 Okay, so good afternoon and good morning, good afternoon, good evening to all of you spread across the globe. Thank you very much for joining us in this. Let's hope we are returning to some kind of regular seminar here. I'm really delighted to say that this is a hybrid seminar. So we have a small audience here at ICTP and we also have audience logged in from Brazil to Uzbekistan from many different countries. And it is a particular pleasure for me to introduce first speaker of this Returning to Normality Era, Professor Akhtam Zalilov from Turin Polytechnic in Tashkent, Uzbekistan. Professor Zalilov was a student of CNI and has since been doing a very good research work in Uzbekistan on very sophisticated results on circle homomorphism. He is going to give us a little bit of a glimpse of that today. He has been visiting ICTP for more than 20 years. He has been here many, many times, he has participated in many conferences. He has been here for six months at a time sometimes. And he is now an ICTP senior associate, so he is visiting in that capacity. And we are very, very pleased to have him. And so without further ado, let me ask Professor Zalilov to start the seminar. Thank you very much. Good afternoon. I am very happy to be in ICTP again. I like to be here. Thank you very much to Professor Lozato for warm sign, speak atmosphere. Let us begin my talk. That FB orientation circle homomorphism, orientation present in circle homomorphism, such kind of circle homomorphism first was studied by Henri Poincaré at the end of the 19th century. And then first, fundamental results also belongs to Poincaré. So let FB orientation circle, orientation present in circle homomorphism with the irrational rotation number. Roy's rotation number of this homomorphism, which is all time well-defined. Now, HR's homomorphism can be defined as mode 1 for any X from a circle. So F of X can be defined by some function capital F of X by this formula. And where F of X satisfies these two conditions, F is dysfunctional from R1 to R1, and this continues strictly increasing on a real line. And the second, F of X plus 1, F of X plus 1 equal to F of X plus 1 for any, for any X. This function uniformly defined by this formula, F of X satisfies these two conditions, F of X, usual function, real line. Capital F of X is called lift function, is lift function, or simple lift function of homomorphism F. If we know one of them, one lift function, then other ones we can define as F of X plus K, where K is any integer. So the class of all, if we have one homomorphism, then we can define by the class of such kind lift functions. We will give one of them, other ones we can define by this way, this countable set. I want to introduce the one important notion, this dynamical partitions. Dynamical partitions of the circle. In the circle we can define many kinds of partitions. Now I give the definition of dynamical partitions, which is very useful in the proving many important theorems in circle dynamics. First, I need very short information about continuous functions. So if there are continuous fractions, so if we have rotation number all time between 0 and 1, if we have rho, then we can express it as continuous fraction in the following form, 1 over K1 plus 1 over K2 and so on, plus 1 over KN plus so on. And this one we can rewrite in the short form, K1, K2 and so on, KN. So if rho is irrational number, in this case it can be written as continuous fraction. It is infinite here because rho is irrational number. But if rho is rational number, then this continuous fraction will finite. This expression is unique. Now we define rational numbers Pn over Qn by using this continuous fraction as K1, K2 and KN for any n. And then these numbers Pn over Qn, this quotient is 10 to 2 rho. And then, yes, so this Pn over Qn, 10 to 2 rho and Pn and Qn call it convergence of the rotation number rho. Here I will write the following estimate. This one is main estimate here. The rho minus Pn over Qn less or equal to Qn times Qn plus 1, which shows that it tends to rho, this convergence to rho, which satisfies this bonus. Qn tends to infinity because it is irrational number. Now the numbers Qn are called first return times. First return times. Now we can define the dynamic partitions. So we will take any point and consider it is orbit. Orbit of this point, F, it is the following set. XI is the iteration of X0. If all time we suppose that rho is irrational, then this orbit for maps with irrational rotation number, this orbit all time will infinite. So this sequence is infinite for pure rotation. Pure rotation is this map, F of X equal to X plus rho via mode 1. This map is called linear rotation. For this map it is very simple map. For this map all time the orbit of any point is dense on the circle. But in general this depends on density of this orbit, depends on smoothness of the given map F. Now we take this orbit, X0 is fixed, and then using this orbit we can define the sequence of dynamical partitions. X0 here and then now we will take the part of the orbit X1, X0, X1 and so on, Xn minus 1 plus Xn minus 1. Here exactly Qn minus 1 plus Qn first points here of this orbit. We will take first this segment. They are here and then closest 1 are here QxQn minus 1 and XQn are closest points of the segment which are close to X0. If n is even then XQn lies to the right side of X0 and if n is to the left side of X0. Other points of this set here, a way of this interval. The interval we denote by gn it is XQn minus 1 XQn and this interval is called n's renormalization interval at the point X0. This one is the reason why Qn are called first return times. After Qn steps it is almost back to the point X0. Now these points here they are Qn minus 1 plus Qn and then these points here define some partitions which is n's dynamic partitions and this partition can be described by following way. We denote this one by delta n minus 1 delta n minus 1 0. This is the interval at the segment with n points XQn minus 1 X0. Also we denote by delta n 0. Ah, bigger. Thank you, thank you. Okay, I will write bigger. So X0, XQn. So we denote by delta n minus 1 0 this segment the segment which with n points XQn minus 1 X0 and this one we denote by delta n 0. They are here. This one is delta n minus 1 0 delta n 0. Now other intervals we introduce by following way. So delta now we denote by XI n of X0 the set of intervals delta n minus 1 0 delta n minus 1 1 and so on delta n minus 1 Qn minus 1. This here Qn intervals of rank n minus 1. This one is initial interval of rank n minus 1. This one is initial interval of rank n. These are iterations delta n minus 1. I is iteration of the initial interval of this one. The same here also. We can write about delta n G also. And the union also the intervals delta n 0 delta n 1 and so on delta n Qn minus 1 minus 1. So now we denote this system by XI n of X0 and here Qn intervals of rank n minus 1 and Qn minus 1 intervals of rank n. This one exactly gives us the partition which we get here using these points. So this partition which we defined using this by these points we can get by this way. It is proved first by Danjoua. So this one is called XI n X0 of X0 is called n's dynamical partition of the circle. Of course it depends on X0 of initial point X0 and if we change X0 it is also changed. Then also it depends on F of course. I don't write it yet but it depends on F. What happens if we pass from Xn X0 to the next partition XI n plus 1 X0. All intervals of rank n this one is big intervals for linear rotation. This one is really big intervals. This one is small intervals. So each interval of rank will preserve it. This means each interval of rank n is element of XI n plus 1. But each of the intervals of rank n minus 1 divided to Kn plus 1 plus 1 intervals by the following way. Each interval for example I will write it for the initial one delta n minus 1 0 is delta n plus 1 0. And then as once here delta n delta n qn minus 1 plus s times qn where s changes from 0 to Kn plus 1 minus 1. I can draw it here also. So if this one is X0 Xqn minus 1 this one is the interval delta n minus 1 0 here the interval delta n 0. Then the big interval the interval n minus 1 0 divided this one delta n plus 1 0. Other ones are the intervals of rank n. So each interval of this set divided exactly to Kn plus 1 plus 1 intervals of the next partition XI n plus 1. So we get by this way the sequence of increasing partitions of dynamic partitions. This means each interval of XI n is union of the next partition. Now about behavior of the length of this of the elements of the dynamic partition. This depends on smoothness of the function f. For pure rotation for f rho for linear rotation delta n the length of delta n. We denote by this way delta n's length equal to qn times rho minus pn. This one is in the case of pure rotation the length of the interval of rank n. In the case of pure rotation they are equal the length of these intervals of all these intervals equals this one also. And then it tends to 0 because if you rewrite this as qn times rho minus pn over qn then for this one I give them bounds. So we have here qn times 1 over qn times qn plus 1 and then we get here 1 over qn plus 1. So qn plus 1 this tends to infinity exponentially. So this means the length of this one tends to 0 exponentially fast. But this one is for pure linear rotation. In general I want to formulate lemma. Let f be circle diffeomorphism. Circle diffeomorphism with irrational rotation number rho. f is invertible f belongs to the class c1 and inverse map also belongs to the class c1 on the circle. Now we denote by v total variation of derivative of f. Then we suppose this one is finite. So total variation is finite of f prime. Then delta n the length of delta n less or equal to some constant times lambda degree n for n bigger or equal to 1 where delta n of x0 is any element of psi n of x0. So in the case of diffeomorphism which smoothness a little bit better than c1 because this condition in this case the length of intervals of dynamic partition tends to 0 exponentially. What is lambda? I will write lambda is 1 over square root 1 plus x0 degree minus v. v is total jump which is positive. So this one is less than 1. So lambda depends on total variation. So for diffeomorphism with this condition the length of the dynamic partition tends to 0. So now we consider also the circle homomorphism with breakpoints or piecewise circle homomorphism with finite number of breakpoints. This means we consider the following kind maps. It h but here the left derivative there if it is positive right one also is positive and then they are different but so at this point the graph has here this has corner here also. This one is breakpoints. Piecewise smooth map with finite number of breakpoints. If we have f which satisfies on h interval of continuity of f' here it is. And then on this interval between consecutive breakpoints f' is continuous more on the circle if total variation again of this function is finite. In this case this lambda is true. So we can generalize this lambda for piecewise circle maps with finite total variation. In this case also we have this bounders. What this bounders give us if we have this one we have for all n. So if we consider the sequence of dynamic partitions which end points are the points of this orbit. We use it all time with the points of this orbit. This bounders or this estimates show that this orbit is dense on the circle. It is dense on the circle. So we have orbit which is and if f satisfies this condition which orbit is dense on the circle using this fact also. So we have this map. Now we can using this fact we can construct the conjugation map also. In fact we are proving the classical denjuo theorem. So we have here orbit which is dense by this fact we suppose f satisfies this condition. Then we will do following. So we will take two copies of the circle. Here we can see on the second one we can see the f row here f and then take here the point x0. And then here we can see the orbit of x0. Here we can see also the orbit x0 by the linear rotation. Then we denote by xn bar the f row of x0. So this orbit also here dense because it is linear rotation. And now we can define the conjugating map phi by following this. So f phi of xi was set as xi bar here. For example if x1 is here at the point x1 we define phi as x1 bar. Orbit to orbit. Passed orbit to orbit. So we define here this function for all points of the orbit. Now this function defined on the countable set which is dense here. And the second this function is increasing. This function is increasing because the orbit of x0 by f row has the same order. And this gives us the monotonicity of the map phi on the set o on the orbit. So now using the continuity using this monotonicity we can extend the map for two circles to the 01. By this way we construct the map phi on the circle. And then moreover we can check first we can check for the points of orbit. And because at other points away from the orbit we will construct by limit it satisfies the following condition. Phi of f phi composition f equal to f row phi. So this means they are topologically equivalent. Phi is called conjugating map or conjugation. So by this way if we have this estimate which gives us the density of the orbit we can construct the conjugating map phi. So and then it satisfies this condition. So this means if f is diffeomorphism which satisfies this condition then f under f row. Rows rotation number of f they are topologically conjugated by conjugation phi. In fact we proved the sketch of proof of the classical dengue theorem. So now this theorem also is true for such maps. For such maps this fact we can find the Hermann's book in French. Now about but from the theorem Dengue. This one is called we proved the classical theorem Dengue. We can decide only about continuity of the conjugation phi. No information about smoothness of phi. So the smoothness of this conjugating map phi is one of main problems of circle dynamics. And then this problem present time well studied for the class. Circle diffeomorphisms. Here main results obtained by Arnold, Hermann, Jokos, and Moser, Katzinson, Arnstein, Kannen and Sinai and others. So now I give the definition of invariant major and this conjugating map can be there is one remarkable fact. First I will give the definition of invariant major. So if we consider on the circle boron sigma algebra and then the probability major mu and then f is circle homomorphism. Now invariant major is mu such that mu of a equal to mu of f minus 1a. This one is pre-image of a for any a from sigma algebra b. Boron sigma algebra it is minimal sigma algebra containing all intervals on the circle. I give this definition for circle. This also can give for any transformation on major space. But I concentrate on the circle. Now this one is definition of invariant major, probability invariant major. Now in the general case if we consider maps on metrical space, there is classical theorem, Bagoliou-Krylov's classical theorem, which states that if we consider compact metrical space and then on this space if we consider any continuous map, then this map has at least one invariant probability major, the general theorem. But in this case, in the case circle maps, I formulate the following main theorem. So let f be circle homomorphism with irrational rotation number. Then f is strictly ergodic. This means f has unique probability invariant major, mu f. So in the case of circle homomorphism with irrational rotation number, f has only one probability invariant major. Now using this invariant major, we can construct or define the conjugation phi. So this one is a remarkable fact. In the circle dynamics phi of x, we can define as invariant major of the segment x0x. So x0 is a fixed point. We can take 0 also. Then this one we can rewrite also as integral by segment x0x as d mu. Phi of x, the conjugation, we can define by this formula. So this shows that phi of x is, in fact, the distribution function of probability major mu. But this one probability is called the distribution function. This formula shows that if phi is smooth function, then mu is absolutely continuous. The derivative of phi gives us the density of mu. And also if mu is absolutely continuous with respect to Li-Berg major, then phi is smooth function. This formula shows that, yes, that one. But in fact, this formula is true for any circle homomorphism. It doesn't depend on smoothness. If it is sufficiently smooth, then phi is homomorphism and it can be defined by this formula. Yes, to the original, yes. Now, why phi is continuous, yes? Because if rho is irrational, all times mu is continuous major. This means it is non-atomic. The major of h points, point set, is 0 mu of the point z0 equal to 0 all time. It is continuous. We say it is continuous major, non-atomic, they are same. Using this fact, we can see this phi is all time continuous function. Phi is continuous function. Without condition to smoothness, phi is all time continuous function, but not homomorphism. We get homomorphism only in the case if f, initial map, f is minimal. This means the orbit of h point is dense on the circle. But if we have these estimates, in this case it holds. Yes, if f is minimal, then phi is conjugation. Then t at tf is topologically conjugated. Now, I formulate our theorem about invariant measures of such maps. As I said that in the case of circle diffiumophismus, this problem of smoothness well studied. And then last theorem was proved in the end of 80 years using the renormalization method. And then last theorem belongs to Hane and Sinai, also to Katzenesan and Ardenstein for diffiumophismus. And then for diffiumophismus, for typical rotation numbers and for sufficiently smooth diffiumophismus, all time invariant major is absolutely continuous with respect to Lebesgue major. Now, I formulate some results concerning to piecewise smooth homomorphismus with several breakpoints. Let f be piecewise smooth circle homomorphism from class C to plus epsilon. If f has breakpoints, then on each interval, here on each interval of continuity f prime, derivative of f, f belongs to the class C to plus epsilon. This means f is two times differentiable. The second derivative satisfies Halder's condition with epsilon. On each interval between consecutive breakpoints. And rho is this irrational rotation number rho. Also, we suppose that derivative is positive. It is bigger than some positive constant for any x from this set. So b1, b2 and bm are breakpoints of the map f. We denote, I will write here, we denote by bi, the jump of this fashion at the point ba minus ba, bi. This one is left derivative at this point. It is too small. Sigma of bi, I will definition this one, is derivative f at the point bi. This one is left derivative divided by right derivative. This one, this one is called the jump of the f at the point, at the breakpoint bi. Because this one is breakpoint. Now, let f be piecewise smooth homomorphism from this class with rotation number rho. Derivative is positive, bi is well defined. And the total jump, I will write here, total jump is the product sigma bi, i from 1 till m. So if f has m breakpoints at each point, we define jump ratio. And then this product we call total jump of f by all breakpoints. The product of jumps. If total jump non equal to 1, then invariant major of mu is singular. Singular with respect to...