Transcript

Equidistribution of Expanding Horospheres in the Space of Translation Surfaces - Omri Solan

From Institute for Advanced Study · Episode page · Watch on YouTube

· 1 h 18 min · 8,552 words · 91 timestamped sections

This is the full, timestamped transcript of "Equidistribution of Expanding Horospheres in the Space of Translation Surfaces - Omri Solan", an episode of Institute for Advanced Study published January 22, 2026 (1 h 18 min, about 8,552 words). It mentions 9 people, works and ideas, including Ratner's measure classification theorem, SL2R, HG, GT, S2R.

What is discussed, and when

  1. – 10:00: Ratner's measure classification theorem, HG, U1
  2. – 20:00: Ratner's measure classification theorem, John Smiley
  3. – 30:00: Esen Maser
  4. – 40:00: Ratner's measure classification theorem
  5. – 50:00: U1
  6. – 1:10:00: C^2

Notable quotes from "Equidistribution of Expanding Horospheres in the Space of Translation Surfaces - Omri Solan"

  • The analysist I like measure classification and as a homogeneous analysist my basic theorem that I know and my the bread and butter is ratner's measure classification theorem that classifies environment measures in homogeneous.
    about Ratner's measure classification theorem · at
  • Almost all points of the surface are smooth. However, and they have a they have an atlas by R by R2 line the as the octagon is inside R2. But the vertices have more than 2 pi…
    about C^2 · at
  • And we have one the group of invertible 2x2 matrices SN2 G2 to act on this space. So we have the matrix we can apply the entire matrix on the polygon and get maybe another polygon. So here we applied matrix 11 1 the octagon and we got.
    about SL2R · at
  • Only in important matrices and here the invarian for this to work we need the invariant matrices we need also the diagonal matrix I need to show you those US and TT For the energy classification term we need not only measures in variance to.
    about GT · at
  • Is weird for me if yes this result is with John Smiley and Barack this it's not it's not it's smiley we have a related result with him okay.
    about John Smiley · at
  • I can focus okay I want to talk about one application of this one new application of this film inspired by made by a paper of Esen Maser from 2001.
    about Esen Maser · at
  • How do we do that? So just before that we have U1 invariance only under the Z action. Now we want to get involved which is yeah it's the real un important not only.
    about U1 · at

Who and what is mentioned in this episode?

People

  • John Smiley — Person · Collaborator on a related result; referenced researcher · 1 mention · first at
    Named as a co-author (with the speaker and 'Barack') on a result about unusual phenomena for unipotent (horocycle) dynamics in strata: orbit closures with non-integer dimension / fractal behavior and atypical equidistribution properties.
  • Esen Maser — Person · Referenced author of prior work (billiards counting) · 1 mention · first at
    Credited with a 2001 paper inspiring an application discussed: counting trajectories/angles in rational-angled polygonal billiards with holes, with asymptotic growth ~ c·d^2.

Concepts & topics

  • Ratner's measure classification theorem — Concept · Core topic · 3 mentions · first at
    Central theorem discussed in the podcast, classifying invariant measures in homogeneous spaces.
  • SL2R — Concept · Key argument · 5 mentions
    Group of invertible 2x2 matrices acting on the space HG, crucial for dynamics.
  • HG — Concept · Key argument · 1 mention · first at
    Space of translation surfaces of genus G, central to the discussion on dynamics.
    Any genus and we call the and we collect all the translation surfaces of genus G in a space that's called HG for this talk you should think of this HG as a manifold though.
  • GT — Concept · Key argument · 5 mentions
    Diagonal matrix action discussed in the context of invariant measures.
  • S2R — Concept · Key argument · 5 mentions
    Group of invertible 3x3 real matrices mentioned in the context of semi-simple groups.
  • U1 — Concept · Key argument · 2 mentions · first at
    Unipotent matrix action discussed in the context of Ratner's theorem.
  • C^2 — Concept · Key argument · 1 mention · first at
    Mentioned in the context of the space of translation surfaces.

Full transcript with timestamps

I'm going to take over. Okay. Okay. Thanks for having me. I'm talk about expanding distribution of expanding whole spheres in the space of translation surfaces. So, but I won't start from expanding whole sphere or translation surfaces. Um I start from measure classification as the analysist I like measure classification and as a homogeneous analysist my basic theorem that I know and my the bread and butter is ratner's measure classification theorem that classifies environment measures in homogeneous spaces. So rat's ratner's theorem deals with is lies in the following setting. We have a semi-imple group G. And if you don't know what semi-imple group is, SF3R is great. The group of convertible 3x3 real matters one. Inside it we have a discrete subgroup gamma.

You can you can think of S3 Z. The space G mod gamma is what's what I will call homogeneous space and it has an action of G. Ratna's measure classification theorem deals with an action of specific elements that are called unipent the action of U elements U and G. So that all the values of of the adjint action of u on the algebra are all one which would mean that the map takes n to u to the n is polinomial in the in these adu coordinates. uh in this situation ratner classifies all measures that all probability measures that are invariant and erodic under the U action and declar measure on subomogeneous spaces so that there's some subgroup of G and some subgroup of G and a periodic orbit of this subgroup so that the measure is le on this h this talk is not about homogeneous homogeneous spaces but in a nonomogeneous setting and the nomogeneous setting I want to tell you about is the space of translation

surfaces as the title suggests what is a translation surface so a translation surface is obtained by taking some polomial say an octagon uh and gluing all its edges in a parallel way. So we glue the yellow to the yellow, the green to the green, blue to blue, teal to the teal in a parallel way. Those edges are all parallel and you obtain a surface in this in this circle. It is a genus 2 surface. Almost all points of the surface are smooth. However, and they have a they have an atlas by R by by R2 line the as the octagon is inside R2. But uh the vertices have more than 2 pi degrees around them. You can see that this uh this vertex is glued to this vertex by the teal identification. Then glued to the uh this to this edge here to this vertex here by the blue then glued to the this vertex. And so this this this all the vertices are same. And we see that the total degrees is six pi around uh around a single point.

So this is a translation surface when two translation surf is the same. Uh then they're the same. If we can take the polygon and cut it uh and cut it and glue it together and we when we cut we are allowed to move only by translations because we want to keep the the local geometry locally over near each point. We want to know an R2 coordinates after translation. So, so this is a operation we're allowed to make and so we get a lot of insertion services this as I said genus two but we can get any genus and we call the and we collect all the translation surfaces of genus G in a space that's called HG for uh for this talk you should think of this of this HG as a manifold though maybe or fall is more accurate or but I don't want to get into for this talk it's a man so and okay we want to do dynamics so we need a group action on the space HG and we have one the group of invertible 2x2 matrices SN2

G2 to act on this space. So we have the matrix we can apply the entire matrix on the polygon and get maybe another polygon. So here we applied matrix 11 1 the octagon and we got some skew some skew octagon. One would think that if we apply big matrices we would get thinner and more skew polygons and some of the dynamics is not interesting because polygons go are getting more and more skew but we have the tool of cutting and regluing. So if it got too a skew we can we can perhaps find a way to cut it and glue it again in a way that's less less skewy.

And so now now it seems possible that if we take a big matrix and apply to a certain uh polygon we can cut and glue it to make it simpler. So to make it smaller. So this is a so we can have returns. If we apply big matrices we can return to small polygons again. And this is a theorem and this is formula is result and of mas independently they show that on the space hg of genus g translation surface there is an in okay we need to add volume one because the volume parameter is not instant we just rescale and so in face h of volume one there is a smooth slr environment probability measure which means if we sample randomly a surface with this measure and then we apply a huge matrix we will still get something that's distributed via this measure. So there is a current we can expect to return to the same smallish set again.

So now we can ask all the questions we like. What are the invariant measures on these things? And you have an amazing theorem by Alex asking here and honey. This states the following measure specification theorem. uh we studied a group B of upper triangular matrices inside SN2R and they state that all beam variant and probability measures on on our states HG genus one transient surfaces are smooth on certain sub manifolds and our first slant and smooth on certain sub manifolds. So uh there is a collection of some manifolds this space hg1 so that uh for and for every manifold there's a unique probability measure mm on this manifold so that and those are the only in probability being invariant erodic probability measures

you wrote no varieties is it manifolds is it variet algebraic I'd have to break

I think they are but it's not they didn't show that if Syrian film show that yeah Philip show that it's not yeah I don't want to study the algebraic structure in any in any capacity I I won't talk about this at all for me and some is the same thing please talk Okay. And here we study the invariant measures and they got a result which is very similar to R the uh all invariant and robotic measures are smooth on certain subvariet sub manifolds. Okay. Another result it's maybe a consequent but the year is earlier because of publication uh years states the following for every point x we can take the a little piece of un important of length one u of I should write big u of 01 of X and that's the X us here.

Then we can apply we can choose T randomly from zero to T and apply to our small important orbit a large diagonal and we get a large surface that's getting larger and larger. together showed that that this shape equidistributes inside the minimal the minimal manifold containing X. Now the limit of this measure is being invariant. It's it's GT invariant because we apply GT. So it will it will only move in this direction very in a very small amount compared to this t and it's union variant because most of those expanded uni potent are very long or very long and so the uni potent will shear them uh very few very in a very small way compared to their length.

This limit is being variant. So you would think it follows from asking the highest term but this limit not necessarily alic and so it may give positive mass to some manifold of m and with with the uh they excluded this possibility.

So you meant the average over t I suppose. Yes, we we average over little t and yes this integral from 0 to t ah one over t. You're right. Yeah. Um okay but this is somewhat there's an unsatisfying point here that ratner used only in important matrices and here the invarian uh for this to work we need the invariant matrices we need also the diagonal matrix ah I I need to show you those US and TT For the energy classification term we need not only measures in variance to the unipent matrix but also to gt so that we get the the entire bit.

So that's somewhat unsatisfying and one can ask what happens if we only interested in the u uh in the u action. In fact, the situation is very weird, I would say. So, John Vice showed that that there are closed sets that are closed invariant sets and even orbit closures of non- integral dimensions. So, they're certainly not manifolds. There are some fractiles and from the measure perspective they show the orbits that the orbits that make distribute inside there some unificant orbits that actually distribute along a measure who such that x the original point is not in the support. This is another phenomenon that doesn't happen in homogeneous dynamics and is weird for me if yes

this result is with John Smiley and Barack this it's not it's not it's smiley we have a related result with him okay sorry Um okay. So what do we know about Okay. Um yes there are U invariant measures that are not that are GT invariant that are sorry this is a problem. There are gin sorry there are measures in mu that are u invariant but not g invariant this is a possibility however all known examples are algebraic so so the all the examples we know are algebraic so we currently don't know as chaotic behavior as as we have for closed sets where we have sets of non- integer household dimension And what do we know about those U in variance measures?

Estani Muhammadi shows that if we integrate the take mu we push by GT and and average the push that would converge as t goes to infinity to some a fine me to some to some smooth measure n m. But what about individual GT? So for noticed that that um uh for density one that that in fact it would follow from estim that for density one of of values of t this would for density one this would most of density one of values of t were in fact very close to mm by the of the measure MM and Chik and John and Busama showed that in fact not only most of the time GT mu is close to this measure m but the deviations from mm are of bounded length. So if I I draw here is mm And so the orbit the this GT mu will most of the time will be here but any time that it leaves then the then the set of key is where it's where it's epsilon away is of bounded length. Every excursion away from mm is of bounded length.

What do you mean by that exactly? for every epsilon if we for every neighborhood of m

what's the metric now some we start topology or earth mover distance well the minimal distance of an average joining that's my favorite metric Then uh the set of points where t lives can be is a union of bounded intervals where where the measure is away from mm is a union of intervals of bounded length not the points the times the the the times t where this stuff is more than epsilon from this of boundary length. And today I want to talk to you about a new work in progress where I show that where I show that in fact those excursions do not exist and we have a convergence to an M. And even in the expanding expanding horosphere situation, if you take a horseere and you look on extended widget, those uh extended hores would distribute along m where m is the minimum. Where where's the topological behavior bad here in the counter examples of try and smoking what does it look like in terms of

you're saying measure theoretically seems okay so can you maybe give a tuition maybe tell us the counter example

I don't understand the counter example I understand the count example. I have this crude picture that I can try to tell you. And John, if I'm wrong, you can correct me. I have some maybe keep it question for later. We'll have we'll we'll discuss very close details later in the just one more comment. This serum should have a capital T.

It's not yet eaten. I use the convention that when I'm not if it's not okay I small okay

actually m introduced the theorem with a little t still writing it up I can focus okay

I want to talk about one application of this one new application of this film inspired by made by a paper of Esen Maser from 2001.

Is that a pentagon? That's a pentagon, not a regular pentagon. Okay. Discusses in billyards with rational angles. So suppose we have a polygon a polygonal billyard table. So that the angles are rational. Then the angle is rational and we have some holes in our table and we have a ball inside. We want to count how many ways we can hit the ball in how many angles we can hit the ball. The ball will move and reflect and around the table and after distance less than three would hit exactly one of the holes. Um the this caller states that that the number of angles we can hit it will be c plus little of 1 * d^².

So asintotics t squ is is is somewhat to be expected. So how's this different to counting periodic? I mean like if you have a pentagon where beach computes this asmtoic using the beach surface so it's related

I believe that this is not a regular yeah I understand for a surface for a surface you could use ratnos still right but it's not that

which surface correspond to sun orbiting there no no surface the counting of what he was doing uses that series Yeah, it's even earlier result but and yet this would correspond to some point in some HG that we want to this service would correspond to some this billion table would correspond to some surface in HG and the question is can you prove a similar liquidity distribution theorem for this surface that's coming from the bill Here there are certain surfaces where the where the orbit closure is very simple and so you can comput you can prove the distribution result you want via simpler thems

I guess uh No, no, but even in that case it's not easy to compute. As far as I know, there isn't any other way, right? Alex,

no, you can do you can use distribution of spheres. But can you t maybe not with a remainder?

You don't get you don't get the remainder. Yes, but sorry just just to go with this. Is this analog for periodic orbits known or not? If you have any any any

question I ask is it I didn't get it now. Is it yes or no? Um I don't think it's

it's only for specific periodic orbits. I don't think it's known. Okay. But only for specific it's known.

For specific bill I think it's known but yes for periodic orbits. Why don't your the gives it?

No the gives it the gives periodic orbits. I don't I don't think it's Yeah, it follows from the film in the same way.

It's holes and periodic orbits. Yeah, that was my question. You what is these the usual question people ask periodic or you're saying distance to go to the holeations connections connections if you with marked points saying the mathematics behind it's not difficult.

Okay. Okay. All right. Okay. Any questions to go before we go a bit to the proof? Okay. So the proof is going to rely on on the proof is rely extends RNA's proof for Ratner's proton classification theorem. And let me show you how what's what's the relation between uh my proof and RNA. So Rottner had a dual approach in mind where the goal was to get extra inv to start with measure mu on GM mod that's U environment and get more and more involv

tries to get more and more and more invariants and if we get enough invance say G invariance would be would deduce that our measure mu is the hard measure. But even with less invariance, right? Right now we know how what how to deal with even less variance but okay but we want more environments. That's the that's the approach and has an argument that produce extra invariance but this argument doesn't always work and now this is a dual approach. If the argument fails, Ratner analyze its failure and leverage the failure to get to prove that the measure is supported on some sub manifold. This is also okay. If we show that mu is actually you involved in sub in smaller homogeneous manifold, we would also make the problem simpler.

Yeah. either support manifold or when we push GT mu yeah when we push mu by g minus t it it's it diverges that's another option here and we follow somewhat similar approach now we start for a measure that's us invariant on some manifold m so we would like to say to get more invariance but we have only SL2R acting on on on our space. So we can't have so there's not much more environments to get. However,

along the action the extending directions extending directions and contracting directions I won't draw them what we're trying to do is find some for is instead of invariance that we instead of invariance we will find some foliation of the expanding direction and try to prove that along those foliation the measure is is is the deck and if prove and if and this will be our replacement for invariance and then call it invariance and even though it's not statistic invarian we call it invariance in the foliation direction.

So the first step is obtain expanding and unst unstable expanding directions is all the directions that when we move the dynamics back they contract to zero and stable are all the directions that when we move back death by GT they don't expand. So you want to get more and more invariance in directions which are expanding or unstable. Now we we can also fail as argument may fail our argument. Okay, sorry before failure that what if we succeed if we get enough invariance we get GT invariance and GT invariance means we have both US invariance and GT variance we can use estimates of honey and classify and say that the measure is mm so now what if we fail what if we cannot find a for more advanced selection. In this case like uh like in rat's theorem we would show that taking gt of mu back in time with g minus t will reverge and now there is final case that's not doesn't really appear in ratner

that's what if the unstable case the unstable foliation is just is generated by the by gt and the expanding foliation. What happens if the invariance we have has both is is not contained in the external direction? It projects non-triving to GT but it does not contain GT. In this case we would also we would show that when we take g minus t of mu back in time g minus t of mu it will converge to sum m prime for some m prime which is inside strictly inside. Those two cases cannot happen for expanding for for expanding horospheric for expanding for limits of expanding horospheres.

And so we only get and so the argument only gives us the only outcome is the measure is an empty. Today we will focus only on on the first part how can we possibly get excellent variance in the strength. So we will focus on idea of how to get more invance and we'll explain how to adapt it to our case. Yes. Sorry just in possibility four you also obtain GT and variance of the back limit.

Yes. So today we'll not talk about this uh pos options at all. We'll only talk about how to get more involved in ratner and how in the homogeneous setting and how to get more invariance in our setting. The argument that produced more invariance is called polinomial drift. It was extended by and it was variations of it were used before Ratner's measure classification theorem after by and variations were used by in in paper to prove procification and we'll use a different variation of the algorithms. So to introduce the argument in the simplest possible case uh in the homogeneous setting we work on we work in the setting where we have SL2 mod gamma we have equations of SL2 and in it a measure that's in to U1 to the to the infant matrix U1 and our goal would be to show that U is either are all or supporting a closed on the closed orbit. If we show that it's hard on the SL2R mod gamma then we'll finish. If we

show that it's supported on closed orbit then closed US orbits are circles and classifying environment measures in circles is easy. Okay. The main tool that we use will be built theorem that states that for mu almost every x if you take the orbit the nstep orbit of x it distributes along the measure. So if I yeah have a space we have nstep orbits and for most points the nstep orbits distributes along mu points x that satisfy this condition are called generic and most points are generic the name makes sense how The polomial drift element works that was on purpose.

Okay. Ignore this quickly. Okay. Does it happen on my computer? No. Okay. I don't know why it happens. So we have a point P and a point Q and we draw the orbit along P and Q and we have the P orbit in black the Q orbit in red and we see that they start to drift as in the picture uh they start to drift uh I I draw it like that in the beginning they're very close but as n progresses uh I say that gets further apart. Now if we look at the end of the orbit or the end of the pieces then still the black parts distribute VMU the red parts distribute VMU and they differ by the drift and if this drift was constant and and so the black part is mu the red part is mu but they differ that mu differ from you by the glue errors by the drift And that implies that mu is environmental drift.

So you should read everything with almost. The black part is almost mu. The red part is almost mu. And so mu is almost invariant drift. When we take certain parameters to zero, we'll get actually invariance. This is the idea. And let's try to actually write this in SL2R and see that we get a drift and we get excellent variance. And yes, for this to work, we need those blue arrows to be the same to be multiplication by the same by close to the same matrix. If they somehow explode and some drifts are much some drifts are large and some drifts are small, we would get nothing.

So let's do it in SL2. Uh we start with two close generic points that are Oh no, this picture is wind. I hope this painting technology is not wind. Okay, we start from two points P and Q. These are close. So inside S2R inside S2R gamma they can be related by small matrix and we want to compute how the orbits drift. So we have a computation let's follow it slowly. UN Q is just U N * ABCD * P. Now I before the between the ABCD and P I can end U minus N and U N and so in the middle we got a conjugation of ABCD and here is an explicit explicit I wrote the conjugation.

So and and so we we we got how U and Q is related to U and P and this is by this polinomial matrix in N. And since this is a polinomial it's called polinomial drift. So when does this start being significant? When when this drift starts being big for which n this drift is big. So the diagonal start being so a is close to one d is close to one. So diagonal starts being far from one when n is roughly one / c. And when l is n is of minus 1 / c the diagonal deviates from one. What about the one two coordinates? Then we have here we have here a polinomial a polinomial b is very small.

So the polinomial starts being big when either n * d minus a is big which happens when n is roughly 1 / d minus a or when n^ 2 c is big which happens when n is roughly 1 / square root c. So this happens when n is roughly 1 / square c. It's at most when n is 1 / c which is much earlier than when n is 1 / c. So the first deviation start would start at the one two coordinates.

Okay. So actually there may be a possibility there's no deviation. What if if this is a constant polomial which means that D minus A is Z which means first C is zero and D equals A which means that ABCD is a triangular matrix with A and D close to one and equal which means A and D are one then the polinomial would be zero. However, in this case, we got that there are two generic points P and Q are related by unident elements. If all pairs of closed generic points are related by unident element, that means that the measure is supporting a closed orbit.

If this does not happen, I want to show that we get uh US environment in the US direction for FDS. Uh how do we do that? So just before that we have U1 invariance invariance only under the Z action. Now we want to get involved which is yeah it's the real un important not only the integer important so you take n to be the maximum value such that b times so that this polinomial is less than s for this n h between for for little n between 1 - epsilon n to n since this since this is a polinomial and in the end it obtains s polinomials don't change too quickly we obtain that this polinomial will be close to s for all these values of n and we reduce the measure and and so and so we get in this picture those drifts are all close to us.

So the measure is is US environment. So that is how um yeah so this is how we we computed the drift and we saw that since it expands polomally uh it behaves well and we can get extra environments. So here we only showed how to leverage U1 invariance to U S invariance integer un important to really important. Same ideas can leverage really important to the really important to get diagonal environments and and in larger semi in larger GM guns it would get there would be more steps and for this argument however we're here for the nonhomogeneous case. So let's try to see what happens when we try to apply this for anomalous case. Well on its face it seem ludicrous. Here it works because it's polinomial. It's called a polinomial lift. We won't have anything like that. So what can we do?

Okay, before I go to the next slide, the space of the space HG is too scary and has annoying linear algebra. And so for the rest of the talk, I'm not going to talk actually on the space HG because we want to avoid annoying linear algebra. I want to so I will present some baby case some erh some other manifold that satisfy all the conditions we need to explain the arguments

m we'll call it m. So what conditions do we need on the manifold to make the brief work? or to see the failure of the polomial lift and then to fix it. So suppose M is a fivedimensional function and suppose that with an action of SL2R since it has an action of SL2R at the tangent space we have the tangent space of the manifold and inside them the tent space of the acting group which is SL2R. So inside the t base of m we have the time base of s2 r and the the le group the le algebra sr and we assume there is a a direct summoned a which we call vx and and we assume that vx is is invariant under the action. So sorry we assume that vx is invalid under the action and we get that tm deco composes as the acting direction and some transversal direction to to you to the way we'll use estim result is that we will assume that on space m we have only one ban variant measure which we call M a unique B invariant

measure which is also SL2R invariant and and smooth and smooth so it's it is obtained by a constant volume form on SL2 on SL2 and so it's smooth it requires a volume form on TN which is SL2 plus Vx so it's obtained by some constant volume form here and an invariant volume form Finally we need to assume that the gt action on vx has positively responds. So vx de composes uh as vx plus plus vx minus and this de composition is gt invariant and and there are the exponents which say that for every vector in one of in vx plus minus uh the size of the vector exp uh gt expands the vector by e to the plus minus lambda 30 but there is a little little of one here that I going to assume given the unique environment measures there is always such a um always okay I assume this that this expansion holds to almost or um and the goal will be to use the to use some drift argument to show that every US environment measure is in fact MM

is the fact that your low dimensions important in fact what five dimension no it's just the minimum case that I can apply the drift. Our enemy is going to be V. Vx is going to be our enemy. The non the the variation in how fast V expands is going to be our enemy. And let me explain what that means. If we have a vector in the expanding direction, we can ask how how does the how fast GT expanded. So we know that it's on average uh lambda t. However, I want some something some almost sure answer which is not lambda t. So what do I want to claim that if we assume some uristics we should get here lambda t plus o squ t and so this size is lambda t plus gshian and why is that uh this uh this logarithm can represent the sum of smaller of of consequent quotients. So uh gtv over v uh log gtv over v is log gt g1 v over v plus log g2 v over g1 v and up to gtv over g tus 1 v.

If the original point X is random, those have equal distribution. And if we assume there are somehow that far apart, they're independent, we would get a gshian result. This is just heristics but it will but they actually okay in the setting of uh of translation surfaces in some cases it was proved that the deviation is uh lambda t plus gshian. So this is not an emptyistic and let's show how this ruins the polinomials. So what do we want to do? We start from a U which is U invariance and we and our first attempt in the polinomial drift will be to get invariance in the V plus direction. We want to get X this extra invariance and we want to generalize the polinomial drift.

Now apply to normal lift means we start from close from two close u generic points in n that are by a small by a small vector on the times man. So we'll call them. Okay. Now we need to draw this. We start from two generic points P and Q. So this is Q. Maybe it's also not there. This is P. This is Q. And they differ maybe difference by a vector D no. And now we're going to compute the drift between the U orbit from P and the U orbit at Q. The U orbits are of size E to the T. So let me draw out the orbits. We have some orbited P, some orbit Q and we want to comput the drifts.

Specifically, we want to compute the drift at at u of e to the t s at u of e to ts uh times t and you get a vector which is u of e to t s * v drift. So as we say we can decompose u of e to e to t s is de compos the product of matrices uh gt us s g minus t which in the picture the gt action contracts uh contracts the hospheric spheric orbits. So if we here we have an orbit size e to the t here we apply g minus t we get an orbit of size one. So here okay so we want to compute this gtus g minus TV. So it starts with understanding G minus TV then US G minus TV and then GT us G minus TV.

So G TV we denote by uh V prime. This is V prime and here we'll have U of S V prime and this U of SV prime will be expanded by GT to this vector to U to TSV. So you would have this expansion. Now to understand how U of SV prime expands we need to decompose it into we need to decompose it into U of S V prime. We compose it as the corresponding part. The expanding part plus uh the contacting part.

Now the expanding part uh would uh would expand the now we have we need to show it's it's it holds that the expanding part is the the expanding part and the contracting part of similar size or at least the expanding part is larger and so we'll have

so so we have that when we apply GT T to that H GT to that we'll have GT times U S E prime plus this expands at speed e to the t e to the lambda t plus okay of square So we say that those deviations so it's this times the size of the problems. Okay, we see those deviations have a square root t um square root uncertainty in the size. They maybe there is square variation in the size. though we always point in the expanding direction. So the direction part of the argument holds but the size completely fails. The size is there is a squ variation.

The way we fix the size is by changing the space on which we act. Okay. of exercise by not taking all values of S but somehow we will condition on the on the subset of S's we take and take only S's on which we have control over the the expansion of the of V plus component the way we do that will be by changing the states we act on so Okay. So let me introduce you the new space as we act. We start from the space F. F will be just the projectivization of V on M. And this since V is S2 environment. This st has the access to our action has an asset to our action. Now mu can be lifted to a measure mu prime on f. However, the way to lift mu to to envir measure on f is by just selecting in every point in every point the direction which is the expanding direction.

So, mu prime is just the listening to mu by selecting the expanding direction. But now comes the real the real space I want to introduce. The space is obtained by not taking the union over over the spaces over view over the projectivization but over the spaces themselves without zero. So we think of the length of the vector. We we think of this in logarithmic coordinates. We think of this as f * r where r is log of the size of the vector. So this and this will differ only by a bit by by a sign. And again this has an SL2R action on this space.

On this space we can and now we can't simply lift mu prime to another probability measure on F because it's very infinite. There's it's F * R. is an entire space of escape and deviation. This r was the entire reason we constructed fail. So what we do for every direction we define the every direction in our space. So this set is isomeorphic to R star and we find and we put on it the measure the how measured R star here and more explicitly every interval of the form V prime alpha V prime gets measure log alpha and then we we denote by M the infinite measure which is integral integral over those measures over over n prime.

Now this in measure is going to be the measure we use on the measure we use the the the brief argument to prove that mutil is invol isn't violent and more explicitly we decompose mutil to us erodic measures mutil is while infinite it still has an erodic position it can be demposed is integral of measures that are involved and are not thoroughly decomposible. Now this no okay so the first problem to applying the the the polinomial drift will be that build of sing theory doesn't work because those erodic components are likely infinite luckily we have hops rational theorem hopic theorem states that in our It would say that in our possibly infinite measure here, this is Fil in our possibly infinite measure for any set that we take. If you restrict infinite measure mux, mutil x to this measure and take almost every orbit, the restriction of the orbit to the spite measure set distribute along the

restriction of the measure of mutx to the final measure set. Now when we try to apply the the the drift argument over over this space fill and not over and not over m. those red arrows that originally the size deviated between. So first first returning to a bounded sets near P. Return to bounded set near P will cover only a little will be the the part on which we return to will be our conditional set S on which we have some control. And over this set the red arrows won't have the this slow t randomness that we had before because in our in our new space f sizes of vectors in the expanding directions are well behaved.

Okay, that is the solution to the polinomial drift problem that that we had and on this space f one can make the polinomial drift argument work. Thank you. Thank you questions. state. Would you say prove about periodic orbits? This

periodic orbits for what? The horosc flow or the um translation flow on a translation surface.

I'm an old guy. I think I'll ask a question in between and then people can ask a question.

Okay. Um, you also use this work of uh, Philip and Alex, right? And Alex, can you explain well this fits into the unit?

Yes. Thanks. So here we claim that the size of V prime is roughly the size of the expanding direction. We we okay we had this V here. It moved to V prime here. It moved to us V prime here. Here we decomposed it uh g of s v prime to the plus direction and the minus direction and we related the plus direction to to the size of the original vector which was the size of v prime. But the relation between so this de composition is governed by the expanding conducting direction here. The expanding conducting direction here in V.

The V plus the expanding direction is is is fixed along the uh the whole the whole sphere. But the V minus direction changes. And the V minus direction is the important one because if we want to show that a constant vector projects with almost it full size to to the plus direction, we need to show it's not close to the minus direction. So what we need for this argument to work is that the minus direction changes along whole spheres and that so in in our baby case it just follows from the atom of exponents and the uniqueness follows from the exponents but in but in general in H in HJ one would the a similar result would follow from Philip, Alex and Alex. And but here what we need is even something stronger than that. We need that uh this contracting direction is not constant when we restrict to this S's where we have certain returns here. However, there is also a consequence of the result and

uses uh the results as a black box. Do you want to ask your question or I just remember Alex that one of the big problems that you were trying to understand years ago was for periodic and for rational billions this constant ups t² periodic orbits.

Yes. And it seems to me you're saying something new about it. Yes. What? I think it's solved now.

Okay. So, yeah. Okay. So, you just stated it in terms of holes. Yes, I said it in terms of

and it's you've solved it in complete general. It's always constant times C^ square. The concept depends on the on the shape but other than that it's it's a C plus little of one.

That's wonderful. Can you say something about the constant? No. Um okay it it will follow from certain integrals of certain functions on the on HG but I can't say anything on this.

This is for any rational bill. Yes. Why did you state in terms of these holes? That's what is mathematical.

So it depends only on this depends on project. Yes. And yes, John, do you have any understanding of what um are the obstructions to a horror cycle of a given length being close to being equidistributed

of a given size being close to? So should I say a more precise question or is an erotic argument it's hard to push it to to fin thesis but if I just say it's really really long then it's almost a measure almost involving measure and and in my understanding like the the US orbits that are The U invariant or the U invariant measures that are not M invariant that are not S2R invariant can come in the following way the so there are two reasons for them and uh that's actually okay and that's what I stated in this in this uh strategy section. So there's one reason that when we take them back in time they they diverge and one reason when we take them back in time they distribute along the smaller subvariet and and those are the reason they I can think of the obstructions of a USO it's not to distribute inside. So, you think there's a chance that some statement like that follows from your work with

the appropriate? Well, not so hard, right? It's probably just some kind of um limiting argument that you say that uh it's just this argument. I I'm not sure uh if it follows directly from expanding horses but I believe it follows from the from the argument as well. Yes. Just to connect. So you have these examples not yours but the examples of these orbit closures which are weird. We have like integral dimension. If you start from one of those points on such an orbit and then you push it by the measure, you get by your results like a nice limiting measure.

Sort of these examples probably although your examples if you push them with contrast probably don't behave pathologically. I think so. So I I think Omry is saying the following which I don't think there's any examples that contradict which is you take a horoscotic measure you push it forward by the geodessic flow it would distributes to a horoscotic measure you pull it backwards it equidistributes to a horoscotic measure that's possibly the zero measure and the forward guy contains the backwards guy not necessarily properly but it contains it and I think that's basically basically what he said and I think that works with all known examples if that made sense.

I think you answered my question that I asked at the beginning but don't know what you told me. Um, so let me try to answer them to say to draw on the to draw what John said and to draw what John what John said and and uh so suppose we have two measures two a fine manifold M prime inside M. So this is okay this is m and this will be m prime. So this is m prime and this is m and we start from a point which is extremely close to uh to m prime but it's not in m prime and I take a horosphere on it and push if I understand correctly. Uh so when we take this horse and push this horsefield would deviate from m prime at third speed and this speed of deviation from m prime is something that's very similar to this v that we saw to vectors expanding in v vectors expanding in the comp in the complements to emp prime and if I understand john's examples They happen when tiny proportion of the the orbit does deviate from m prime

but uh does devate from n prime and we see a fractal behavior of deviations from n prime of deviations from n prime but it's only tiny tiny proportion of the originals and those are the fractile examples if I understand them correctly. Um I that's the mental image I have of them and when and if somehow the deviation is fixed is is was deviating a constant speed then we would actually get an invariant measure. But since we deviating constant speed, it seems more likely all the examples that I and and from what I know the examples are algebraic. And so those deviated measures are algebraic if the deviation still is constant. And those are the intermediate those are the intermediate things between m prime and n. Yeah. Anyway, this this is not quite a very well posed question, but like if you look at the applications of random SQL distribution theorem of a uni pool and then there is basically never used instead what she

uses this I mean so she does say that you know for almost for every point the distribution uh but um there is a ser but some even if you Say burkov if you just forget whatever I'm saying just use burkov theorem then burkov theorem says that for almost every point you have distribution and you can make it uniform by saying there is some sort of set of almost full measure in which you have uniform liquid distribution and uh now the dynamo theorem is essentially does that for unifor but they they ex explicitly give you a set of almost full measure they tell you what it is basically you have to avoid some low orbit with lower dimensional things and I wonder if we like you know how close can you get to some theorem like this in this setting that it would solve all possible problems you know just try to find the formulation of the dynam

I don't know I I don't know anything about that so in your nice uh t² result there if you want to prove a polomial error term There is this like maybe as negatively hard as proving like a polomial error term ops conjecture or something like that

for what I know yes I you need effective distribution and that would give you here a polinomial

probably at the moment completely hopeless in this setting yes nothing is hopeless people thinking I mean people are definitely thinking about these things

but um I I held completely open it is completely open. It's not completely hopeless. Um probably we can continue asking questions over dinner or coming days but I have no idea what the plans are for dinner. So

yeah. So we made a reservation at Metro North um 6 at 6:30. Yes. Any show of hands who would want to come?

What time did you say? 6:30. Okay. or think again.

Questions about this episode

What is discussed in "Equidistribution of Expanding Horospheres in the Space of Translation Surfaces - Omri Solan"?

In order: 0:00 Ratner's measure classification theorem, HG, U1; 10:00 Ratner's measure classification theorem, John Smiley; 20:00 Esen Maser; 30:00 Ratner's measure classification theorem; 40:00 U1; 1:00:00 C^2.

What is mentioned in this Institute for Advanced Study episode?

9 people, works and ideas are mentioned, including Ratner's measure classification theorem, SL2R, HG, GT, S2R, U1. Each one on this page links to its own page with every episode that mentions it.

How long is "Equidistribution of Expanding Horospheres in the Space of Translation Surfaces - Omri Solan"?

The episode runs 1 h 18 min; the transcript is about 8,552 words in 91 timestamped sections.

More from Institute for Advanced Study

About this transcript: the episode's spoken words as captioned on YouTube, split into timestamped sections. Reclip.Me locates every person, work and idea in the text, keeps the time it is first named, and quotes the words around it verbatim. Last updated January 24, 2026.