 Okay, so maybe I will start. So in the last lecture, Jens introduced the hyperbolic plane and described some of its basic geometry. And then he showed how the unit tangent bundle of the hyperbolic plane can be identified with the group PSL2R and how this is very useful if you want to give simple algebraic formulas for the geodesic flow and for the horocycle flow. And at the very end, he briefly mentioned that all of this transfers down to any hyperbolic surface where now the hyperbolic plane will be the universal covering surface of that. But the formulas for the geodesic flow and the horocycle flow transfer down to the hyperbolic surface. And today I will start doing that a bit more carefully. So GE today will be the lead group PSL2R. Recall that this is the special linear group of order two, so the group of two by two matrices, real matrices with determinant one, but you mod out with sign changes. So you identify any two matrices if they are just the negates of each other. So this is modulo plus minus identity matrix. And then today we will also fix some discrete subgroup, let gamma be some discrete subgroup of G. So this is simply some subgroup which as a set is discrete. The one example which will be very important for us is to take gamma equals PSL2Z, the group of all integer matrices with determinant one. And then we will be interested in the homogenous space X equals G modulo gamma. So this is the, as you saw in the introductory lecture, this is the, as a set, this is just the set of all left cosets gamma G where G goes through the group G. And locally since gamma is discrete locally this looks just like the group G but we have to identify various points of G. G modulo gamma, sorry, okay. And the point now for this special group G equals PSL2R which can be identified with the unit tangent bundle of the hyperbolic plane. It follows that this homogenous space can also be viewed as the unit tangent bundle of the hyperbolic plane modulo the action of gamma. I will say a bit more about what this means very soon. So this is using the identification that G can be identified with the unit tangent bundle of H. This is the map, I think Jens called it PSI in his lecture. And then this is almost the same as the unit tangent bundle of the hyperbolic surface H mod gamma. It is the same at least if there is no torsion in gamma. So this is if there is no torsion in gamma. Otherwise they are almost the same but if there is some elliptic element in gamma that fixes some point in the hyperbolic plane then this hyperbolic surface will have a conical point and it's a bit maybe not well defined to speak about its unit tangent bundle. But I want to say what is this H mod gamma. So that is a hyperbolic surface and it is by definition it's what we get if we take H but we identify points which are equivalent under the action of gamma. So here this sim is the equivalence relation saying that any two points in the upper half plane are equivalent if there is some element in gamma that brings one point to the other. So this is the hyperbolic plane but we have identified certain points. But locally unless we are at such an elliptic point that is fixed by elements of gamma this locally this looks exactly like the hyperbolic plane. So all the structure of the hyperbolic plane transfers to this hyperbolic surface. We have the hyperbolic metric giving the hyperbolic area measure and so on. So let me just show an example of what this looks like in the example which will be most important for us when gamma is equal to P cell to Z. If gamma is equal to P cell to Z then a fundamental domain for the action of gamma on the hyperbolic plane is given by the following set. So I will say what the fundamental domain is. A fundamental domain for gamma acting on H and this is the standard choice of a fundamental domain that is always done in the literature in the last century or so. So the following set, consider the following set in the upper half plane. The set of all points in H such that the real part of Z is less than or equal to one half and the absolute value of Z is larger than or equal to one. So here's a picture of that. So we consider all points which lie outside this circle and they should also have real part between one half and minus one half. So it's this domain and I've taken it to be a closed set. So it includes all the boundary. This is F and what fundamental domain means that basically it contains exactly one representative for each equivalence class here but I've included some boundary so there is some over representation but only at the boundary so to state that in precise sense. So H if I consider all the translates of this F by elements of the group gamma then I will get a perfect covering of H. So it's almost a disjoint union except I have some overlap in the boundary so this is a union of all the images of F by elements of gamma and there is only overlap in the boundary. So if I take any two images of F gamma one F and gamma two F then the interiors of these are disjoint for any gamma one gamma two in gamma. So this is what it means to be a fundamental domain. I mean depending on the context sometimes one considers open fundamental domain sometimes closed ones sometimes abstract ones and then one would insist that they are truly disjoint and exactly covering H. Okay so now to see what the hyperbolic surface actually looks like we can also point out what are the identifications. Well there is one map in gamma which identifies which is just translation. This map is in gamma and it acts on the hyperbolic plane by just taking Z to Z plus one and that clearly identifies these two sides and then there is also another map which is the rotation of angle pi around the point I and that is the map zero minus one one in gamma which gives this identification. And then it turns out that so this shows that the point I is actually an elliptic point of order two and it turns out if you compose this in the right way actually this point here is also an elliptic point of order three. But now you can if you do this folding you see that this is a surface of genus zero and so topologically it's just a sphere but we have these two special elliptic points and we have a casp also so it's not a compact surface. And the picture in general for will be kind of similar. So always if gamma is a lattice meaning that this hyperbolic surface is finite area then we can always find a very similar domain some convex domain convex with respect to the hyperbolic geometry with a finite number of sides such that the hyperbolic surface you get it by just identifying the sides in pairs and folding. So now we have this understanding of the homogenous space X. So this by this I mean just I take the unit tangent bundle of H and identify points according to the action of gamma so similar definition as here. And then as I said if gamma doesn't have any torsion in it there are no elliptic elements in gamma then we can identify it completely with the unit tangent bundle of H. But even in the remaining cases we can say that it's we understand what it looks like in the elliptic points and we can see that they are similar. So now we have a very good picture of this homogenous space which I said is a really important homogenous space for us. So in this special case when gamma is equal to PSL2Z then X equals G mod gamma is up to some well up to this minor problem with the elliptic points it's the same thing as the unit tangent bundle of this hyperbolic surface here. But let me also point out that this is the same as SL2Z without the P. So SL2R modulo SL2Z without the P and the reason for this is that the minus the identity element is in here so you get exactly the same quotient because minus identity element is in SL2Z these are the same. So as I said in the first lecture this homogenous space and also for higher order so SLDR modulo SLDZ is going to be a very important space for us. In that lecture I said that this space can be identified with the space of all lattices in R2 in this case so that's a completely different way of viewing this space that we will come back to in a later lecture. But the point here is that when D is equal to 2 we also have this picture of the space it's the unit tangent bundle of this hyperbolic surface. But now when I continue talking gamma will be a general discrete group not necessarily the PSL2Z. So now let's remind ourselves about the flows which Jens introduced in last lecture. How do they look like? Are you want to see a picture of this tessellation? Yeah, so obviously if you translate you get just this but I guess somebody can Google and you will see nice pictures of these tessellations it's very. And then another copy if you rotate it by this you get something like that or you get this. But then is it, doesn't it look like, well I shouldn't embarrass myself by, I suggest you Google and you get this tessellation or we can do it, sorry for. But obviously it will look, when you look down here it will, since the geometry of the hyperbolic plane is as it is it will look really fractal like or something. Okay, so now let's remind ourselves about the flows which Jens introduced. However let's immediately define these as flows on the homogenous space g mod gamma and recall or that means the unit tangent bundle of our hyperbolic surface. So we had the geodesic flow. So in Jens lecture this was a flow on the group g and it was defined by taking g to the point g of at where at is this diagonal matrix. However, now we can define this also on the homogenous space by just saying that the point, the coset gamma g is mapped to gamma g at and you can very easily check that this is well defined. It doesn't depend on which representative of the coset I choose. The reason being that I have gamma on the left but I'm acting on the right by 80. And similarly Jens, we had the horocycle flow and now on the surface or on the homogenous space this will be given by gamma g n t where n is this unipotent matrix 1 t 1 0. And we have the opposite horocycle flow given by gamma g n t minus where n t minus is 1 t 1 0. Okay. And Jens showed you yesterday that this is really the stable horocycle flow for this geodesic flow. So it's a flow along the stable manifold for phi of t. Now I will draw a slightly different picture or I will just point out the following fact. Note that if I take, well, this is a formula and this formula captures very easily, captures in a neat way the fact that this is the stable horocycle flow for this geodesic flow. Yeah, it's just a formula. Phi t composed with H s is equal to H of s e to the minus t composed with phi t for any t and any s. So this will be an exercise to check it. Clearly it's some commutativity relation between the matrices a t and n s. And what it means is the following. If I take any point in my homogenous space or in my unit tangent model gamma g and I flow with the horocycle flow for time s, I get the point H s gamma g. On the other hand, I can flow with the geodesic flow, which is really chaotic, so I want to have some red alarming color. But yeah, I can flow with the geodesic flow for time t. I can do that to both these points and then I get phi t gamma g. And then I could flow for a certain time with the horocycle flow. And the point is that I get back to a certain point here in very short time. I have to flow with the horocycle flow only for time s times e to the minus t here. And then I get the same point as I get by taking phi t flowing time t with this point. So the point is that this horocycle, which may be fairly long, has shrunk exponentially here after. So it's just, maybe I'll write it out. That's H s e to the minus t phi t gamma g. And this is true for any point gamma g in the unit tangent bundle. So I won't do it now, but now you could also consider the derivatives along the flow directions. And this will give you at any point gamma g, it gives you a splitting of the tangent bundle of the unit tangent bundle of the hyperbolic surface into a direct sum of three one-dimensional subspaces. And this gives a splitting of the bundle that is invariant by the flows and such that if you take a tangent vector in the flow direction of H s, that is contracted exponentially by the geodesic flow and so on. So this splitting gives you the fact that this is an unosso flow. I just say that without doing it. So here we have the stable manifolds for that flow. Okay, so now I will give the first theorem. Simply saying that phi t, this geodesic flow, but also the horocycle flows are ergodic and are also mixing. This was proved first by Hopf, 1939, using this geometry involving stable and unstable manifolds and then Birkhoff's ergodicity theorem and it's a very nice proof and it's fairly short. But I will not give it, I will just state it and then I will, I hope to say something stronger. So theorem one, all these flows. Phi t and ht, both of the, both of the horocycle flows. They are mixing and hence ergodic. So I won't talk about what ergodicity means, but I'm sure that many or most of you are well aware of it. They are mixing as flows on, or sorry, I should have introduced the volume measure also. So I have to introduce Mu before stating this. They are mixing on X provided with its Liouville measure Mu. So, okay, before stating the theorem, let Mu be hr measure on G and hence it discussed hr measure yesterday. You have explicit formulas for it on the, on PSL2R. And we will assume that gamma is a lattice. So, we assume that the homogenous space, if I take a fundamental domain for it in G, it has finite measure with respect to Mu. And this is equivalent to assuming that the corresponding hyperbolic surface has finite area, easy to see. And when we have done this assumption, we can just as well renormalize the hr measure. So it's not going to be given by exactly the formula, which Jens had before. But you change it by a constant. So we actually assume that Mu is a probability measure on the homogenous space. And if I didn't do it, let me stress that I write X also for this homogenous space. So now we have X provided with a probability measure. Liouville measure for, okay. And so, here's the statement. These flows are mixing. And let me remind you what mixing means. It means that for any functions, any test functions, I can take them to be L2 functions on X. If I consider F1 composed with Phi t and take it in a product with F2 over X. Then as t goes to plus or minus infinity, this tends to just the average of F1 times the average of F2. Okay, so the reason why do we call this mixing? Well, think of the case when F1 and F2 are characteristic functions of some sets. Maybe take some nice sets. And F1 and F2 are the characteristic functions of these sets. Then what this says is that if I let one of the sets flow by the deodesic flow, it will spread out evenly all over the surface because it's integral against any other nice set tends to become proportional to the area of that other set, so to say. I mean, so if you think about F1 and F2 being characteristic functions, it's clear why this result should be called a mixing of the flow, okay? And I have a couple of remarks, so I hope for this using these properties of expanding and contracting submanifolds and so on. But one can obtain this with a very good exponential rate. In fact, one can have a really explicit understanding of the rate in this particular case when we have just a homogenous space. So let me give one formulation of this. So if you want the rate in this result, then you have to take F1 and F2 to be fairly nice functions. There is no possibility of having a rate for arbitrary L2 functions. It is to show constructions that you can't have any explicit rate for this in this space of functions. But so here, let me take just to have a simple formulation. Assume that F1 and F2 are compactly supported and smooth functions. Then the error, so I mean the difference between this and this side as will be bounded by some constant times. So we will have exponential mixing, the decay is exponential. And we can even get a very precise understanding of the constant here. The constant is typically just minus one half, unless we have small eigenvalues of the Laplace operator on the surface. So it's one half plus epsilon. And then if the, so I just write something out. If there is some small eigenvalue, so lambda one. Let lambda one be the smallest non-zero eigenvalue of the Laplace-Beltrami operator on the surface. So where lambda one is the smallest eigenvalue. So smallest non-zero eigenvalue of the Laplace-Beltrami operator on the hyperbolic surface. So if there are no small eigenvalues, and small eigenvalue here means an eigenvalue less than a quarter. If there are no small eigenvalues, then this is just minus one half plus epsilon. But if there is some small eigenvalue, then the rate is worse. And if the spectral gap, so lambda captures the spectral gap. If lambda is close to zero, then this constant will be small. We have minus one half plus something near one half. So the constant is small. But it's an exponential rate, all the same. So if I get time, I will try to show how one, I mean, just give a brief outline how you prove such a result later on. So one can prove it using representation theory. And this is a result of a decay of matrix coefficients. This is really a matrix coefficient for a certain representation. And also, another remark, so if you have mixing for the geodesic flow, then it is easy to get mixing also for the horizontal flows. And what you do is you note that this matrix NT here, you carton decompose this matrix. So carton decomposition is a general decomposition of any element in the group G. And it says that any element can be decomposed as some element in the compact group SO2 times an element in the diagonal group times some other element in SO2. It's more or less unique. And so these are in SO2. And I think Jens introduced a notation k for SO2. This is in SO2. And this is a diagonal matrix. And it turns out if you compute this, you find that when t is large, the value of t prime here will be approximately 2 times logarithm of absolute value of t as absolute value of t tends to infinity. So this shows, and these are in some compact groups. So there's no problem dealing with this. You can have a result which is uniform with respect to both of these. So it's not hard from that to show that if you want a result with a rate, you will get a polynomial rate for the horizontal flow because you have this logarithm. But you can get hold of exactly the polynomial rate of decay for the mixing for the horizontal flow using this relation. OK. So as I said, I hope to say a few words about how to prove mixing with a really precise rate. Or I should mention that, of course, you can get exponential decay for mixing in much more general situations. For example, Dolgo Piat has proved that you have such an exponential mixing result for any unosso flow with some conditions. But that is much more difficult than. So it's a very different method for this result. On the homogenous space, we have access to representation theory, which is really a powerful tool where you can get a really precise understanding of everything involved here. OK. So I wanted to turn to an application of this mixing result to prove an equidistribution result, namely equidistribution of expanding translates of horror cycles. And this is something that is used over and over again in homogenous dynamics, the fact that you can get such equidistribution fairly easily. And it has numerous applications, as you will see in later lectures. This fact extended to higher dimensional settings. So here's a theorem. So I will consider a piece of a horror cycle. And the starting point will be, I call the starting point P. P is a point in my unit tangent bundle of the hyperbolic surface. And then I fix the length of this initial horror cycle piece. And then let curly H be the horror cycle starting at P and having length u. And the claim is that if I take this piece of a horror cycle and I transport it by the geodesic flow, as in the picture which I just removed, then as T goes to minus infinity, this tends to become more and more equidistributed in X. Clearly, T has to go to minus infinity because when T goes to infinity, it becomes more and more just like a point as this picture shows. Now we are going the other way. So phi Th equidistributes in X mu as T goes to minus infinity. And what I mean by that is that if I take an arbitrary test function, a bounded continuous test function or equivalent, it is a compactly supported continuous test function. Then if I integrate my test function along this horror cycle, so I take the average of the test function along this transported horror cycle, which is again a horror cycle. So phi Th up du. And I take an average. So I take 1 over u here. As T goes to minus infinity, this tends to the volume average of F. OK, so a picture. I had a picture over there. The point is we are now, let me do a picture on the, here's my three-manifold, the unit-ended bundle. And I have some horror cycle H. And I'm transporting it with a geodesic flow, but backwards. So if I would go this way, I would shrink the horror cycle exponentially. But now I'm going the other way. So the horror cycle becomes a longer and longer curve. And if I go sufficiently long, now in the picture, I don't go very far, but it is phi Th. But if I go very long, then this curve will spread out all over the surface in a more and more evenly distributed way, as this shows. I'm writing too small, maybe. So it's F phi Th U of P. Maybe I rub it out and write it in larger signs. F phi Th U P. Sorry? No, no, U is, I integrate always. U goes only between 0 and my fixed number, capital U. D U. D U is the parametrization. P is a fixed point. Yeah, yeah. So I don't know. Maybe it's better to write it something like this. So this is the average along the horror cycle. It's H T P. This is my horror cycle. And then I want to plug this. If I write this and I take some test function and plug it into, then it is a one-dimensional measure along the horror cycle. Now I actually plug it into F composed with phi T or something like that, then I average. Say it again. T goes to minus infinity, very importantly. And do I repeat the same stupidity? No, I didn't. So it's U here, of course. In this, in the picture, thanks, thanks. So P is here, just a starting point of a horror cycle. P is the starting point. And then this is the horror cycle I get by following the horror cycle flow for a fixed time U. This is H capital U of P. So it looks the same, I guess, more or less. So I just told you that this is a three-dimensional picture if you want to believe it. But if you want to see this as just the hyperbolic surface, I would draw the same picture. OK, OK, but with the unit. OK, so I see what you mean, maybe, as Jens wrote it. Here's the horror cycle. And then here's the geodesic flow. And OK, so this is really some, I'm not sure if this is a good, this would be one. If it happens to be, if the point P happens to be a unit tangent vector pointing upwards, then the horror cycle would look like this. And I would have all these unit tangent vectors. And then I follow the geodesic flow but backwards so that this is expanding. And I will get a really long horror cycle curve down here. Yeah, I was actually going to mention, yeah, OK, say again. OK, I didn't hear. But I'm considering the orbit, the horror cycle orbit of the point P, but just for a finite fixed time. So here is phi t of P. And here is phi t of H u of P. Yeah, clearly I must learn to present this in a better way until next lecture. I mean, next time I give the course. I will come to that. Here I stated just for some interval. But OK, I will point out the corollary. OK, so yeah, maybe we can, I will be happy to try to explain it more in the tutorials. Yeah, I wanted to point out that one application of this, one special case would be if H is a piece of a closed horror cycle. For instance, in the modular group as I had before, then perhaps I have my fundamental domain of the modular group sitting here. And then since I have a casp in this case, there is an associated family of closed horror cycles. If I have a completely horizontal horror cycle here, then it's clearly a closed curve on the surface. I come back to the same point after having flowed a certain time. And so there is a one parameter family of closed horror cycles. They can be parameterized by the length, for instance. And when I'm flowing in this direction with the geodesic flow, the length of the horror cycle is increasing exponentially. And I have a, well, so this says that a really long closed horror cycle sitting down here. And if I want to see it on the surface, I have to pull back. I have to take each point on this horror cycle and pull it back by the appropriate element in PSL2Z. I will get some curve, which I can't draw. But I will get some curve inside this fundamental domain. And the point is that this curve, when the horror cycle is really long, it will look almost equidistributed according to the hyperbolic area measure. And even if we lift it to the unit tangent bundle, we will have equidistribution also direction wise. Date, ah, OK, I think Selberg has it in unpublished work in this case, at least in some setting. But Sarnak, the paper by Sarnak, which we have in our list of references, 1981, well, he proved it. OK, no, no, no, he didn't prove it. He proved it for closed horror cycles. So who should one? OK, Eskin and McMullen, they do it more generally. So maybe Eskin and McMullen, 96, I think. But I'm not sure if there is a bet. Again, Margolis, ah, in Margolis thesis, it is, ah, yes, of course. I'm going to explain the proof of this. And the proof, you can prove it using a certain thickening technique, or it's also called Margolis trick. And he actually pointed this out for more general and also flows in his PhD thesis. So sorry, this is way, yeah, it was much earlier. Sometimes late 60s maybe, or early 70s. Margolis thesis, so this is rubbish. I mean, no, no, no, the papers are not rubbish. Sorry. So I want to outline the proof of this. But I first want to point out that from this, you can actually generalize it more or less directly by noticing that we get the same limit for any piece of a horror cycle. Then this means you can, using that just, you can, one can see that instead of having Lebesgue measure along a piece of a horror cycle, we can have just any absolutely continuous measure on the real line. Any absolutely continuous probability measure, say. And we will get the same limit. And we are also free to let the test function f depend on this parameter u. This is technically very useful in many applications. So I will leave it as an exercise to prove it. And I'm happy to discuss it and so on. But I want to state it in a precise way. Corollary for any absolutely continuous probability measure new on the real line. So absolutely continuous with respect to Lebesgue measure. And again, say for any compactly supported test function on x. We have that if I instead, I take the same average as here, but with respect to the measure new instead. So f phi t, h, u, p, d, nu, u. And this again tends to the same, the volume average of f as t goes to minus infinity. And even, as I said, for any test function, and now it's a compactly supported function on the space r times x. And I do the same integral, but I now have two parameters and I let f depend also. I have u, phi, t, h, u, p, d, nu, u. And this again tends to the same thing. So f depends also on the parameter u. And the proof of this is quite simple. Basically, you decompose. OK, so we will discuss it as an exercise. But by approximation, clearly, we can replace the then, because f is a bounded function. So I can approximate the density in L1 sense. So I can easily reduce to nice, compactly supported, continuous density. All right, I don't get the same limits. Thank you. Thank you. Clearly, you have to integrate now. So r times x. And then f, u, p, du, no, no, no, no, d nu, no, d, and then the other order, d mu, p, and d nu, u. The same average here. So that is no good notation. So thank you. Q, maybe? Sorry if it's not legible. So this is an exercise to deduce this corollary from the theorem. But now, in the last six or seven minutes, I hope to just outline the proof. One proof of this theorem, this equidistribution theorem, using a so-called Margulis trick or Margulis thickening technique, which he introduced in his PhD thesis. And it will use the fact that the geodesic flow is mixing, outline of proof. So we would like to apply the mixing theorem with F1, OK, I just erased the mixing theorem, but you have two arbitrary test functions, F1 and F2, in the mixing result. You would like to take F1 to be equal to the test function F. And then you would like F2 to be a one-dimensional measure along the horizontal. If you were allowed to do that in the mixing result, you would get this directly. So I just write this out. Want to apply theorem 1, the fact that the geodesic flow is mixing, with F1 as the given test function F. And F2 as one-dimensional Lebesgue measure along the horizontal piece H, the fixed given horizontal piece H. So the one-dimensional Lebesgue along H. However, this is a distribution. It is not an L2 function. And the mixing result doesn't extend to arbitrary distributions or arbitrary measures. It's easy to show by examples. It doesn't. For example, if I would take an average along the unstable horizontal cycle instead and then flow backwards in time, it clearly I would get some result which is false. So I'm not allowed to do this. But still, OK, this is forbidden. But still, one can start asking, what happens if I try to approximate this Lebesgue measure by some characteristic function, some tube around H? And it turns out that since H is an unstable curve for the expanding deodesic flow, I mean I'm flowing in the direction such that it expands, since I'm averaging along an unstable piece of, since I'm expanding along an unstable sub-manifold, this approximation will work. It will work for me. So OK, so the trick is to thicken the horizontal cycle piece H. And OK, maybe I want to leave the statement of the theorem, so I move here and discuss. So OK, we want to thicken H. So we fix some small epsilon and let H sub-epsilon be an epsilon neighborhood of H. Now, this is not completely, this is a bit vague because I've never introduced a metric on G. One could introduce a metric and this would make sense. But I'm outlining the idea of the proofs. Then I hope to say a bit more about how you can make it more precise. So this is an epsilon tube around the fixed horizontal cycle piece H. And let F2 be the characteristic function of this tube. So characteristic function of H epsilon. But let's normalize it so that it has total mass 1. So I divide by the volume of the tube so that F2, the total volume average of F2 is 1. OK. And now we want to see what happens if I flow this tube. I have this initial horizontal cycle H. And now I have fixed some epsilon neighborhood of it in some three-dimensional neighborhood, which is H epsilon. Ah, yeah, of course, thanks. OK, so I have H and I have H epsilon. And we flow with the geodesic flow. And I know that the curve H, well, by definition, that becomes the curve phi th. So t is negative, and the geodesic flow is going in this direction. But the point of this proof, and I'm not going to prove it, I'm just stating it. The point is that actually also phi t of H epsilon is a nice neighborhood of phi th. It remains so when t goes to minus infinity. And the reason is that the horizontal cycle itself is the unstable direction for the flow. And so when I go out, go away from this horizontal piece, I can see it as moving along the geodesic direction and the other horizontal direction, which is stable, so contracting for the geodesic flow. So that approximation, when I transfer it by phi, will remain a small approximation. Of course, some point sitting here, say, one-third between the starting point and the end point, that point can move quite far along this unstable direction. And it can. But if you write down this with coordinates, it turns out that that averaging remains nicely. It spreads out nicely, so that averaging remains good. So that's the point of it. I'm going to write something. It's basically epsilon. OK, so you have to introduce coordinates. And the point is that you don't have to take epsilon neighborhood. You can even take some fixed one neighborhood along the unstable direction, but it will remain epsilon if you have fixed the metric and so on. It will remain. So in this proof, we can fix epsilon and we get the result. And then only at the very end, we will let epsilon go to zero to get a good approximation, a bit vague. And I want to write at least a few lines to say something. So the point here is that phi t of h epsilon is near the transported horocycle piece. And this means that here, now, to conclude the proof for this hand-waving proof, here's the thing that I wanted to compute. I wanted to compute f of phi t of hup, the u, one of the u. And since this, here I'm moving along this transported horocycle, and since the tube phi t h epsilon is always near this, and the averaging is as it should be, this turns out to be approximately equal to f phi t p and f2 p. Remember, f2 is the characteristic function of h epsilon and d mu p, because this is an indicator. This kicks in only when p is in the epsilon neighborhood of hu, and that means that p is in hu, is in the original h. And then I get exactly this. So how good this approximation is depends on epsilon. But now, here I can apply the mixing result. This, as t goes to minus infinity or plus infinity, if you like, this tends to the volume average of f d mu times the volume average of f2 d mu x. And I chose this, so it's equal to 1. Otherwise, this approximation, that would be a constant here. OK, so that's really hand-waving. I was, yeah, hope to give a bit more details, maybe, in the tutorials. Thank you.