 Okay, so like everyone others, I would like to thank the organizers for this nice conference and give me an opportunity to give a talk here. And also I would like to thank the audience here for coming. Now it is not very trivial, I think. So this time I would like to speak about the problem of periodic orbit counting. So we consider some flow on a manifold, Ft, m to m. And we assume that, so this is the flow. And this is non-singular, okay. And we count the number of periodic point whose period is less than given number of some t. So we count the number of periodic orbit. And here gamma is prime periodic, so periodic. And we denote by this the period of gamma, okay. And we want to see how, what is the asymptotic of this number. So for example, if you take the timeline and maybe after sometimes you find one periodic orbit here. And then you have, if the flow is hyperbolic, say Anosov, then you will find more and more periodic orbit, right. And we want to see how the number increase, okay. And there are several works on it. And maybe one of the beginning observations that in say, in some flows of some high publicity, then maybe this is maybe proportional to, or if you take a log and divide it by t, then it's like, sorry, okay, sorry, I'm confusing. So maybe this is what is well known. So basically this grows exponentially first and this rate is topological entropy. But if you restrict ourselves to some more specific system, then we can get better estimate, okay. And one well known theorem is something called prime periodic orbit theorem. This is first observed by Marguerite for geodesic flow on negatively curved manifold. And then improved in much more general setting by parry and polycode. So, so for example, if we assume that the flow is answer flow, and assume that it is mixing, or topologically mixing, or actually the theorem by parry and polycode treats a more general setting, but here we restrict ourselves to this thing. Then the number of periodic orbit is given in this formula. So there's some small error term or one, but basically the, so rest, pi t is given by this one. So basically, okay, the ratio of the left-hand side and this that integral is close to one. So it gives a very precise asymptotic for the number of periodic orbit, okay. And here h is an topological entropy, and it is very well known theorem. And it's a kind of analogy of prime number theorem in dynamical system setting, okay. And more recently, so if you restrict also further, so to the contact and so forth. What is contact and so forth? So if the manifold n carries one contact form, and we assume that answer flow preserves that contact form. And this, if you don't know this, maybe you can imagine that this is a geodesic flow on a negatively-carved manifold. Okay, then in that case, actually this error term, this error term is actually exponentially small with respect to t. So we can get, actually this term can be written like this. It is proved by poly-cotton shaft in two-dimensional setting, and no, three-dimensional, I think. And then in higher-dimension, proved by 2.5 more recently. Okay, so this is quite sharp result for number of periodic orbit. It's already very satisfactory, I think. But I want to go further, and if you consider the case of geodesic flow for hyperbolic surface, then we can use some formula called cell bug trace formula, and give this very precise formula. So this is much more precise now, formula for the number of periodic orbit. So here, so the main term is the same, but now we have a few resonance term, and then we have some error term. And error term here is e to the rho t, and rho is 3 fourth of entropy. So actually this is big, so it's exponentially big, but compared to this, it is much smaller. And this resonance term has some intermediate growth. Okay, let me add that. So in this case, because the curvature is constantly one, so actually entropy is one. And these numbers, so that appears in the resonance term, is related to the eigenvalues of the Laplacian in this way. So these are real numbers. Okay, and my question is that how general does this kind of asymptotic formula holds too? So I want to show this kind of formula up to some resonance term for more general anosoflow, or a contact anosoflow. So let me, so now throughout this talk, I want to mention this formula. So this is sometimes called full bus formula, so this is the main term, t to the e, h, t, dt, plus some resonance term, plus some error term. Okay, and actually I have one result in this direction. So actually I have this formula, but for general contact anosoflow, for example, I have one formula which holds for geodesic flow on the negatively curved surface, especially in variable curvature case. And in this case, the error term is of this form, so this form, so we actually take, sorry, I have to write t here, sorry, so the error term is like o exponential, and we take average of h and half of maximum depth of exponent. So this is what I, actually what I can prove, and okay, but okay, one remark is that now this resonance term may be complex, but anyway, so we have this, so this is raw, so we have asymptotic estimate like this. And I like this result very much because if we consider the constant curvature case, then maybe, so this maximum depth of exponent coincide with topological entropy. So this error term, or this row becomes one fourth of, three fourth of entropy. So this is just the same as the case of Hoover. So it is very nice, but actually if this ratio, so usually the maximum depth of exponent is bigger than topological entropy, so this is bigger than one. So then this estimate is very good if this ratio is close to one. But if it becomes greater than two, then if you look at this, then the second term, so lambda max over two becomes h, or bigger than h. Then this error term is bigger than the main term. So this is quite backwards. And so in some cases it gives a very good estimate for error term, but sometimes it's very bad. And maybe in the purpose, one purpose of this talk is how this happens. And maybe I want to give one remedy in a rather simplified setting. Okay. Now, so this is a motivation. Now I want to consider, so actually I'm speaking about rather technical problem. But I want to emphasize that this kind of problem is somewhat universal in the, in the, actually in the field of quantum chaos or wave chaos. And it's some, it's some very similarity because you see that, okay, here we have some condition that the maximum depth of exponent is bigger than twice of entropy. And this kind of condition sometimes happens in this area. And in one side, everything goes well. And the other side, we don't know anything. And maybe we can think of this problem as a counterpart of that kind of problem in dynamical system. So I want to explain the problem and then give some solution for it. Okay. And since I want to extract the essence of the problem, I want to consider things in a very simplified setting. So we don't consider the analysis of flows, but we consider some very simple expanding semi flow. This is an suspension flow of an angle multiplying map on the circle. So we consider mapping tau, which is just an angle multiplying map on the circle. So maybe if you like, you can just consider angle doubling. And then we consider some loop function. Okay. And it should be smooth. And then we can consider this suspension semi flow of that kind of mapping. So maybe this kind of mapping flow has appeared in this conference twice or three times. So the flow is like this. So the phase space is this one. So S1 cross R. But in this R direction, we have this restriction. So the orbit goes up by constant speed. And if it reaches the upper bound, it goes down. But when it goes down, we apply the mapping tau. Okay. So basically it is an expanding flow, but actually this is not one-to-one, so expanding semi flow. And we consider things in this simplified setting. Okay. Then everything becomes much rather simple. And to consider this kind of problem on periodic orbit counting, actually one secret is to consider transfer operator. And it's spectral to get spectral picture of it. So this time, so the related transfer operator is this. So this is a transfer operator acting on the function on the phase space. And this is simply defined by this formula. So this just push forward functions by the semi flow. And if, but of course this is, this map, this flow is, this is semi flow. Okay. Several point comes to one point by, and if the time is t is large, then this, for each point z, the point satisfying this condition is very large. Number of such point is very large. But anyway, so we defined the transfer operator like this. So locally it's a very simple transfer operator. There's no co-action. And the relation to this transfer operator and periodic orbit counting is given by the so-called Atiyabot trace formula. Okay. So we consider the trace of this transfer operator. But usually LT is not a trace transfer operator. So here we write down LT as an integral operator like this, kt, y. And this integral kernel is rather singular, so kt, x, y is delta ft minus 1 x minus y. So this is a short kernel of this transfer operator. And the flat trace is defined, so this is the definition, is the integral of this short kernel along the diagonal. This is not a usual trace, but anyway, this is a kind of trace for operator LT, and we can compute it. And if you compute it, then you will get this formula. So first maybe you can just easily observe that this integration is related to the periodic point, but because this takes non-zero value only around the periodic orbit. And if you do some a bit more careful computation, you will get this formula. And then there's one remark, so here D gamma is the expansion rate along the periodic orbit. So if you go along the, so this is the periodic orbit, then if you consider Poincare map at some point, then you consider the derivative of the prime Poincare map at this point. And then you will get D gamma. So we take this sum, so this is actually not a usual function of t in usual sense, this is a distribution. So at some point, so when t is a multiple of period of some prime periodic orbit, a prime period, then it takes, we put some direct measure on it. And we take sum over all the periodic, prime periodic orbit and take sum with respect to n, natural number n. And maybe it is easy to observe that this effect of this factor and also sum over n because of 2 is relatively small. And if you do some estimate, then we can show that actually we can drop this up to some small error term. And you will get this formula. So number of prime periodic orbit is given by this integration up to error term of this size. And this is actually much smaller than that we will have as an error term. So basically we can ignore this. So we can express the number of periodic orbit by using this integral. Ah, yes, in this case. But note that this is not a trace in the usual sense. But now I ask you to admit that actually this is related in some sense to the spectrum of LT. So for example, if LT is a finite dimensional matrix and one parameter family, then I think you can easily imagine that you will find this kind of formula for the number of periodic of this integral. So if it is finite dimensional and if you take this as a trace, then maybe this chi i should be, chi i and h should be the spectrum of the generator of LT. Okay, so if the thing is a finite dimensional, everything goes well and we don't have error term. Now we extend this idea to this setting. So for this, we introduce a notion of essential spectrum radius. So if we are given a bounded operator L and suppose that it is a compact perturbation of L0 and L0 has an operator norm smaller than or equal to L lambda, then we know that the spectrum of L on the outside of the disk radius lambda consists of discrete eigenvalues. And the infinimum of such lambda is called essential spectrum radius of L. We try to consider arbitrary compact operator and we consider this decomposition and we see that how small we can make the operator norm L0 small. Okay, and so basically we can reduce the problem of periodic orbit counting to the spectrum problem on the LT and especially reduce to this question of estimate of essential spectrum radius choosing some appropriate functional space. If we know that essential spectrum radius is small, then maybe we get this spectral picture and then we get this asymptotic formula for the periodic orbit. And basically the eigenvalues outside this essential spectrum radius appear as this resonance term. So maybe it should be true that the largest eigenvalue should be h or e to the ht for LT. And essential spectrum radius should be related to this number of. And to tell the truth, by some technical reason, this number, rho does not coincide with the essential spectrum radius. Actually we have to take the average with entropy. This is a rather technical problem, so I want to avoid this. But anyway the truth is that by technical reason we have to take average. So for example, in the case of constant geodesic flow on the hyperbolic surface, actually essential spectrum radius should be e to the half of entropy times t. And the Foubert theorem gives a, here we have a 3 4th of h. And so at this moment we cannot avoid this technicality. Anyway, if we get very good estimate on the essential spectrum radius of the trans-operator, then we get good estimate on this error term. This is true. Okay, now I want to go into more detail of the structure of the trans-operator. So we are considering some semi-flow. And as we saw, the flow is locally just a translation go up. So it is very natural to consider the frequency decomposition of the function with respect to the frequency in the flow direction. Because this is, the frequency should be preserved, right? And because there is some recurrence and the manifold step is compact. So we cannot do this decomposition completely. But mentally it's possible. So suppose that it's possible. So we decompose the function on the space xf with respect to the frequency in the flow direction. Okay, it is virtually true. Of course, in detail we have to do something. But for this moment let us consider that this is true. So then the LT preserved that decomposition. It implies that LT is basically decomposed into countably many operators. Okay, and each of the operator is just the action of LT on the function space, on the space of function who have fixed frequency in the flow direction. Okay. So it seems like this. So if you consider this operator, L omega t, and omega denotes the frequency. Okay, so we consider some functions. Okay, we just look a local picture of the flow. So in some neighborhood of the point, there is some function. And maybe I'm writing this black line, the point which has the same phase, complex phase. Okay. And then since our mapping or semi-flow is expanding in the direction transversal to the flow. So after a long time, actually this circle, this neighborhood food is expanded in the horizontal, the direction that is transversal to the flow. So we will get this kind of picture, at least in the universal covering. And then finally some point comes to one point. Because, okay, this is a picture on the universal covering. But now we finally have to go down to the manifold XF. Okay. So the operator is somewhat simple. So locally it looks like this. Okay. So we consider some function of this form. It have a fixed frequency in the flow direction. And we expand it. And then we cut off some part, cut up some part and superpose it. This is the operator we are considering. So essentially the problem is the following. So suppose that we have some function like this. So this is a wave function. Simple plane wave function in the direction CK. And I suppose that there is complex space. But of sort of body is just A or just one. Okay. And we superpose those functions. Okay. So we will get some functions. And maybe there are some interference between them. And we ask what can we say about the result? Of course it depends on the choice of these numbers. For example, for choice of these directions. And maybe one question is that can the result, the result of superposition can concentrate on very small subset? This is the question. Maybe let me explain, go back again by this setting. Okay. So here, okay, let us try to estimate the operator norm of LT omega. So first we consider L infinity norm. Okay. Then from this process. So, okay, this is just for expansion. And, okay, so if you look things in L infinity norm, there's nothing. Just the norm is preserved. Okay. And then we consider superposition. Actually, this number is proportional to e to the ht. Right? e to the ht. This is a number of backwards of it here, coming to here. So if you consider the L infinity estimate, then, okay, we get this estimate. But this is trivial. Basically, this gives the main term and gives nothing else. Okay. Because this is not a good estimate because it does not consider any cancellation between the functions. Because there's a complex phase. So if you add, there should be many cancellations between the functions. Okay. And to look at those cancellations, actually L2 norm is the best one. So, actually we call it the L2 norm. And we support some transversality condition. So we support that these frequency directions are transversal to each other. Okay. Then, okay, in this process, actually we have this estimate, this growth of L2 norm. Because the area is expanding by this rate. Okay. Sorry. This should be lambda max. But lambda max, t. So since we consider L2 norm, maybe we have to divide by 2. Then, we superpose it. But then, actually, if the frequency directions are different, then they are orthogonal in L2 norm. So basically, we don't have any increase of the operator norm. So finally, we get this estimate. This is the estimate that we have in the contactance of case. Okay. But the problem with this L2 norm is that, okay, flow is expanding in horizontal direction. But the expanding rate depends on the point. And sometimes, there is some small factor like set in the phase space in which the flow, semi-flow is strongly expanding than other place. Okay. So, suppose that there is some ergodic measure because depth-noth exponent is very big and bigger than twice of entropy. Then, maybe that should be concentrated on this, on some small subset with small half-dolph dimension. Okay. So, and if we consider L2 norm, okay, we can capture the cancellation between the function. We don't know what in which place the result of superposition can concentrate. So, after this, it becomes a problem because if the result of this superposition, if the image is concentrated on this bad region, then it should be expanded strongly later. And this is a situation we like to avoid. Okay. Now, what we can do? So, this is basically the essence of the problem. Okay. So, we consider the superposition of plane waves. And if the image is concentrated on very small, but bad factor like set, then it causes problems. And I don't know how to... I don't really know how to avoid this situation. And one idea of mine is that we consider Lp0 because L2 norm captures cancellation between functions very well, but actually we do not know much about the result. L infinity norm gives only trivial result. So, we like to go in between. And if you consider that Lp0, we can get the following result. So, this is the main result of my talk. So, before we consider L2 norm, but now we consider Lp0 and p depend on the rate... Okay, we can choose p, any p. And the result is following. So, for each p, actually there is some sub-banner space which is contained in L2p, such that... So, LT, this transformer for sufficient elasticity is bounded on this banner space. And spectral radius is what we expect. And the main statement is the following. So, for any option, actually there is this open then subset of seeding function or root function. So, in general case, then essential spectral radius of LT on this banner space is bounded in this way. And this row to pf is defined in this way. Okay, let me examine what this means. So, for example, so this is a formula. Okay? And if p equal 1, we can choose p. So, if we take p equal 1, so actually this is L2 case because we consider L2p. So, this is L2 case. And then, okay, p is 1. And this is bigger than 1. So, actually we always have this and this part is cancelled. So, we have this formula. But, basically we can choose any p. And maybe one of the good choices is this choice. So, we consider the ratio. So, this is bigger than or equal to 1 by real inequality. And depending on this number, we choose p. Then, okay, this number p is slightly bigger than this one. So, actually max is just p and we get this formula. Right? And what I want to emphasize is that now this number is smaller than the political entropy. So, our formula is not a vacuous. Okay. And as a consequence, so this is an estimate on the essential spectrum radius. But now, we can derive the consequence for the periodic orbit counting for this simplified case. And in this case, so we have this, a simple formula. And this roba is just an average of h and rho p. Sorry, I have to write rho to p here. Okay. And we can take minimum. So, basically this number is smaller than this one. So, if this is smaller than or close to 1, maybe this is close to the Foubert's formula. But if this rate, so basically this is a non-uniformity of the expansion. And if this becomes bigger, then our estimate becomes worse. But anyway, we have some exponentially small error term and some resonance term. Finally, I would like to speak about the idea of the proof. But the proof, the idea is not difficult because, as I said, in estimating L2 norm, we consider the transversality and the transversality gives the orthogonality in L2. And if you consider L2p norm, maybe we consider this kind of U1, U2, Up and V1, V2, this kind of integral. And we see how often they disappear. And, okay, so this is rather, of course there are some technicality, but either itself is not difficult. Just replace L2 norm by L2p. Okay, I stop here. Thank you.