 Welcome back everyone, and it's really a great pleasure to introduce the second speaker of today, Anieska Zelović, who will talk about inducing schemes and a role in thermodynamics. Thank you Anieska. Thank you to the organizers for the invitation. So I would like to talk about my joint work with Farouk Shahidi. Our results are not that new, I think the paper was published in 2019, so it's not the newest result, but I thought it was very appropriate for the conference, and maybe it's good to advertise our work because I didn't give that many talks about what we did. So our paper is partially a survey, so the idea was to put together all the known results that involve inducing schemes and towers, specifically in particular to study thermodynamic formalism. So this was our emphasis on studying thermodynamic formalism and equilibrium measures, but we go a little bit beyond that. So the idea was to put together all the known results and point out the relations in the intermediate systems that are used in the techniques of inducing towers, and also fill in the gaps. So find out what is not known and see if we can say something, like fill in some of the gaps. So this should be a rather easy talk to follow because a lot of it will repeat from what already appeared in other talks and also during the mini courses. So we start a little bit more abstractly. So we just start from a continuous map on a compact metric space. So we'll try to do the setup at least at first a little bit general, so not even the thermomorphism, no smooth structure for now, just a continuous map. And then we imagine that we have some inducing scheme on that map. So what is the inducing scheme? So we call it inducing scheme of hyperbolic type. I think this was introduced, at least we use the setting that was introduced by Pesin, Sentie and Zhang in their paper. So we imagine that we have some inducing domain, which is actually there, which is made out of a countable collection of this joint borrower sets J, right? So we take the inducing domain, which is also will be referred to as the base of the tower. This is still there is a countable union of this joint borrower sets, and then for each of those sets, we have inducing time. So we require that for every set J in this inducing domain, the appropriate image corresponding to this inducing time will return back to the base. So this is this condition I1, and then if I take all the appropriate images of all the sets in this union, then they will cover the whole inducing domain. So this is the first condition. And then the second one is that, okay, so then we can, once we have something like that, we can consider the induced map, which is F tilde, which is defined on the inducing domain, so only on the base. And the second condition that we want is we want this inducing map to be conjugate to a full shift in a countable set of states. However, we can weaken this condition a little bit so we don't need to require that it's conjugated to the full shift, but we require that there is some set, some shift invariant subset of the full shift on which we have this conjugation. And then I'm not stating it now, but later I will assume that this set is in a sense big, that the complement, I think I later I will assume that the complement of this set doesn't see any nice measures. But for now, let's just say that there is some subset of the full shift on which the induced map is conjugated to a shift, okay? So we have the important thing is that we have symbolic representation on the base of the tower, on the base of the tower, right? And then it will be also useful for us to study the tower map which is defined like this. So the tower map, what it is, it's a simplified abstract model of our original map. So the tower map is trying to mimic our original map, but it is a little bit more abstract and a little bit simpler in a way, right? So now what we are going to do, we are going to take our inducing domain and above every set J, we are going to stack a column, a corresponding column which is made out of copies of J and how many copies we put, exactly tower of J, right? So the tower of J is the return time. So above each J, we are going to put tower of J copies of itself and this will be our tower and then the corresponding tower map which I will denote by f hat is given by just going up by one, right? So from point X which was in J to a copy of X in the J, in the copy of J which is above, right? So in the tower, I will just keep going up until I reach the end of the column and then I go by, I do what the induced map tells me, right? So then I go f tilde of X and so I'm back in the base of the tower, right? So the tower map is like an abstract model for the original map, but it will be later useful to study the original map, okay? So we have really three dynamic systems here and so our goal was to study the relations between the three of them, all right? So okay, an example of course, this is a workshop on Young's Diffo-omorphisms. So it should immediately look similar that this abstract setup resembles a Young's Diffo-morphism. Yes, the Young's Diffo-morphism is definitely an example of what I just described. And so if we have a Young tower, then the base of the tower would be this countable horseshoe, right? So our sets J would be nice rectangles whose boundaries would be stable and unstable manifolds, right? So this is, and then we would have symbolic representation of course, because the base of the tower is a countable horseshoe, right? So this, we have this Markov property for the, for the Young, for the Young Diffo-morphisms and this property ensures that the base has symbolic representation. Okay, so that was an example, but our setting in principle is more general because we don't a priori need a smooth structure, okay? So now what I want to say is, okay, so now we want to study invariant measures and the relations between invariant measures for the three systems. So let's say that as a starting point, we have a measure which was invariant under the induced map. So here I will follow the convention that whenever I talk about the induced map, so the map which is on the base of the tower, I'm going to use notation with the, with the wave, right? So if I have mu with the wave, it means that it's a measure which is invariant under the induced map and if I have a hat, then it means that I'm talking about the tower and if I have nothing, then it means I'm talking about the original map. Okay, so let's say that we have a measure which is invariant under the induced map and then I integrate the return time with respect to this measure and it is important for our analysis that this integral is finite. Okay, so one thing that I should point out is that the induced time, we do not require that it, we do not require that the induced time is the first return time. This is kind of important generalization that it doesn't have to be the first return time, which makes the analysis a little bit more complicated. On the other hand, the way we defined the tower map, for the tower map, the induced map, the induced time is the first return time. So that may be a little bit helpful later. Okay, so anyway, so yeah, so we do, because we do not require that the first, that the return time is the first return time, we don't, apriori know that the induced time is integrable. So we have to assume that, okay, so we assume that the integral of the return time is finite and if we have that, then we can lift the measure from the base to the original space using this formula and we can do the same on the tower, on the tower, okay, so maybe the formula itself is a little bit more complicated, but it's very easy to see what we would, what would happen on the tower, how we would extend the measure from the induced base to the tower. We would just, because the tower is made out of copies of subsets of the base, right, so we would just move the measure, right, so we would take the measure restricted on J and put it in the copy of J, right, so we know exactly what the measure looks like on the tower, how we can lift it to the tower and then you would, you could see that you would get exactly the same formula if you tried to write it down, okay, so we can, we have this. Now the relation which is very easy to see is that if the measure that we started with the measure defined on the base was ergodic with respect to the induced map, then the lifted measures are also going to be ergodic. Of course they will be invariant for the corresponding maps and they will be also ergodic, but this is very easy to see. There is a little bit, there is a little bit of risk that our mechanism will not see all the measures, because for example on X, not all the measures which are invariant under our original map can be obtained as a measure lifted from the base, in particular because, okay, but here this relation which, so we will only restrict our attention to measures which can be obtained by this procedure, so we will only consider the class of lifted measures and those are all the measures which were lifted from the base and what this class is for the original map, maybe it's not so clear in particular if the return time is not the first return time, however, for the, if I look on the tower map, because on the tower map the return time is the first return time, then we can see that all the measures on the tower which were lifted from the base, those are all the measures for which the base of the tower has positive measure, so all the measures invariant under the tower map which give positive measure to the base, all of them can be lifted from the base and this is because on the tower the induced time is the first return time, okay, and then there is a relation, the tower map and the original map, though they can be related through the base, right, so if we have two measures which were both lifted from the base, we can relate them by, so if I take, if I have a measure for the original map which was lifted from the base, I can project it back to the base and then lift it in the tower, right, so that's how I can relate the measures on the original map and on the tower through this projections and lifting again, so I will actually call them, say that, that mu is a projection of some mu hat, if I can obtain mu hat by doing this, projecting and lifting again to the tower, okay, all right, okay, so yeah, so okay, so natural question to ask is since we have a way of obtaining measures, invariant measures for the original map and for the tower from measures on the, on the base, the natural question to ask is what kind of properties will also transfer from the, from the base to the tower and to original map and also since we can relate the measures, the measure, the original map and the tower map, they seem to be very closely related, what are, what are the relations, what are the properties of the measures on the tower and on the original map, so like this is exactly the kind of questions that I will be asking and we will see that beyond just ergodicity, so we can clearly see if we have, if we started with an invariant ergodic measure, we will obtain an invariant ergodic measure if we, after lifting it, but if you ask about any other properties, since turns out it's very difficult, so let me start with the first thing, which is the Bernoulli property, so we imagine that what we have, okay, so we start with the measure which was on the base and then we lifted, then we lifted to the tower and to our original space and now we are going to ask about the Bernoulli property, so the first thing is I'm going to ask if I can prove Bernoulli property for the tower map, does it imply Bernoulli property for the original map and this is one of the positive results that we obtain and I even included the proof because the proof is very easy, so here we can see the way that we define the tower map as I said, oh I froze, did I freeze? No, we can hear you, it's fine. Okay, I see my face is frozen, alright, okay so, so as I said the tower map is supposed to serve as an abstract model of the original map, in fact what we can show is that our original map is isometric to a factor of the tower map and why is it so metrically isometric, so why is it not actually isometric or isomorphic, sorry isomorphic, why is it not actually isomorphic to the tower map and the reason is because on the tower map we have the first return time which is the first, the induced time is the first return time but it's not necessarily a case for the original map, so what can happen is that when we take the tower and we stack copies of elements of the tower one above the other, we always obtain disjoint images this way but in actuality for the original map if we take, if I keep iterating sets j under my original map the images may not be disjoint, right, because the induced time is not necessarily the first return time, okay, so the original map and the tower are not necessarily isomorphic but if I account for this relation, right, so I can introduce, I can, I can introduce this relation that will say that two pairs on the tower xk and yl I identify them if they are actually, if you have this relation that fk of x is the same as fl of y on my space x, right, so if I do this identification then I definitely have isomorphism between the tower and the map after doing this identification, right, so, so then if the tower map was Bernoulli then the original space as a factor of it must also be Bernoulli, because every factor of Bernoulli is Bernoulli, can you hear me because something strange is going on, I think? On this side it seems everything fine, actually, although we cannot see you, okay, let me see if maybe I can, but we can hear you fine actually and we can see the slides, yeah, that's fine, I think I came back, all right, okay, so this was Bernoulli property but now also from this you see that it's not so clear if we can have the implication the other way around, right, because the original map is a factor of the tower, so that's why if the tower is Bernoulli then original map is Bernoulli, this doesn't necessarily seem to happen the other way around, right, okay, but this was very simple but now what we are really interested in is let's say that we have a measure which was Bernoulli on the base and now can we say that this is Bernoulli on the tower, because this is really the direction that we are interested in when we use the methods of inducing, right, we would be able to show that, yes, that the measure that on the inducing domain is Bernoulli and can we say that it's Bernoulli when we lift it and this is much more complicated and in general it is probably not necessarily true but we can prove that it's true if we have those two conditions B1 and B2 for the measure on the base and those conditions they look very strong and they're probably in general very hard to check but there are some situations in which I can imagine that they could be verified, so the condition B1 it should be rather easy, it just means that the measure on the base had a product structure and that should follow from product structure but condition B2 is much more restrictive but how can it be used? Well the only advantage of this condition is that I don't have to, I can stick to just analyzing the measure which was on the base, right, so if I am able to prove that the measure on the base is Bernoulli with respect to the induced map in order to show something like this I would probably have to show condition B2 with replacing F hat with F tilde, right, so if I had induced map so in order to show that the partition by cylinders is weak Bernoulli and so in order to show that the induced map is Bernoulli I would probably need a condition like this if I replaced F hat with F tilde, right, so if I am able to show that the induced map is Bernoulli most likely I am able to show this condition for the induced map and so our result says that if I can replace here the induced map with the tower map but still only analyze with respect to the induced measure then the tower map is Bernoulli, okay and I will show later when I will move to more concrete classes of measures how this can be verified, okay but in general we would have to check something like this, okay, all right, now another question that one could ask is what are the relations between mixing rates and maybe central limit theorem is it true if induced map satisfies central limit theorem is it true for the for the lifted measures as well and those aren't our results, I mean this proposition follows easily if we just put together results of Young and so this is one paper which was done for I think expanding maps and then there is another paper by Melbourne Therese that extended to hyperbolic maps so if we put together those results those two papers we obtain a result like this about decay of correlations so of course you always expect when once you know mixing rates for the induced map in order to say something about the the tower map you have to know something about the decay rate of the tail of the rate so you have to study tail of the measure with respect to the induced time and that's really the only way that you can claim something for the tower but this is itself not enough we also need to know something about the measure the measure has to have some nice properties too so the result that is known is that if we have a measure which has this nice property which is like the push forward Jacobian is held there continues pretty much this is what it says so we have this condition for the measure where the g of mu tilde it's just the push forward Jacobian of the measure so if a measure has this form then we have the following so for the tower map we have that if the induced measure had exponential has exponential tail then then the tower map has exponential decay of correlations right and the same with polynomial yeah okay and yeah and about the central limit theorem we can claim central limit theorem for the for the tower map for observables that have certain form that they don't go too far into too high into the tower so if they are supported kind of in this lower portion of the tower this finite portion of the tower then we can also show central limit theorem all right so this was for the tower map and then in order to claim the same for the original map we need a little bit more we need those two conditions f1 and f2 that pretty much f1 pretty much tells us that we have contraction and expansion respectively so here you would expect some kind of hyperbolic structure on the on the base of the tower so for example for young's the film morphine this will be satisfied and then there is another regularity condition f2 and so if we have those two conditions then we have the same result for the for the original map okay all right so this is as much as we can tell in general in order to verify those conditions and actually say something about particular measure we need to know something about the measure and then there is a big class of measures for which those results can be established and so now I will restrict my attention to just equilibrium measures and study equilibrium measures okay but for this we also need additional conditions so those conditions that we borrowed from the paper by passing sentience yang so passing sentience yang they wrote a paper in which they use those inducing schemes of hyperbolic type to establish existence of equilibrium measures for certain maps and so we are exactly going to use this and then establish some more properties so what we need is so I mentioned that at the beginning of my talk that we need on the base of the tower we want the base of the tower to be pretty much conjugated to the full shift right so I said that we assume that there is some set a which is a shift invariant subset of the full shift on which we have the conjugation and so now we add this condition that the set that the complement of this set supports no invariant measure which gives positive weight to any open set so basically the idea is that because we are aiming at we and our goal is now to find Gibbs measures for the induced map and so the idea is that no Gibbs measure can be supported on the complement of this good set a okay so this is the first condition and then we also need an existence of at least one periodic point in our inducing domain and the last one is arithmetic condition that says that gcd of the inducing times is equal to one and that will give us later that the tower is will be a topologically mixing mark of shift okay all right and so now we want to study equilibrium measures uh again for the three uh for the three maps so we start with the some potential which is defined on the original space and then we define two corresponding potentials one on the tower and one on the inducing domain right so on the on the base we uh we induce right so we sum up along the columns of the tower pretty much um okay and then we study equilibrium measures but we restrict our attention to only measures which can be lifted right so by equilibrium measure in this setting I will mean the measure that maximizes the sum of entropy and the integral in the space of all lifted measures right so this doesn't necessarily immediately give us equilibrium measure but in many cases we can show that those lifted pressures defined in this way they are actually the same as topological pressures okay so we have this and then on the inducing domain we actually have this will be the same as the Gurevich pressure later okay so this is it you should speed up okay and then we have collection of conditions for the potential and again this is just the list of conditions that guarantees that so this is going to guarantee two things it's going to guarantee that on the base of the tower where we have symbolic representation if we study thermodynamic formalism for the induced potential we can apply all the results of that were established by Sarig so we will have unique equilibrium measures and they will have really nice properties but also those conditions guarantee that we can lift those equilibrium measures to obtain equilibrium measures for the on the tower okay so those are conditions and then here is the first result which it's not ours so so the result is for the induced map so under those conditions on the induced map we can apply results of Sarig to obtain that there exists a unique equilibrium measure and in fact it's a Gibbs measure for the induced potential okay on the base and then since it's a Gibbs measure it's a well-known result classical result that this measure is going to have exponential decay of correlations and satisfied central limit theorem and also another result by Sarig is that this map has Bernoulli property so this map that this measure has Bernoulli property so we have all the nice properties on the base okay now in order to lift them to the tower well here's what happens yes we can lift them to the tower so if we lift the measure under those conditions we we obtain ergodic equilibrium measure for the potential on the tower and this is a yeah and now about the other properties we we have that if this lift that measure is mixing then it is Bernoulli so this is the result we would have to check mixing independently but if we can check that the lift that measure is mixing then it is Bernoulli and then we have decay of correlations but only but only under depending on the on the tail right so if the tail of the lift that measure decays exponentially then we have exponential decay of correlations if it decays polynomial then we have polynomial of correlations and we have the central limit theorem but with this if we restrict our attention to observables that only remain in this finite part of the tower so we have this and then for the original map so the existence of a of an equilibrium measure this was the main result in the paper by passing sentient shang and then we added to it several things so so we added the relation that that the measure on the on the original map is can be obtained as by using this relation between the map and the tower right so all the three measures they are related in this nice way and and then we also have the same that if if the measure is mixing then it is Bernoulli and under those conditions f1 and nf2 that I introduced earlier we have the same results about decay of correlations and central limit theorem okay and I wanted to talk a little bit more about how we obtained Bernoulli property because this was really our main contribution here so in order to obtain Bernoulli property we we actually prove a more general result in symbolic setting and then extend it so we use so on the tower we can we can have a symbolic representation so we we define the following mark of partition of the tower which says let me see if I can maybe what happened oh no I messed up everything's fine we can see you and here okay no I exited the full screen oh yeah now it should be better no I just wanted to be able to mark something okay so we have okay so we introduce the mark of partition on the on the tower in the following way we treat the whole we we treat the whole base of the tower as one element of the mark of partition and then other elements of the mark of partitions have this form so for k strictly positive we take each j k right so each copy of each j as a separate element of a of a mark of partition but then the whole base as as just one element of the mark of partition and then we have we obtain a countable mark of shift with the following allowed transitions so from base of the tower I can go to the copy of any j in the first level right so to any j but on the first level and then from from each j k I can go to j k plus one and then from j once I reach the the top of the tower the allowed transition is to go back to the base of the tower right so we have those allowed transitions so now this is a mark of shift okay and it's topologically mixing because we have this condition condition ify all right so now what we are going to do we are going to prove I mean in this talk I'm only going to state but what we prove the following result in the symbolic setting is that so imagine that we have a topologically mixing a countable mark of shift and we we choose one element right so one one letter right one state and then we are going to consider the induced shift on this on this one letter okay and we assume that we have a potential which is only defined on this on this one cylinder okay and we assume that this potential is locally held there with respect to the induced shift uh huh right so we have a potential which yeah so we assume that we have a potential which is induced which is induced on this one element and it's locally held there with respect to the induced shift and has finite garbage pressure and so then we have unique ergodic equilibrium measure for this potential and we can lift this measure to the full to the to the mark of shift that the countable mark of shift that we started with uh and then we have the following if we can prove uh that this lifted measure is mixing then it is burning okay so this extends the result of sarik so so if I assumed um that the original potential on the on the original countable mark of shift was locally held there continues that um it is a result of sarik that that the equilibrium measure is uh Bernoulli however so we extend it and we only uh so when we are on topological mark uh mixing mark of shift we now induce uh on one element and then we assume that the induced potential is locally held there and we can still approve that the lifted measure uh so I I do all the analysis for the induced shift and then I lift this measure and it is still going to be Bernoulli the lifted measure okay so this is uh this is what we prove which I thought maybe was interesting to see um and what else oh okay we also talk about phase transitions um so so so so now it's separate because we no longer talk about measures but since we we are talking about equilibrium measures it also makes sense to talk about pressure itself um so now we are going to study uh regularity of the pressure so we assume bunch of conditions on the potentials now we have two potentials uh and we require that they satisfy various conditions and then under those conditions uh we can show that the pressure on the tower uh which is the same as the pressure for the original map uh is a real analytic and so again um the fact that the pressure for the induced potential is is really analytic it follows from the results of Sarik and we extend it to uh to the tower and the original map um and do I explain how I do it I don't think it's oh no now I just apply to okay um okay so maybe it's worth mentioning how we can extend results about the analyticity of the pressure from the induced domain to the tower um we do a nice trick so um right so so Sarik proves analyticity of the pressure on the countable mark of shifts under certain conditions he proves that the Ruel operator has a spectral gap and it has all those nice properties and what we do is we obtain the pressure on the tower implicitly as the as the zero of a certain potential so we would have potential so we have an induced potential minus omega so the pressure is zero right and the and the omega will be pressure on the lifted our lifted pressure so we obtain our pressure on the tower implicitly as a zero of a certain equation which was set up for the pressure of the induced potential and then uh we obtain analyticity of our pressure using the implicit function theorem so we use this uh kind of this using the simple trick we can uh we can obtain that which by the way reminds me of how we actually prove those properties to obtain earlier the Bernoulli measure the Bernoulli property for the equilibrium measure we also use operator for the for the for the for the induced measure uh and verify this condition which I stated earlier okay um all right so now just uh applications of like maybe some I thought maybe would be good to actually give some concrete example of when our all results play out uh so let's go back to uh Young's defilomorphisms so now we have nice hyperbolic structure nice smooth structure defilomorphism and nice hyper with some hyperbolicity uh but so I I'm listing all the conditions here that I'm assuming and so some of them are standard conditions which are just part of the definition of a young defilomorphism but I also added the conditions that we need for for our results to play out so uh I think I I don't even remember which are which I think yeah condition y4 it doesn't always appear in the definition of a young defilomorphism but we need it for our results right so that you see this arithmetic condition so this is the first set but like the first two are just standard conditions for the young defilomorphism and then we have more right condition y7 is the standard condition for the for the distortion but it's also usually the most difficult one to verify um okay and we also need I think it's not usually a part of the definition of a young defilomorphism but we have this condition y8 which will help us establish the tail of the um yeah the decay of the of the tail of the measure with respect to the inducing time so we we do need this condition in our setting all right so we have all those conditions and then we consider a potential which is a geometric potential um and then under those conditions um it has been verified by passing sentient rank in their paper that a young defilomorphism admits an inducing scheme that satisfies our our conditions that abstract conditions that I introduced earlier um i2 and i4 which I forgot what they are but condition i3 has to be checked independently condition i3 was to check that um that the complements of of the set where we have conjugation to the full shift doesn't support any nice measures so under this condition we have unique srb and then this condition guarantees that the geometric potential satisfies the conditions that that were needed for the for the existence of equilibrium measures right um okay so what is actually the result so this is so this is just saying that the young defilomorphism satisfies all the conditions uh in that that we listed earlier for our results to hold and so what is the actually the result when we put everything together we have the following if we have a defilomorphism that satisfies all of the conditions that I gave for young defilomorphism um and in addition as satisfies this condition i3 uh then we have uh some interval in t uh so so we have some t0 and we have inter on the interval from t0 to 1 we have a unique equilibrium measure uh for the geometric potential um and uh the pressure is real analytic and then we have exponential decay of correlations and central limit theorem if we in addition verify uh this condition for the for our map and we have that the equilibrium measure has Bernoulli property property okay so we have all this um and I think that was it okay all right so this is all I had to say thank you very much but thank you very much and yes thank you um are there any questions so I have a question that I've always actually been very intrigued by this last condition you mentioned about the cardinality of the sets that I have a certain return time right yes this s and yes this one yes um because so what do you is it is there a way to explain why this comes in or somehow what the role of this condition is because in some sense the cardinality of the partition element is not necessarily an intrinsic it's not necessarily intrinsic part of the tower like because you could well yeah it's not clear to me what that why that condition is used uh yeah I think this condition is supposed to use to verify uh the condition p4 for the potential which was I think yeah the hardest condition to check so let me maybe go back and see what the p4 is yeah so p4 is this condition uh this technical condition which is uh if I remember uh correctly in the language used by Sarik I think it's called um positive records or some more than a strongly positive record the Omri is here so maybe he can have us yeah so this is the condition that is usually the also I think the harder to start the hardest to establish uh that relates to strong positive records yeah so so this this uh bound on the cardinality is just a sufficient condition to prove this is this what I say I say so the might be so so is it do you think um it is possible to relax that condition you know to use different ways to verify this other than having that can you can are there examples where you do not satisfy the bound on the cardinality but you do satisfy this condition any examples like that I mean of course if you could verify this condition uh independently then uh then all the results should follow because I think other conditions can be verified without it but uh verifying this is not easy I mean there is another work I think by Ben Kohl where he does something I don't know which methods he uses but he obtains the whole so we have so we not only have that we need this strange condition to verify p4 but even with this condition we only can obtain that there exists this interval from t0 to 1 right and it's not clear like why this t0 appears if it's uh but like it appears in the estimate that we cannot do it for all the t's like less than 1 but there is this t0 which this is where we run out of good estimates but I think there is a work by Ben Kohl where he also studies geometric potential and he doesn't have this t0 so I think he has results for all the t's but I'm not sure what results about the methods he's using so maybe he found like a better way but yeah the short answer is I don't know okay okay okay thank you are there any other questions for Nieshka okay well then let's thank both of today's speakers on Nieshka and Omri thank you very much and thank you to all the participants for being here today it's very nice to it I recognize the names every day so it is a little bit of feeling of the of a regular conference in some way of seeing the same people every day and so we will be here again as Yuli said same time same place tomorrow for the last day and the last day of talks okay so very best wishes for the rest of the day everyone bye bye everyone thank you