 Thanks a lot for the invitation. It's a pleasure to be here and I really love that you keep it running even though Corona is officially over. So, let me start by saying what we will talk about kind of a very broad terms. So what I want to consider is on a hyper. Sorry, I first need to activate this year. Okay. On a hyperbolic surface. I want to consider automatic functions. And these are here really understood at the moment kind of as unfolded objects. So they live on the upper half plane with values in C. They are invariant by the group action. So this G is somewhere in the gamma and the Z is in the H. And they should be eigenfunctions of the Laplacian, the standard Laplacian here. I will always split as usually the eigenvalue into s times four minus s. So we talk about spectrum parameters. So these hyperbolic surfaces, they don't have any properties besides being geometrically finite. So I do allow elliptic elements. I also allow infinite area surfaces as well. What I want to understand is how I can study automatic functions via dynamics. Or if we go back a bit in history via the geometry of the space. And of course, we all know there are a lot of results. It's really history which goes back more than 100 years. So let me just mention two results, which will be also of interest for today. The one is the Zebik zeta function, which as we think we all know is given by this infinite product, this double infinite product, the one going over an auxiliary parameter here, then having the Euler product in here. And this L, which comes here in the exponent, runs through the set of primitive length of periodic geodesics. And of course, it also goes via multiplicity. So if there are several periodic geodesics with the same primitive length, we just have this in here several times. So what is the Zebik zeta function, which is a purely geometric object, helps us to understand automatic functions. So this is a result which in the first instance goes back to Zebik himself. And then later on was generalized to other surfaces refined and so on. So I would just omit references for this is just too many references. So it says we have a zero of the Zebik zeta function, if and only if this x is a resonance of the Laplacian on the surface. So a resonance here is a pole of the resolvent. And for this purpose here, I do need the splitting into spectra parameters, because otherwise I do have problems with meromorphic continuation of the resolvent, or these zeros might come about for a different reason, which is called then a topological zero. So it's there by reasons which are not of spectral nature. We do understand very well where these topological zeros are, how many they are, is three of minor order. So for the purpose of this talk, I will forget about these topological zeros. Also this if and only if it's not 100% correct, there is a minor set of exceptions. But for the purpose of the talk, it's okay to just pretend it's an if and only if. So what do we see here on the one side, we see a formula up here, which is formed only using a geometric information or if you want dynamical information, namely the length of periodic geodesics putting it into some object, which is then by its very nature of definition, a purely geometric object. This year has an axis of convergence, which is, I would say, easily proven. And then it has a meromorphic continuation, and in the meromorphic continuation, we might have zeros, and these zeros give us spectral information of our surface. So the question what one can have here is, is it possible to go deeper into this kind of relation that we can say we do not only see resonances, we do not only see the spectral parameters, not only the poles, not only the eigenvalues, we see more. We also see the eigenfunction, the automorphic functions belonging to these eigenvalues to the spectral parameters as well. And indeed, this is possible by now, at least for some surfaces, and it's getting more and more extended these kind of results. So let me recall this, or show you this in a first example. So this here, which I'm stating now, actually goes back already by now, 20 years probably, where it's possible to assign to automorphic functions something which is called a period function. We will see an example in a second, or actually functional equations, which are defining equations for period functions. So how does this here go? It's kind of first the abstract way, what we want to expect here. So on the one side, you take an automorphic function, and you assign to it another function, which is typically living on some real interval, and the way to go from the one side to the other is that it's typically given by the integral of a certain closed form. I will give you a precise example in a moment. This form takes as input the automorphic function itself, and of course there is parameter t, and then it integrates along a well-chosen geodesic, which makes it reasonable to call it a period function. So what we will be mostly interested in is not the period function itself, but more the functional equation they satisfy. So for example, the first example where this here was worked out in precise terms is for our favorite modular group. So this function on the period function is satisfying this functional equation here, and this t here is just running through the positive reagents. Okay, so what is more interesting than even the functional equation itself is how it comes about or how one can say it comes about. Originally, this bijection up here was proven by Louis Zagier in number theoretic terms, so they used harmonic analysis, a lot of analytic number theory, but after that there was developing a different approach to that, which goes via dynamics, and which from my perspective gives a bit more insight about why it's working, what we should expect for other focusing groups, and so on. So the participating entities which we have here are actually like this. The functional equation, it looks at the moment when you look at this functional equation, you would say it's kind of arbitrary. Of course, you see something here. When you think about the modular group as being generated by this element here, and then of course also by this element here, you do see the action of these elements in here. It would be kind of an action of this element, and here you do see the action of this element. And this t here is something which is very built into the system, which comes from a dynamical approach to this, and we'll try to tell you what is the dynamical approach here. So what we have is how we go to these functional equations is this. You take your surface. This here is for the example of the modular surface. On the surface, you have the geodesic flow. So let's say here's a geodesic, it's moving on your surface, and you try to encode your geodesic in a way that you can handle it in a much better way than just having kind of this continuous flow on your surfaces. So what you do is you pick a so-called cross section. A cross section is a set in the unit tangent bundle, which is intersected by every periodic geodesic, and whenever you have an intersection, it is discreet in space and time. So it's sufficient to think about discreet in time when you travel along the geodesic. In this setup here, I would take a cross section which is based on a complete geodesic. And the way you should really think about this is you go to the upper half plane. So you unfold the situation. You move this here over to here. It's all these unit tangent vectors on the imaginary axis, which point to the right. Let me draw in the classical fundamental domain, which would be here, somewhere there's i, somewhere here's minus one half, or more or less. And what I did instead of unfolding it completely, I took over the part which is originally here. I brought it back down here, which is possible by the action on the surface. So the second thing I want to do is I want to change now. Let me restart the explanation here. What I want to do is this. I take a geodesic, so a starting point here. Let's say it is this. I move along this geodesic. A cross section is chosen in a way that it intersects again. Otherwise, it's a non-well chosen thing. And also this second pink vector will be somewhere, let's say it's here. I follow this geodesic here. It's the first one. So let's give them some recognizable things. So this is the V, this is the W thing. And I do the same with the second geodesic. And I ask myself, how do I move from this point here, the endpoint of that geodesic to this point here? It needs to be possible within the group action, because on the surface, so on the quotient, it's the same geodesic, just time-shifted. Good. So how do I do this? The way to do it is you take the green thing here, and you move it across the whole space with the group action. So you consider this set here. And when you play around with the fundamental domain and all this side-pairing properties, what you notice is it's better to think about this fundamental domain here, this here. And move that around. You will end up with something like this, this triangle. And then the only places where you have to look for the next intersections is either here or it's here, because all the other vectors, all the other tangent vectors are not in your transported set. So in this situation, which we are in here, the vector of interest, the next intersection would be this. Let's call this, I don't know, GV. And you also know which element G have. This here is just this group element acting on the full set of cross-section here. So you move this GV, the GW, of course, back to the W by applying G inverse. So this element, let's call this T. The T inverse, which also means down here, you have to apply the T inverse to go from the X to the X prime. And this is something you just do for all tangent vectors, for all geodesics which are there in your cross-section. And then you look very closely at the boundary down here and you notice as soon as your endpoint of the geodesic is above 1, you will always use the element T inverse. When it's below 1, it's somewhere between 0 and 1, you will use this element here. And then it gives you on the boundary, so on R plus, it gives you a map, let's call this capital F, which goes like this. So when you're above 1, you're mapping here with the T inverse and when you're in here, you're also mapping to the set here. And here you apply, let's call this otherwise, this other element is T transpose and then the inverse, so it looks like this. Okay, so what do we do with this for getting the function equation? Now there's coming an idea from statistical mechanics or egotics theory, thermodynamic formalism, what we want to have is an operator, which is an evolution operator of this map F. So the way to think about this is this, you have a little function F, which is living on the domain of the capital F, so on 0 infinity, pretend it to be a density or distribution function, or something you want to evolve with time, and time means one time step for the capital F. So what you do is you take one point and you ask for all the pre-images you have and you accumulate all the information at these pre-images and in front of here, you have to take, because of the COBI determinants, the derivative of the capital F to the minus one and because of statistical mechanics, you want to have something which is kind of an inverse temperature, which is then getting into the spectra parameters instead of the one you take the S in here and then you just evaluate it, so you see for every point which you can have you have two pre-images and if you go through the full calculation, you will find that you get exactly the function equation from above on the notes. Okay? Okay, so what is going on here? So what is the way of thinking about this? This encoding of geodesics, you can understand as discretizing the geodesic flow that you have kind of a discrete dynamical system which is reflecting certain information of the geodesic flow and exactly that information which we want to use. This unfolding procedure with inducing a map on the boundary is kind of a, again, a discrete system. Here, of course, you're losing some information about what happens in backwards time, but forward time is sufficient in particular if you only want officially or honestly talk about periodic geodesics because a forward direction of a periodic geodesic knows what's happening in the backwards direction. And then you're trying to find kind of an equilibrium state and this is done by this evolution operator that it asks you for how is life being for the functions or how it's distributing mass and in the end, you will ask for one eigenfunctions here and this parameter s which goes in here is deeply motivated by physics telling us, okay, somehow we need to have some additional property in here kind of thinking back to correspondence principle between classical and quantum mechanics. So somehow spectrum information needs to be built in here as a kind of inverse temperature which is then for mathematician, just a spectra parameter which goes into this exponent. Okay. So the way you can think about this approach here compared to the Zebik-Zeta function is this. When you think about the Zebik-Zeta function, you find spectra informations on your automorphic functions but only the eigenvalues of the spectra parameters what we have down here with this bijection, so this bijection here is we get information on the automorphic functions themselves by using also the flow, the dynamics, not only the length of periodic geodesics, we also use the periodic geodesics themselves. This is an approach with this transfer operator business which is by now I would say about 20 years old. It goes in a lot of names. So Louis Jagje from the beginning on from the number theoretic side and there's Dieter Meyer who is a physicist and there's this doctoral student Shang and then also I did some parts about this by simplifying the system here. The question one immediately has here is of course how is this here related to the Zebik-Zeta function and there's another interesting object coming about here. So when you think about this dynamical system which we have in here with this one step situation, it feels a bit like continued fractions but way too slow, right? Because whenever you have something which is up here, let's say it's a 10 and 10x plus an epsilon, you would move 10 times backwards just with a one. You can speed up this thing here by saying, okay, instead of having the next hit I want to count as one object all the successive uses of the same kind of intermediate action which we give you a second discrete dynamical system which is then actually the continued fraction algorithm which also comes with an evolution operator. I will call this fast because it's speeded up. This here is then on a good space in nuclear operator which means you can take a freedom determinant of that and this is on the nose the Zebik-Zeta function for the system. So which means you do get information back of the Zebik-Zeta function situation via this approach. But this approach, it has kind of deeper information even though at the current state of art we don't know how to use all the information which is built into the system. So there's still a lot of research to be done to get information out of the system about automatic functions or distribution of resonances or whatever you want to do. But what I want to focus on today is another question namely when you have your automatic functions in this unfolded situation with which I started you have the standard question why you want to work with this and not with something more generic. So what I would like to do instead of going only into C I would like to be in a vector valued situation so some finite dimension vector space here and I also would like to have a representation here at the moment a unitary representation which allows me to have an equivalent here which is not trivial. Why do I want to have this? So of course there are a lot of advantages when one sees, when one works with this time by time so also for me it took quite some time until I understood how powerful this can be. So the first thing is when you want to work with one group you can go through the whole business what you want to do and then the question comes what happens if I take a different group then officially you have to do the whole business again but if it's a subgroup of finite index then you can bid in the finite index so the subgroup into this business by using the induced representation from the subgroup to the group you already worked with and just consider this vector valued setup so you can study both of them simultaneously and honestly whenever you have this representation in you study anyway a whole bunch of representations so you have even more of these subgroups built in. Another reason why one wants to study is this think about having not only one representation but a continuous family of representation which starts at the identity and you want to understand how something is behaving by using a perturbation so you have some property I don't know maybe finding zeros finding poles and then you modify the system and you ask how stable is being a zero under this perturbation and one way to do these perturbations is by putting in representations into the system slowly moving along a family of representations so this is something I would call perturbation then of course there is also a historical reason Zellbeck himself promoted to do exactly that to study vector valued automorphic functions simultaneously with the classical ones to see what are the differences what are the similarities what stay stable under turning from classical to vector valued ones to understand the properties and by now they are also used somehow in physics don't ask me about the physics I only know that physicists ask for these representations so they will have a good reason to do this when it comes to these questions above with the Zellbeck-Zeta function there is a Zellbeck-Zeta function also for unitary representations you just beat in the representation into the formula I will show you the formula in a minute and this if and only if statement about the zeros stays true it just carries over at least for finite area hyperbolic surfaces there is still a gap in the literature about infinite area hyperbolic surfaces but from my perspective this is more like just a gap that it's not that new methods are needed also this second part with how to go from automorphic functions to period functions if you follow through these old papers and also some newer papers which I will come to you see that you can beat in unitary representations into this whole isomorphism and it also stays true it's not really written up because we worked with more general situations but one sees that it carries through and before I state the Zellbeck-Zeta function let me ask a new question and the question is why do we want to restrict to unitary representations and when I started to ask myself this question which is by now some years ago my honest answer was I don't know why should be and then I looked around a bit and found that people started working on exactly this question what happens if this representation is not unitary anymore how much is still valid so why do people do this there is a motivation coming from physics and again here I only know that there is motivation from physics I do not really understand why and how it's needed somehow it's used for conform a feed theory there's also quite some literature about vector-valued-automorphic functions with non-unitary representations by Knopp and Mason they call it generalized-automorphic forms so they talk about forms of perturbation above here with perturbation theory this of course stays and also when you want to talk about finite index subgroups of course this stays in the picture because it was there before the effect what we see here in addition how much more flexible it is is this so let's say this is a complex plane and let's suppose this is our hyperbolic surface so for the experts in the audience this is a hyperbolic it's a Schottky surface with one generator so I have one hyperbolic element which is generating the full group the set of resonances looks like this it's on a grid for the classical setup so this is for the situation that you have no twist here if you now have a twist in here and you allow the twist to be unitary then you're moving and you start at the identity and then slowly move up and down this is a one-dimensional situation so it's a character up and down in this direction but when you allow this to be also a non-unitary family then you can move wherever you want so if I tell you I want to have this point moved to that I can give you a family which would do this and then some of the others have to adapt if you tell me you want to have this here move to that I would just give you a family of representations we go through non-unitary and I can move it in there so kind of the perturbation which you're allowed to do are much more flexible I'm sorry I don't understand what's the relation between the lattice picture on the left and so the quotient of H not the one-parameter description so I'm asking the question if I consider this surface here this this hyperbolic cylinder what are the resonances on that surface you can also say the same thing what are the zeros of the Zebic Zeta function for this situation the Zebic Zeta function is just this sorry this becomes unreadable the Zebic Zeta function here is a square and then it's minus this problem S plus K and then it's the length of this geodesic which is going around here so the left is the zeros in the complex plane of the Zebic Zeta function and then you say when I transform by A of T so an eigen I've got an eigenform an automorphic form U and then I may change the spectral parameter and get to another zero is that what you're saying almost so the black dots are just the zeros of the Zebic Zeta function and then I ask for a different function space so then I ask for in terms of function it becomes then this Zebic Zeta function with the representation in here and then you have this E to the minus L and then honestly you have a plus and minus here because the square resolves and then I'm asking for the zeros of the second one and I ask how do the zeros of the Zebic Zeta function belong to this pink one so how the zeros are moved along the space another way to say it is this I'm asking for the spectral parameters of the Laplacian of the invariant functions and I ask if I have an eigenvalue there or a spectral parameter there and I now turn to the space of the equivariant functions is this eigenvalue survived or is it moved somewhere okay please say yes or no yes thank you and what we see is that when you if you stay with unitary representations you are quite restricted at least in this example and this is something we observe in general but if you allow arbitrary or quite arbitrary you are flexible in moving around and this might help you to understand properties which are there so and then the question of course was if we want to do a Zebic Zeta function for non-unitary representations or this bijection between automorphic functions which are now equivariant and some type of functional equations can we do this to make sure that there are results and we started at this site where we said first the Zebic Zeta function needs to exist otherwise it doesn't make sense to kind of start with doing something and we understood after some time what is the real problem what one has there so the question is as a question which representations which non-unitary representations can I allow can I allow or having a Zebic Zeta function and then as soon as this question is answered we can ask can I go on and see all the other properties which I would like to see in addition okay so let me draw a picture which hopefully explains what's the problem we expect a Zebic Zeta function of this form so here's our representation here are these two products here's the determinant of kind of the Euler product and we'll explain all the things in a second yeah so we have the Euler product this is here now the norm of an hyperbolic element instead of talking about periodic geodesics I convert them into equivalence classes of primitive hyperbolic elements because it's much easier for any purposes of doing something really here and the unitary case showed us that the representation needs to go here to give us reasonable results so we do expect that we have in the end the same formula but only with a non-unitary representation and the first question is is this infinite product able to converge somewhere and this is really the problem one has to think about so this is your surface this is surface with a casp here is somehow the rest of your surface you only have to focus on one casp to understand what the problem is you think about again the hyperbolic elements as being periodic geodesics and because there is a casp you all know that there are these periodic geodesics which can go deep into the casp as deep as you would like to have them then at some point they turn around they go back and then they pass their life down here and they close up and all these periodic geodesics will have an impact on your formula on the periodic geodesics in terms of the hyperbolic elements then for the picture it means this you have one group element which kind of transports you into the casp area then you have a parabolic element which is doing the rotation, the winding around the casp area to with some multiplicity here so this is the number of windings and then the geodesic does whatever it would like to do so and the M can be any number all M's, so any positive number all M's appear and they all need to be handed why this infinite product so which means when you it's better to have it this way when you now have this H as one of the H in there actually in HM in here into your product you have something which is more like something you can control you have a parabolic element to some power you have a representation of some element you do not really care about but this this part here this appears in your formula for all M's if you now just pretend it's a matrix if you have a matrix with a large eigenvalue this would give you an exponentially increasing thing in there which shows if you have a matrix which is not really good behaving so when eigenvalues are too big you have a factor in which explodes and then the infinite product cannot converge so there's one solution around this is your restrict the representation of representation which you want to work and you say okay you only want to allow those representations for which the windings don't cost us anything so in terms of mathematics it means whenever you have a parabolic element here that the eigenvalues of the xenomorphism which you have here they all need to have absolute value one because then it means increasing it's also not decreasing so it kind of is just a bit rotating or in other words the Jordan normal form of all of your parabolic representations look like this there's some eigenvalue you are allowed to have any kind of length of Jordan blocks and all the eigenvalues here have to be absolute value one so representation of this form has a name it's not given to this by us it's called non-expanding custom monogamy abbreviated as Neckum so at the moment we only have heuristics which tells us we would like to have these type of representations in our business and indeed this is possible it's a result which is by now some years old with Xenia Fedosova oops where we could prove that for all geometrically finite gammas with no restrictions also infinite area is perfectly fine with this is if the representation I use in the Xenia Fedosova function has this non-expanding custom monogamy property then the Xenia Fedosova function converges as soon as the real part of S is sufficiently large you can give precise bounds on what this means but for our purposes this is sufficient and if you do not have this property then you also have no convergence nowhere which means this Neckum domain of representations is exactly where you can expect a Zebic theory which has a Zebic-Zeta function which is given by an infinite product if you want to work with Zebic-Zeta function of course the question is do convergence is not sufficient do I have a meromorphic continuation and then we took an approach which is somewhat very unconventional we said we want to go via the dynamics we do not want to go via this classical theory via I don't know macro-local analysis, harmonic analysis we want to use that we have a very good feeling about transfer operators, evolution operators can handle representations we want to take advantage of that and the answer here is if we have a transfer operator approach so let me say this easier with if there exists the transfer operator such that the Zebic-Zeta function so it's for this gamma and for the XI sorry, not for the XI for the trivial representation so if it exists such that the Fretaum determinant is the Zebic-Zeta function the untwisted Zebic-Zeta function on the nose then you can put in this representation at all levels so which means as soon as you managed to find a good coding for your geodesics which gives you a nice kind of continuous fraction algorithm for which you can have this equality here which is sometimes very helpful you get for free the built-in representation situation of course then the question is what about this if here and this is some new work this is with my former PhD student, Paul Wapnitz who finished last year so paper came out this year what we can say is for many there's still some technical problems but for many non-uniform geometrically finite gamma we have these operators this is the result which at the moment when you see it as a statement looks like there is an existence result but actually the result itself is constructive so if you give me your favorite gamma I sit down maybe 10 minutes maybe 2 days and produce you this operator here and there is work in progress with a Pfeiffer so this is almost done and then if there is nothing wrong anymore then we would have an all here ok then the question is if we think back about the classical theory the Zebik-Zeta function the zeros are important in the meromorphic continuation and here then the question is do I have also here a spectral interpretation of the Zebik-Zeta function with these non-unitary representations of course there are as usual two approaches to try this the one is going via classical process via harmonic analysis and try to do it somehow there to mimic kind of the classical proofs but the way we took is we wanted to understand can we do it via having this transfer operator which gives us the bijection that functions and this is something which is still in progress but it's so much done that we are absolutely sure that there is no problem anymore so this says this for certain I will tell you in a minute what means certain for certain hyperbolic surfaces of infinite area so we wanted to have also with casps and then honestly some restrictions on spectral parameter we have a result of this type we have automorphic functions which are vector valued with your representation which you consider plus some properties which concern like how they decay at casps and funnids they are isomorphic with an explicit isomorphism to the eigenfunctions of these transfer operators which you build in a similar way than for the SA2 our situation SA2Z situation here you have some conditions on regularity of your functions on how they decay at boundaries of interval and in between you have a second objects mediating between the two worlds this is a homology so here the the module on which you represent is essentially function spaces for the serious representation but you need to have some additional conditions to make things work and also on the core cycle sometimes need to have some conditions so it all comes about very natural but it takes quite some time to state it let me say one word about the idea what we do here is this before when you thought about the SA2Z situation you were going from automorphic functions to period functions for integrating along way-chosions geodesics which are somewhat related in a way which I didn't explain to the coding which you have here you cannot do that anymore so in this situation you need to go from certain points let me say it basis for coding but you still have the old form the old closed form which allows you to do this you put in your automorphic function you get out a function on some parts of R you then see is a function of the transfer which you developed via the coding but in between to prove all the things you see what you honestly do is something like this instead of going via the one part where you did the where you put in your cross-section you exchange this into a second part and how should I say it the easiest way to say it is the transfer rate has mimic kind of a simplistic homology which you use in this whole business to get things working okay what we did what we did then is we asked ourselves what are good spaces of automorphic functions which you can single out here for nice properties we came up with three classes of functions it would take some time to write it down but the classes which we have is one class which is quite arbitrary which just says add a funnel so these spaces with which we work here they look typically like this you have a gasp they have a funnel where we want to say add the funnel the decay the growth is not too bad somehow controlled then we have a then we can go through all this here and really write down what are the properties we need at every level what are the precise conditions then there is a second subclass which tells you it shouldn't grow too fast at the gasp so it's still not a gasp form but it's kind of controlled it has polynomial growth into a gasp also then at that level you can at every instance see what you have there and of course in the end we have something which is honestly gasp form so you have a decay at a gasp with a very fast rate also then you can characterize again at every level what is what you want to have so with that I would like to end here thanks a lot