the equation of computation of corresponding [INAUDIBLE] numbers.
So this is the flat spheres which you are constructing. You're given 1 zillion [? patterns ?] like this, and, well, the question is, how many connected creatures with prescribed corner angles you can construct. Thanks, Gabby, for the photo.
OK. So there is a theorem, and the whole theory is for this slightly more general flat surface. Absolutely parallel. Everything works. You have to correct slightly the formula for the Lyapunov exponents. You see this rational function slightly different.
The trouble is that, for quadratic differentials-- well, there is an analogous formula for the Siegel-Veech constant, but there is no formula for volumes. Well, there is formula for volumes. There is a work of Eskin, Okounkov, and Pandharipande. But unfortunately, the formula-- well, it's rather an algorithm. And at the current stage, the algorithm is sort of inefficient. We don't have values. This algorithm doesn't give numbers.
One can bypass this trouble, in one particular case, in the following way. Consider this formula in the case when g is equal to 0, as in the previous slide, when we have a sphere. Sphere does not have first homology.
So on the left, g is equal to 0, and on the left, we have 0. So we get an immediate expression of the Siegel-Veech constant in terms of this thing. This is the expression. On the other hand-- [GRUNTS]
Yeah. And this expression, and the fact that we have, also, expression in terms of volumes, allows to compute the volumes. So let's define a function like this. So again, double factorial is product of all even numbers of all odd numbers.
So just simple function of integer number n. And here's the formula for the volumes in genus 0. Just the product of all of these functions with respect to all the centers, which encode the [? cone ?] [? angle ?] similarities.
This formula was conjectured by Maxim Kontsevich about 10 years ago. And using approximate values of Lyapunov exponents which were obtained by computer, he first guessed formula for the volumes of this thing using the Siegel-Veech constant, and then made ambitious guess for general case, and yeah, we proved it.
So the proof is that, on the one hand, we have this expression for a Siegel-Veech constant, and on the other hand, we have expression in terms of the volumes. So you guess the answer, and you verify that this answer fits this plenty of relations.
And for a long time, we considered this as a sort of a-- the formula for Lyapunov exponent was not proved, and so on. We're trying to prove it [INAUDIBLE]. We knew the answer.
No way. We tried at least six different combinatorial approaches. No way. The only proof which I know is this word proof through Lyapunov exponents, which were not designed to compute volumes. Everything was designed to do vice versa.
And by the way, even proving these relations, it's not completely trivial. To prove some relations between multinomial coefficients, unfortunately, you can write here "generating functions" and use some functional relations on these generating functions to prove this combinatorial relation. So these guys are multinomial coefficients. So multinomial coefficient like this.
And this is just one multinomial coefficient which is spoiled by this factor. And here, the sum of all complementary multinomial coefficients, which-- complementary because, if you sum this thing and this thing, you get this guy. And this guy and this guy get him.
OK. So this is just to show off, and to show that Lyapunov exponents can be useful for computing volumes. And we have just seen that volumes is the same thing as computing some [INAUDIBLE] numbers.
Now, I promised the state of the art, and here's the state of the art. So we write a fantastic theorem which is just proved by Alex Eskin, Maryam Mirzakhani. They prove that the closure of any SL(2, R) orbit is suborbifold, and that in special coordinates on the space of flat surfaces, it's just in a fine subspace.
And even not just arbitrary fine subspace, but some very particular one. There are plenty of constraints. I don't want to go into details, but it's quite limited. And also, the invariant measures are the natural invariant measures corresponding to these [? fine ?] subspaces.
Now, a development over this story is effective methods of constructing these orbit closures. Again, I am not stating the exact result, but there are collection of theorems due to Alex Wright, and Alex Wright and collaborators, and just other people. So it's a story in [? plain ?] development, which really allows you to-- I give you a flat surface. You push the button, you find the orbit closure. Now, this is a dream, but in many cases, this works already.
One more theorem. It's sort of non-commutative [INAUDIBLE], Kerkhoff, Masur, Smillie theorem. For any given flat surface-- I insist, for any given flat surface, and in almost all directions, are Lyapunov generic. So if you know the orbit closure of this particular flat surface, then you know Lyapunov exponents, and all this story about the asymptotic [? flank ?] in almost all directions.
And now we arrive to a solution of the wind-tree problem. So first, you notice that any wind-tree flat surface is a cover of a surface in the hyperelliptic locus in genus 1, and that the cycles h and v we are interested in are induced from this surface. So we use the problem, actually, to genus 1.
Then you prove that your orbit closure is this hyperelliptic locus. And hyperelliptic locus is related to genus 0. For genus 0, we know the volumes and we know Siegel-Veech constant. We can recalculate the Siegel-Veech constant for corresponding hyperelliptic locus in genus 1. And since, in genus 1, there is a single Lyapunov exponent, we can compute it.
So this is 2/3, or this rational number which it was shown in the initial slide. And also, now, I can explain the joke with the old theorem. So in a sense, up to the theorem, after this theorem of Eskin and Mirzakhani, every occasion when you managed to find these 2/3, or something which works, was sort of a lucky coincidence. And starting from this theorem, it's a point of hard working and technology. Morely, this theorem says, you want to know everything about this particular flat surface.
Work hard. Find the orbit closure of this particular flat surface. Work hard. Get estimates or answers for Lyapunov exponents so you have the answer.
OK. What's next? I have another one minute, I guess, right?
First, I would like to suggest a collection of open problems. So some problems are solved. There are plenty of new problems. Very challenging equation is-- Well, now we know that orbit closures are extremely nice, classify them. Nobody knows whether classification is like classification of simple Lie algebras, or simple [? finite ?] [? groups. ?] It might be nice, or complicated. Nobody knows.
But there are indications that it's highly non-trivial, because there is a new invariant orbit closure-- well, new invariant submanifold which is just found by Maryam Mirzakhani and Alex Wright of completely mysterious origin. We do not understand where it comes from. Maybe, meanwhile, they already understand. Estimate individual Lyapunov exponents. And for this, one has to relate the individual Lyapunov exponents and this sub bundle, these terrible sub bundles, measurable sub bundles of Lyapunov subspecies to holomorphic sub bundles of the Hodge bundle.
Prove conjectural formula for asymptotics of volumes, when genus goes to infinity, at least conjecturally. And in the simplest case, this conjecture already seem to be confirmed by very recent work of Dawei Chen, Martin Moeller and Don Zagier, that the formula for the volume of all the spaces simplifies enormously, and in the limit, resembles something which was presented in genus 0.
Very simple formula. It allows to compute Siegel-Veech constant, Lyapunov exponents. But it should be completed.
Also, one has to find values of volumes of quadratic differentials. A very challenging problem is relate everything to algebraic geometry, and not do it analytically, but to express the area in terms of intersection. Construct and interpret-- well, find interpretation in terms of intersection theory, say, in the spirit of [INAUDIBLE] formula for [INAUDIBLE] numbers.
And another problem, forget about Riemann surfaces. Why not consider families of other complex varieties and play the same game? We have Hodge bundle, and so on.
And even more interesting is to go backwards. OK, we have families of, say-- some results are already obtained with experimental results for some particular families of [INAUDIBLE]. But are there equivalent of billiards which are beyond us? This would be nice to find them.
And to finish, I want to present a group photo made at the last Oberwolfach conference in March 14. As you see, most of participants were extremely young. So I think that the average age of participants was, I don't know, 35.
Extremely bright, young people. Wonderful results. Just breath-keeping.
And to my mind-- well, some of these young people probably do not realize to what extent what they are doing relies on the results of Bill, and of what he did. But, well, they continue. They are bright.
So you have young guys, plenty of nice problems. I'm sure that you will create plenty of beautiful mathematics. So you are next. Thank you.
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Asymptotic behavior of leaves of a measured foliation on a Riemann surface is governed by the mean monodromy of the Hodge bundle along the associated trajectory of the Teichmüller geodesic flow in the moduli space. As a consequence, recent progress in the study of the Teichmüller flow (inspired by the fundamental work of A. Eskin and M. Mirzakhani) and in the study of the Lyapunov exponents of the Hodge bundle along this flow leads to new results on measured foliations on surfaces.
Following ideas of V. Delecroix, P. Hubert, and S. Lelièvre, Anton Zorich of the Université Paris 7 Jussieu shows how to apply this technique to description of the diffusion of billiard trajectories in the plane with periodic polygonal obstacles, June 24, 2014 at the Bill Thurston Legacy Conference.
The conference, "What's Next? The mathematical legacy of Bill Thurston," held at Cornell June 23-27, 2014, brought together mathematicians from a broad spectrum of areas to describe recent advances and explore future directions motivated by Thurston's transformative ideas.