SPEAKER 1: [INAUDIBLE] by Hee Oh of Brown University. She's going to talk about circle packings and ergotic theory.
HEE OH: So I'd like to thank the organizers for the invitation. It has such a great conference, and I hope my talk does not ruin it for the continuation of these great talks. Yes, so I'll talk about some counting problems for circle packings. And then these circle packings will include actually those circle packings I tried to draw over there. And in particular, actually the solution of actually this counting problem for circular packings, it introduces actually some new measures for some hyperbolic manifolds, which I think are probably more interesting than this concrete problem. So I'll try to talk about what the new measure is, actually, in my talk.
So let me start with actually talk about what is the construction of these packings. I'm sure actually many of you realize this is what's called Apollonius circle packings. So I'll start with the theorem of Apollonian, which is the basis of the construction of those Apollonian circle packings. This is theorem Apollonian. I think it's about 250 BC.
So the theorem says that for three mutually tangent circles in the plane, there are precisely two circles which are tangent towards three circles. There are precisely two surfaces tangent to all three.
So let me try to just draw this picture. So here is the configuration of three mutually tangent circles. And there are two circles which are tangent to all three circles. You may try to draw maybe one circle here, and then one circle like this.
But we to prove actually we can really actually draw these two circles. And I did a proof of this theorem, which is probably not original proof of Apollonian, using Mobius transformations. So I'm sure everybody knows Mobius transformation. But this Mobius transformation, in fact, actually plays a very important role actually during the whole talk. So let me remind you.
So abcd is element of PSL2C. So this is two by two complex matrix with a determinant one. And it acts on the extended complex plane just by linear press sure and transformation. So abcd z goes to az plus b over cz plus d. And here, z is a point in a complex plane or infinity.
And [INAUDIBLE] of this Mobius transformation is that it maps circles to circles. And whenever I say circles during this talk, lines will be included in circles. And it also preserves angles between circles. So in particular, the Mobius transformation maps two tangent circles to two tangent circles in the plate.
So now, to prove this Apollonian theorem, we can just pick any tangent point, p, here, and choose any Mobius transformation which maps p to infinity. So take any Mobius transformation which maps p to infinity. Then these two circles now will be mapped to two circles tangent at infinity. So two circles tangent at infinity will be those parallel lines. So this image of these two white circles will be these two parallel lines here. And since this third circle, since it's parallel to these two circles, it should be located precisely here.
So this is the image of these three mutually tangent circles by a. And now in this configuration, Apollonian theorem is clear. There are only two circles which are tangent to these three circles, precisely these two. And then we can go back to the original configuration using a inverse, which is again a Mobius transformation. And that proves this theorem.
So now this Apollonian theorem-- I guess I hide this not so great. So using this Apollonian theorem is the basis of the construction of what is called Apollonian circle packing. And to construct this Apollonian circle packing, we start with four mutually tangent circles. We begin four mutually tangent circles.
And here are possible configurations of four mutually tangent circles. So we can have these four mutually tangent circles, or we can have-- OK, I need to [INAUDIBLE]. Or we can also have a configuration containing two parallel lines and then two circles in between. And you can also have configuration one line, and there's three circles.
So we begin with any of these top four possible configurations of four mutually tangent circles. And using Apollonian theorem, we can keep adding further and further circles which are tangent to three of the previous ones. So for instance, if I start with this configuration, I can add this circle. I can add this circle. And then this creates new triangular regions. And then one in each new triangle region will give us exactly this new circles, which are tangent to previous three circles.
We keep adding these circles over here. And it creates actually newer and newer triangular regions. And I keep adding actually circles in each triangular region. We arrive at this [? infinite circle packing, ?] Apollonian circle packing. And then these are actually some of the pictures I tried to draw before this lecture. So we arrive at this Apollonian circle packing.
And if you start with a configuration of this type that contains two parallel lines, then you would have arrived actually at this infinite circle packing, which is also unbounded circle packing in a plane. But there are also actually Apollonian circle packings which are unbounded but which contain only one line, and also which does not contain any lines. But it's much harder to draw, actually, Apollonian circle packing which contains one line or no lines at all because the circles will get enormously large, and maybe only after a few generations. But there are actually those unbounded Apollonian circle packings.
So the counting problem we asked about the Apollonian circle packing is the following. Maybe I'll go to the second blackboard. I picked some bounded Apollonian circle packing. Then we defined the [INAUDIBLE] function N sub T P. And this will be number of surfaces of P of curvature of C less than T. And curvature means that this simply means 1 over radius of C.
And since I am asking for this P to be bounded, there can be only finitely many circles of curvature less than a given number. So this is a well defined function. It's a finite number for each fixed positive number T. But since we'll have infinitely many circles in a given Apollonian packing, as T tends to infinity, this will go to infinity.
So the counting question is actually we want to ask is, how fast does it grow? And can you figure out the precise asymptotic of this N sub T P as T tends to infinity for fixed Apollonian packing? So the question is what is the asymptotic?
And it turns out actually, this question is related to the underlying factors that are called the residual set. So the residual set of P. You can simply think of this as the [INAUDIBLE] over P. And by this, I mean you look at the union of all circles. We have [INAUDIBLE] many circles from each packing. And then you take the [? close ?] of this.
Or in that picture, you can imagine like we just scoop out all the white parts. And then what is left in the plane, the residual is there. So you get some kind of fractal [INAUDIBLE]. And the residual dimension of P, which we'll denote by alpha. By this, we simply mean the [INAUDIBLE] dimension of this set. So [INAUDIBLE] dimension of this residual set.
And clearly, this alpha, the residual dimension, is some number between 1 and 2 because this P, the residual set includes circles. So the dimension [INAUDIBLE], but it's a subset of [INAUDIBLE]. It's some number between 1 and 2. But actually, its precise number is not known. But the only estimate is known. The best possible estimate, actually, which is known is I think 1.305688. This is due to [INAUDIBLE] state.
And one important observation about this residual dimension is that it does not depend on the choice of P. So if I choose two Apollonian packings, and if I choose a Mobius transformation which maps actually three tangent points of one Apollonian packing to three tangent points of the other Apollonian packing, then this Mobius transformation actually maps the whole packing to the other whole packing. And the dimension does not actually change under Mobius transformation. So this is actually alpha does not depend on choice of P.
Now, the first theorem toward this counting question is due to Boyd in 1982. He showed that the log of N sub T P is asymptotic. So if you take the log of this and divide by log of T, then as T tends to infinity, actually, this converges to this residual dimension alpha.
But of course, this does not say quite what this asymptotic is. And he asked if this N sub T P has a purely polynomial growth of T to the alpha, or if there is extra [INAUDIBLE]. And he has written, actually, in his paper-- his numerical experiments actually led him to believe there might be an extra [INAUDIBLE] this N sub T P. But it turns out actually that is not the case.
So theorem was proved by [INAUDIBLE] maybe 1982. He showed that as T tends to infinity, N sub T P is asymptotic to some positive constant depending on P times T to the alpha. And actually, this tilde sign means that the ratio of these two [INAUDIBLE] goes to 1 as T tends to infinity.
And then recently, with [INAUDIBLE] actually, we proved a stronger version of this theorem, which is that, actually, we can also get a [INAUDIBLE] asymptotic. So this is [INAUDIBLE] equals T to the alpha minus eta for some eta independent of P.
So this was actually the answer to the first [? quantum ?] question for bounded Apollonian packing. But we can also ask one final question, or the distribution of circles in a packing. So here, we took some bounded Apollonian packing and we counted the circles from the [? hole ?] packing. But instead of counting circles from [? hole ?] packing, we can try to count the circles maybe from some fixed region like this. So you take any region, bounded region E, and then we can try to count circles only from this region. And if you actually change the problem in this way, this problem makes sense also for unbounded Apollonian packing because instead of counting all the circle packings here, you can just simply cut off some bounded region and then count circles over there.
So the next question is for the asymptotic distribution of circles, small circles. And we formulate this question the following way. So as I just said, for any bounded region E on a complex plane, we define the [INAUDIBLE] function N sub T P, E. So this is the number of circles from P, or curvature less than T. But we only looked at those circles which intersect this region E non-trivially. So C intersection of E is [INAUDIBLE].
So now in this problem, I can take P to be any Apollonian circle packing. It can be also unbounded as well. And the question is now, is there an asymptotic for this counting function? And if there is, what will be the dependence of actually this E? So we can ask if there exists a measure, say omega sub P, such that this N sub T P, E is asymptotic. So we expect the growth of this counting function to be T to the alpha. And the dependents of E we expect to be measured by some measure of [INAUDIBLE]. So [? with a P, you see ?] such that omega P E.
So can you find such a measure? And if you can find such a measure on the plane, which is actually this asymptotic, then you can say actually this measure, omega [? set of ?] P, describes the asymptotic distribution of circles in the circle packing. So if you actually want to find this kind of measure, so where it is actually measured is important, actually. So if I, for instance, take E completely out of this residual set, then this number should be 0, right? So if there is such a measure, then this measure must be supported on the residual set of P, this fractal set of P.
There are several measures, actually, which are defined in such a fractal set, like Hausdorff measure, and then there's also packing measure. And there are also some other measures. But it turns out that actually the [INAUDIBLE] measure, actually, is precisely what we are looking for.
So there is a theorem that has proven with [INAUDIBLE]. So here, P is any Apollonian circle packing. Then we get this for any bounded region E with a smooth boundary, this N sub T P, E it's asymptotic, as I just said. So T to the alpha-- and alpha is this residual dimension-- times-- let me write this H P of E, where this H P denotes the alpha dimensional Hausdorff measure on the residual set.
So this residual set, because the dimension is bigger than 1.3, is much bigger than just counter union of circles. You have this fractal set. And then it turns out, actually, this measure is a locally finite measure. This [INAUDIBLE] is a locally finite measure. It's a well-defined measure. And then we get this.
But there is actually this concept actually in front of this, which I'm going to denote by C sub A. And propose actually to call this constant Apollonian constant because actually, this constant is independent of P. So it's Apollonian constant, and it's independent of [INAUDIBLE], and of course independent of E.
So in particular, this theorem says that if I actually have two different bounded sets, like E1 and E2 with a smooth boundary, and if I look at the ratio of this counting function, then these ratio is completely determined by just Hausdorff measures of E1 and E2. So this Hausdorff measure describes them.
But what's interesting, actually, is this constant here. And if you actually look at this asymptotic, then all this actually, this counting function and this Hausdorff measure and T to the alpha, they have expression in terms of the symmetric [INAUDIBLE] in Euclidean plane.
But our formula for this Apollonian constant, it involves actually some highly singular measure in an infinite [INAUDIBLE] manifold. So [? they even ?] talked about lots of problems about volumes in finite [INAUDIBLE] manifolds. But there are also actually some singular measures. And then this is a ratio of certain single measures.
In fact, actually, this measure, we have not seen this before. So it looks like an interesting new measure to look at. So I will be able to actually define this in the later part of my talk. But it's very interesting to know what this constant is. I mean, I don't know any estimate, like if it's bigger than something or smaller than something. There is no estimate.
I guess the difficulty is that the Hausdorff measure, even though we have a [INAUDIBLE] actually theoretical definition of Hausdorff measure, it's very hard to compute what Hausdorff measure is for a given packing. For instance, for these packings, it's hard to [INAUDIBLE] which Hausdorff measure of these packings is bigger, [INAUDIBLE], for instance. So this is all the question.
And it turns out, actually, that our proof of this theorem works for much more general circle packings than only Apollonian circle packings. So I'll state the theorem for much more general circle packings. Let P be any circle packing on the plane. So by circle packing, actually, I simply mean this union of circles. And they can intersect with each other beyond tangent points, actually. So we are now to the circle packings like this as well.
But we need to impose, of course, certain conditions to be able to count circles here. And the first obvious condition that we need to have is what we call locally finite. So we assume this circle packing is locally finite in the sense that this N sub T P, E is finite for any bounded set, E.
So what this means is that we just want to exclude the case that there are infinite sequence of circles which convert to one circle. We have infinitely many circles which convert into one circle. And if I take this bounded region, then we'll have infinite number of circles of about the same size, right? And then that will actually contribute this condition. But we actually [INAUDIBLE] make by putting this condition.
The second condition-- and this is actually much more crucial-- is that we need actually this P to have a lot of symmetries. It means actually we want to have a P invariant under some Kleinian group. So what's a Kleinian group? Kleinian group is the [INAUDIBLE]. It's a fancy term for this PSL2C.
So [INAUDIBLE] by Mobius transformations. So it makes sense to have [INAUDIBLE] invariant under a Kleinian group, [INAUDIBLE] transformations. And [INAUDIBLE] is non-elementary So non-elementary simply means that it's not [INAUDIBLE]. So it does not have [INAUDIBLE].
And we defined it in [INAUDIBLE] Mobius. So we define it in [INAUDIBLE] Mobius. So in other words, P can be written as union of finite union of gamma C1 and gamma C2. So we have finitely many circles. And then their union actually gives us [? everything. ?]
So this is actually basically [INAUDIBLE]. But for simplicity, I also [INAUDIBLE] geometrically finite. Geometrically finite. This condition actually is not a special condition, but I failed to mention earlier where this condition actually comes into the picture.
But the definition is that it's [INAUDIBLE]. So we have this [INAUDIBLE] Kleinian group on this extended complex plane. And if you look at the [INAUDIBLE]. We get this [INAUDIBLE] here. And the Poincare expansion theorem actually identifies this PSL2C with the orientation [INAUDIBLE]. So we can think of actually gamma actually acting on [? AT3. ?] And geometrically finite means that this quotient manifold, [? AT3 ?] over gamma, [INAUDIBLE]. So it admits finite [INAUDIBLE].
Now before we go on, actually, let me describe some of the examples of circle packings which satisfy these conditions. And as you can imagine, actually, the first example is Apollonian circle packing. So P is Apollonian circle packing.
The first condition is clear. It's locally finite because these circles have disjoint interiors. It's clearly locally finite. But what is the Kleinian group that this Apollonian packing is invariant under? So I take four mutually tangent circles. Then this means that there are three tangent points-- no, six tangent-- one, two, three, five, six-- six tangent points. And these six tangent points determine four circles.
So one more circle, and one of this, and one of this. And then-- my picture is [? also correct. ?] So the others [INAUDIBLE]. I usually do better than this. So here, I have four [INAUDIBLE] circles, one circle and then [INAUDIBLE] circles.
And gamma very simply the [? group generated ?] by [INAUDIBLE] four [INAUDIBLE] circles. So gamma is generated by [INAUDIBLE] with respect to four circles. So as you might have noticed, actually, now I PSL2C as the group generated by [INAUDIBLE] in the plane. But I think the index is only fourth, so I [INAUDIBLE] this difference. I identified PSL2C with [INAUDIBLE] Mobius transformations in the plane.
So now if I actually denote its gamma [INAUDIBLE] circles, it's pretty easy, actually. This Apollonian circle packing will [? preserve its ?] gamma. And it has actually only four orbits under this group gamma. So this P, actually, consists of gamma C1, gamma C2, gamma C3, and gamma C4, where C1, C2, C3 are any four mutually tangent circles from fixed Apollonian circle packing.
And moreover, it has also this important property. The residual set of P coincides with what is called the limit set, lambda gamma. So this lambda gamma is what is called the limit set of gamma. And this is very important invariant of a Kleinian group. And let me give you a definition.
So I take any point, z, in the complex plane. And then I have a Kleinian group acting on this complex plane by Mobius transformation. So we can look at the orbit of this point, z, under gamma. So look at any orbit, gamma z, and the limit set is simply the set of accumulation points of this orbit. So you get some kind of [INAUDIBLE].
It's an easy exercise again. This residual set of Apollonian packing is indeed actually-- you start with this group. It's called Apollonian group. And you take any point in the complex plane. You don't have to take it from the circle. But eventually, by looking at the set of accumulation points, if you look at where this orbit accumulates, actually, they give you the full residual set of P. So this is important observation. So this is one example.
We have also a lot of examples actually constructed in the following way. Take any finite volume at [INAUDIBLE] manifold-- it's always three dimensional in my talk-- with non-empty [INAUDIBLE] geodesic boundary. And if you take a gamma, the fundamental rule of this manifold, and we can put this gamma inside PSL2C.
And now if you look at omega of gamma, which is the domain of this [? continutive ?] but which simply means the complement of the limit set. So it's just C minus the limit set. Then this omega gamma consists of countably many disjoint open disks. So it's the union of [INAUDIBLE] open disks. So it's the disjoint.
And these open disks arise from this boundary component of this totally geodesic soft space over here. So you have countably many open disks in the plane. And then I can simply take my P with the circles in closing these open disks. So we just take the P, the union of this boundary of DI.
And this gives me a circle packing which is clearly locally finite because their interiors are disjoint. But it's also gamma invariant because it's a domain of this [? continutive ?] for gamma. And then it has finite gamma always because of [? Alpha's ?] theorem.
And this example create actually lots of interesting stuff with packings. And Apollonian circle packing can be also understood, actually, by this terminology. For instance, if you take x-- if x is [INAUDIBLE] in this example, then the [INAUDIBLE] consists of disjoint circles.
So in the [INAUDIBLE] example, what is called the Sierpinski curve, consisting of disjointed circles with no tangents here. It's called the Sierpinski curve because it's homomorphic to the well-known Sierpinski curve.
And another set of examples is what's called the Schottky dust. So I take some disjoint, take pairs of closed, disjoint disks. So for instance, if I take here four closed disjoint disks, and then we can find the gamma 1 and gamma 2. So k pairs. This is two pairs. Then let's say Di Di prime [INAUDIBLE]. And for each i-- so let gamma i be our element of PSL2C which moves the exterior of this disk, Di, to the interior of Di prime. So gamma i moves exterior of Di to the interior of Di prime.
And now if I define gamma, the group generated by gamma i. So this is in fact a three group. And now the circle packing, what will be the circle packing? We can imagine [INAUDIBLE] is simply the orbit of actually this [INAUDIBLE] pairs over closed now circles under this gamma. So P will be the union of gamma Ci and gamma Ci prime where the Ci's are just the boundaries of these Di's.
So if I start with actually this circle and then look at the image of this gamma 1 here, the limit of for each gamma i, I'm going to get the same 2k minus 1 circles in each circle. So we get this sequence of nested circles. If I continue, I get even smaller nested circles over here. And you get some fractal shape like what's called this here, Schottky dust.
So in this example, actually, you see the interior of the circles are not disjoint. But it's still allowed because it's locally finite. Since the circles are getting smaller and smaller, even though they are not disjoint, they are still locally finite. And by the construction, it is invariant in the gamma. And then it is finite in [? gamma, always. ?]
So now what will be the measure to describe the distribution of small surface in such a circle packing? So since we actually encountered this P is invariant under gamma and then P is finite in gamma always, do you see where the actual small circles go to? They actually go towards more and more to the limit set of gamma. So we are really looking for some measure which is supported on the limit set of gamma.
So here, I have some circles here. But this is smaller and smaller circles. We move towards more and more to the limit set of gamma. I have some limit set in the plane here. Then what is the measure supporting on the limit set? And there is actually what's called a Patterson-Sullivan measure on the limit set.
And this Patterson-Sullivan measure gives a measure to the limit set for every viewpoint from hyperbolic plane. We'll take actually this measure, uj, which is Patterson-Sullivan measure. It's a finite measure supporting their own limit set. I took the specific viewpoint. It's viewed from this point, j. And by j, I mean this point j, 001.
I'm not giving the [INAUDIBLE] of this measure. But in the case of Apollonian group, this measure actually is precisely the [INAUDIBLE] measure or Hausdorff measure. So this is the general [INAUDIBLE]. So this is the delta dimension of that measure when gamma is [INAUDIBLE] or when gamma is Apollonian group. And then what's delta? So here, delta is precisely the Hausdorff dimension of the limit set.
And now, as you can guess, actually, this measure cannot be the right measure because I said actually it's a finite measure. But when I have [INAUDIBLE] Apollonian packing, the finite measure is not the right measure. So we need to do something.
And the problem actually is this Patterson-Sullivan measure-- I said something over here. It's a delta dimension of Hausdorff measure, but we expect a [INAUDIBLE] metric here, not with respect to Euclidean metric. And I need to change this, actually, to the Euclidean metric, by multiplying this factor. So d omega of gamma will be z squared plus 1 delta [INAUDIBLE] j gamma.
So you can think of this factor as actually the factor you need to transform this delta dimension of Hausdorff measure to the delta dimension of Euclidean measure. This is what you see, actually, when you use this [INAUDIBLE] projection. And when you move it, that's [INAUDIBLE] Euclidean measure.
And our answer is actually this is the right measure. So this measure describes the distribution of circles in circle packings under this condition. And [? this made a ?] precise theorem. I'm getting better. This is [? a good ?] theorem [INAUDIBLE].
So I defined the delta to be the Hausdorff dimension of the limit set. But if delta is smaller than or equal to 1, then we assume that P does not contain [? a bouquet. ?] So [? a bouquet ?] of infinitely many circles. [? By bouquet of ?] infinitely many circles, that simply means that for this configuration, I have this circle, and then I have infinitely many circles getting smaller and smaller, and then the same thing over here. So in the case that delta is smaller than 1, we just simply exclude this case. But if delta is bigger than 1, it's OK to have this kind of configuration as well.
Now remember that actually, this theorem is for general circle packing. It's not only for Apollonian circle packing. So now, remember this function. So N sub T P, E. Let me remind you of this definition. This is exactly the same definition.
So we count the number of circles in P of curvature less than T, but only those intersecting E non-trivially. And this is asymptotic to-- so now T to the i power should be replaced by T to the delta here. So delta is the Hausdorff dimension of the limit set, which will be alpha for our Apollonian packing case because I said the residual dimension is precisely the same as the limit set.
And then here, I have this measure, omega gamma of E. And then this will be actually [INAUDIBLE] dimension of Hausdorff measure in the Apollonian case. So now this constant will be actually my Apollonian constant in the case of Apollonian group. And what is this constant? This constant is what I call the skinny number of P divided by delta times the total measure of what's called the Bowen-Margulis-Sullivan measure.
So now let me explain this terminology. Actually, except for this constant, so I defined what this measure is. But to justify actually this asymptotic, I need to tell you actually these constants are all finite. This is where I need actually-- where I assume the gammas are geometrically finite. So forget my geometrically finite.
The BMS measure, this is notation for what is called Bowen-Margulis-Sullivan measure, defined on the [INAUDIBLE] tangent [INAUDIBLE]. And this measure could be infinite because here, I did not assume anything about gamma except that it's geometrically finite and non-elementary. So gamma does not have to have a finite volume at all. And when gamma has finite volume, its Bowen-Margulis-Sullivan measure is nothing but just the usual [INAUDIBLE] measure.
But here, what Sullivan proved is that if gamma is a geometrically finite measure, then BMS is the finite measure. So actually, this constant is finite. And delta is a positive number. It's the Hausdorff dimension of the limit set.
Now, this skinny number of a packing produces something interesting. So I defined what this skinny number is. But let me just mention that this is finite skinny number, P. This is finite under this condition. So this condition that if delta is less than or equal to 1, then P does not contain this [? bouquet, ?] is precisely to guarantee this condition.
If delta is less than or equal to 1 and if P contains this infinite [INAUDIBLE], then this [INAUDIBLE] number is infinite. So therefore, actually, one cannot expect the same type of asymptotic. And in this case, it turns out actually this asymptotic is not the same form as that. But in fact, this exponent will be different. It will have a [INAUDIBLE] if delta is equal to 1. And if delta is strictly smaller than 1, it will be T to the 1, which is, again, not the same as T to the delta. So this condition actually turns out to be very critical to have this general statement for general circle packings.
So now let me define what is this P number. So I write P as the union of gamma C1, gamma Ck, the finite [INAUDIBLE]. And the skinny number will be the sum of the skinny number of actually the circles of gamma representatives. It's the [INAUDIBLE] the sum of Ci where i is [INAUDIBLE]. So I need to define the skinny number for each circle here.
So here's my circle, C, here. And the skinny number actually is defined from some measure of the hyperbolic space. So we need to actually think of this C as the boundary of the hyperbolic space. Instead of C, actually, I'll be looking at this [INAUDIBLE] hemisphere above C. So this is the hemisphere above C.
And this is [INAUDIBLE] plane [INAUDIBLE] model. And then consider the [INAUDIBLE] over this plane. So we have this union over [INAUDIBLE].
And then here, I have a [INAUDIBLE]. So for each vector here, we have a visual map, which is defined by this vector. So this equals to [INAUDIBLE]. So here, this is nothing but just vertical lines over a circle here. By this direction, we just follow the vertical circle. And then where you land on the plane, that will be your [INAUDIBLE]. And this map is called a visual map. But you can also think of this map as a skinny map. You just kind of peel out these surfaces over here, in and out.
But now, I have a measure supported on the limit set of this plane, which is my Patterson-Sullivan measure. So imagine actually [INAUDIBLE] limit set here. So limit set. I have a measure for limit set which is Patterson-Sullivan measure. And then we are going to actually pull back this Patterson-Sullivan measure on the limit set via the skinny map. And then you give me some measure over here on this-- let me denote by C hat, this [INAUDIBLE] hemisphere of C. So I defined the measure on C hat, which is the pull-back-- multiply something over here-- the pull-back of this Patterson-Sullivan measure, uj, on the limit set.
So now where does this measure fit in? Now this measure is sitting on kind of this [INAUDIBLE], but on this hemisphere. But if I just define this naively, then it's not very defined. It's not [INAUDIBLE] invariant measure. So I need to multiply some density function here, some density function.
And I can write out the precise formula, so [INAUDIBLE]. Then this is what's called e to the delta times B [INAUDIBLE]. And then [INAUDIBLE]. So the [INAUDIBLE] is the point here. So you pull back the Patterson-Sullivan measure. But this is that notation for the [INAUDIBLE] function. But if you don't understand, it's OK. But it's the right [INAUDIBLE] function to make actually this measure, again that invariant measure.
So if I multiply this measure, then actually, it induces a measure on the quotient surface. So I have this C hat, but then look at the quotient surface [INAUDIBLE] combine C hat divided by the stabilizer of C hat inside gamma. So this gives me some surface like this.
But in our [INAUDIBLE], this surface is not going to close, even on the finite theory. If you have infinite [INAUDIBLE], and then maybe some [INAUDIBLE]. But this measure is supported only on here, only on the [INAUDIBLE] level of the limit set.
So even though the surface is [INAUDIBLE] surface, actually, this measure is not going to be the same [INAUDIBLE] measure. It's going to be something different. You can imagine, for instance, even though it's a closed surface, if all the limits that lie on the boundary of this hat, this measure will be [INAUDIBLE] measure. So in general, if I just define the measure like this for any gamma, this measure could be infinite measure and [INAUDIBLE] measure. But there is some measure.
Actually now, let me finish the definition. So skinny number of C is the total measure of actually this [? induced ?] measure here. Let me just write this. Mu s and it's the total measure of this. I hope it's clear. So I defined the measure on C hat on the northern hemisphere.
But there is a gamma, so this measure does not only actually depend on C hat. But it highly depends on your gamma. So [INAUDIBLE] gamma. And then the quotient surface, actually, meaning we define some actual singular measure, which you report to the skinny measure. And this skinny [INAUDIBLE] or skinny measure, skinny number, is actually the total measure of the skinny measure of the quotient surface.
And what we saw in the theorem is that precisely under these conditions, if delta is less than or equal to 1, and if P does not contain this infinite bouquet, then in this case, the skinny number will be finite. And otherwise, it;s infinite, actually.
So in general, we have a complete criteria when this is finite and infinite. And then these two conditions, under the external condition, gamma is geometrically finite actually implies that the constant will be finite in that case. So in general, these two finite [INAUDIBLE] is the only thing we need. So I'm now getting rid of this condition gamma is geometrically finite.
But as long as this Bowen-Margulis-Sullivan measure is finite, and now I'm also getting rid of this condition about infinite bouquet. But as long as the skinny number is finite, then we have the same theorem. The same holds. So you will see actually, the circle packing condition actually boils down to these two conditions. We just need to understand [INAUDIBLE] of two measures here.
So now finally, where is the dynamics of hyperbolic manifolds? Or actually, where this skinny number comes into the picture?
So I want to count the circles, right? So here, I have this bunch of circles, and I want to count circles. But instead of counting circles, I'm going to count these northern hemispheres above circles. Now they become [INAUDIBLE] plane in the hyperbolic manifold. So this is where I go from plane to the hyperbolic space. I can forget about the boundary. But I only consider these hemispheres.
But now where is my curvature? So curvature is exactly one of the radius, right? So now, one of the radius is precisely the Euclidean height of these northern hemispheres. So my function, N sub T P, E becomes now number of actually C hat, the northern hemispheres, that C hat has height. C hat intersects this E sub T non-trivially. And what is E sub T?
So I have this bounded region E. I'm going to look at the cylindrical region above E. So you look at the cylindrical region above E of height between 1 and 1/T. So I have this height here, 1/T. So this is cylindrical region of height between 1/T and 1.
And if my circle has curvature [INAUDIBLE] T, then it means actually this northern hemisphere must intersect actually this region because it must be as high as [INAUDIBLE] this region, E sub T. But actually, by this translation, I can forget about the circles now. Now I can actually count [INAUDIBLE] planes in the hyperbolic space.
But these are all in the gamma orbit of actually something still hyperbolic plane. We assume actually gamma has finitely many orbits. But then I can only assume [INAUDIBLE] has only one gamma orbit. If I think of [INAUDIBLE] C hat actually coming from one gamma orbit of some fixed hyperbolic plane. So I have some fixed hyperbolic plane, and then we are looking at the gamma orbits.
And they're counting actually this kind of problem can be reduced to essentially the following [INAUDIBLE] question. So now I forget about the circles. Now I have some [INAUDIBLE] plane in [? A2T. ?] And we want to understand actually this [INAUDIBLE] plane which intersects this region. And this region is getting larger and larger as T tends to infinity. Even though it looks like it's getting larger only by a little bit. But then since this is the boundary of hyperbolic space, it gets really large when you actually push it down.
And we need to understand this [INAUDIBLE] problem. This is [INAUDIBLE] geodesic plane. And then we will [INAUDIBLE] geodesic plane. [INAUDIBLE]. I look at this [INAUDIBLE]. And then [INAUDIBLE] this [INAUDIBLE] by the [INAUDIBLE]. Or you can think of this as the [INAUDIBLE] boundary of the [INAUDIBLE] of this fixed plane.
So I have this sequence of [INAUDIBLE] this fixed [INAUDIBLE] plane. And then we need to understand actually the distribution of this [INAUDIBLE] plane, not [INAUDIBLE] but in the quotient manifold, in [INAUDIBLE] gamma. Of course, in [INAUDIBLE], if you do this, then everything goes to infinity and then there's nothing to worry about.
But again, this quotient manifold is not finite in general. You can have [INAUDIBLE], but you can also have [INAUDIBLE] like this. And now this [INAUDIBLE] plane in the quotient manifold could either be some [INAUDIBLE] surface, which is not in general finite area. It can have infinite area as well. Let's imagine this [INAUDIBLE]. This is finite area.
And now when you push this [INAUDIBLE] into some distribution [INAUDIBLE]. And then you need to understand this distribution. So the essential ingredient of proving the [INAUDIBLE] theorem is the following distribution theorem in this hyperbolic space, which is we picked S, which is quotient of [INAUDIBLE]. But we assume that this is a [INAUDIBLE] immersed surface inside [INAUDIBLE] of gamma. This is [INAUDIBLE]. So meaning that this invariant is [INAUDIBLE], but [INAUDIBLE].
And then to state the theorem [INAUDIBLE] distribution, [INAUDIBLE]. And then you look at the integer of this continuous function along [INAUDIBLE] these circles, which are moved by [INAUDIBLE] S, and then here is the [INAUDIBLE] of S. And in this notation, [INAUDIBLE] is the [INAUDIBLE]. We need to understand the limit of this integer here. But if you don't understand this notation, you can just [INAUDIBLE].
So I have these quotient surfaces, and then you push by the [INAUDIBLE], and [INAUDIBLE] distribution. And it turns out if you don't do anything, actually, then this will go to zero as T tends to infinity, unless gamma is [INAUDIBLE]. But if you normalize it correctly by e to the 2 minus delta T here, then this is a [INAUDIBLE] distribution, and it converges to some infinite measure, which we call [INAUDIBLE] measure.
And then there's a constant. And the proportion of this measure appears as the limit of this is exactly the skinny constant. So I have here the skinny number of this, S, divided by [INAUDIBLE] Bowen-Margulis-Sullivan measure. Now let me say some words about this BR measure here.
So this BR measure, MBR, this is what we call Burger-Roblin measure. This is a measure defined on the [INAUDIBLE]. So first of all, if we actually assume that gamma is a lattice. Or if [INAUDIBLE] over gamma is a finite volume hyperbolic manifold, then this result is well known, actually. In that case, the skinny number will be just user area of the surface. And then the Bowen-Margulis-Sullivan measure and this BR measure will be the user measure, the [INAUDIBLE] measure or [INAUDIBLE] measure. In that case, this theorem is due to [INAUDIBLE] and also [INAUDIBLE] from about '93.
But in the case that gamma is not assumed to be a lattice, actually, all this actually quantities, they are all for the singular measure and they are all different. And this Burger-Roblin measure, I'm not going to define what it is, but it is an infinite measure when gamma is not a lattice.
And the reason why it's called the Burger-Roblin measure is that this Burger-Roblin, they proved an important classification in that this measure is the unique measure satisfying certain properties. And that property is this. So this is the unique, non-trivial ergodic measure on the [INAUDIBLE] invariant on the [INAUDIBLE].
So here, by non-trivial, I mean that it's not supported on our [INAUDIBLE]. If it's a measure supported on a closed [INAUDIBLE], it will be just some plane embedded in your hyperbolic manifolds, and you [INAUDIBLE] measure. But what [INAUDIBLE] proved is that, except for those measure, so this measure has support everywhere. It has support everywhere if you project down to the hyperbolic manifold. But in the [INAUDIBLE] [? boundary, ?] it highly depends on the directions of these arrows in the [INAUDIBLE] [? boundary. ?]
But it has full support if you project down the hyperbolic manifold. And they show that this is actually unique measure which is ergodic and non-trivial, but for [INAUDIBLE]. And this why actually we call this measure Burger-Roblin measure.
This is kind of the end of circle packing. To me, actually, [INAUDIBLE] problems [INAUDIBLE] very important actually, maybe further directions of research for these dynamics in hyperbolic manifolds. There are lots of work on dynamics on hyperbolic manifolds. But most of this work, actually, has been done under the assumption that gamma is a lattice in the finite [INAUDIBLE] case.
But now we have these dynamics problems for infinite [INAUDIBLE] case. And then the measure you see is not a finite measure. It's an infinite measure. One has to understand actually this class of infinite measures and these infinite [INAUDIBLE] manifolds even to solve these very natural problems like Apollonian packing problems.
So this brings up actually a very important and natural question. So let's assume we have PSL2C less than [INAUDIBLE]. But for simplicity, I'm going to assume it's [INAUDIBLE]. It just means that gamma is [INAUDIBLE].
Then can you classify ergodic? So ergodic simply means [INAUDIBLE]. It's kind of the basic blocks of the measures you need to [INAUDIBLE]. So classify ergodic invariant measures on the quotient PSL2C over gamma invariant under some flows. What flows? [INAUDIBLE] flows. Or under PSL2R, for instance.
So here we have some nice [INAUDIBLE]. And you might actually notice immediately this [INAUDIBLE] looks very familiar. It's exactly what [INAUDIBLE] solved, but in much greater generality, but assuming actually finite measures.
So [INAUDIBLE] classified ergodic finite invariant measures on this quotient space. She does [INAUDIBLE]. Gamma can be any discrete group. But she assumes a very important condition that she classifies only finite measures which are invariant under [INAUDIBLE] flows or under PSL2R. In that case, actually, you have only very nice measures, what's called algebraic measures and homogeneous measures. So you don't see any crazy measures here.
But now, this problem is actually looking at the class of finite measures is not the right one if you are in the infinite space. So instead of actually looking at finite, you need to look at locally finite measures. So in that case, then, this Burger-Roblin measure now will appear here. It does not appear previously because you only looked at finite measures. Now this Burger-Roblin measure will appear, but [INAUDIBLE] other measures.
And this seems to be, for me actually, a very important direction for future direction. So I need to [INAUDIBLE] this problem. So this is what's called the [INAUDIBLE]. You can also ask what is also called [INAUDIBLE] that question, which is what are the [INAUDIBLE] of orbits on the [INAUDIBLE] flows or PSL2R?
Even a very big [INAUDIBLE], like you have this PSL2C over gamma. It's every non-closed PSL2R orbit [INAUDIBLE] over here. Looks like a very simple question, but it's completely open ended. [INAUDIBLE] very hard, actually. Somehow, when you look at this infinite measure class, all you knew about the measures, if you dig into it again, [INAUDIBLE] your measure is [INAUDIBLE] measure or finite measure.
Everything you know actually about these measures fail miserably if you actually get out of this finite measure class. And then it seems it's a very different problem. For instance, for [INAUDIBLE] case, actually, what happens for [INAUDIBLE] flows? It immediately replaces what happens for PSL2R-- not immediately, but there is actually a [INAUDIBLE] going from here to here. But [INAUDIBLE] here actually, they are very different. They have very different [INAUDIBLE].
And my guess is that, if you look at this PSL2R, it may be that the only non-trivial measure-- non-trivial meaning not supported on the closed [INAUDIBLE]-- will be [INAUDIBLE]. So for instance, you don't see [INAUDIBLE]. This [INAUDIBLE] measure is not [INAUDIBLE] PSL2R. But they are invariant on [INAUDIBLE] flows. So if you look at the [INAUDIBLE] flows, then you will see actually lots of similar measures. And at the moment, actually, even [INAUDIBLE]. Proving [INAUDIBLE] some of these known measures seems to be also very difficult. OK. So this is all. Thank you.
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Hee Oh of Brown University discusses counting and equidistribution results for circle packings in the plane invariant under a Kleinian group at the 50th annual Cornell Topology Festival, May 6, 2012.
The festival was organized by the Cornell University Department of Mathematics with support from the National Science Foundation.