 na njih, bolj. Tukaj, bilo. V svojih zelo, that you followed the lecture by Fabio Cicconi and so now you have already an introduction of the main aspect of a dynamical system and now the second part I discussed about the statistical description of a dynamical system Zelo se skupno tegačenja, da se tegačenje vse tegačenje stavili in vse tegačenje. Zelo si videli, da vse tegačenje zjavljene peste, ne? In vse tegačenje vse tegačenje. Kaj je se čest? in iz vsanja, dataministih sistemnih, je tebrne kotik, to je čutko druga, vse je poslutno, poslutno, več nek�o poslutno, nekada nevratim, dokler帶e neč fron. Je to sdi in popril, da si je tako kontroliti katkatilji, se igaj, da je senni tebrni, Tako ima, da odbezx, je težnji taj tih, ki najtej zvukovim v tomesku. Profesivo počk, kar to je revolutnje ideasov. In tih, ki jah jasno se je unika v tomeslju identifičnih in vsega. Sp alike. Kaj ki je nasemilo, dobro pa je zelo počke. If you have a certain uncertainty in the initial state in the chaotic system, you have that uncertainty in increasing time exponentially. Eleveni, da se pridavimo v zelo, skupamo, da b terms pravizamo, o časem na mene jedowno solizavačne. Prejšel prezijom v povdeni. In ešte glade, o ko ampak idem, čiste u vseho, bistv tablespoonsu izponenjava. Kaj je, nešte, vseho, izponenjava desim vse, izponenta... Zdaj, da jih ima... Zdaj, kar je... Zdaj, da dajte... Vse ima, da se biti vse inicijan vse vse. Učinj, da jih ima, da se pripobah vse, da jih ima dogodno, načo bilo jo spicesetno, je zelobane ki začajte delto zelobe zelobe. If you obtain a big grant, you can degrees this. So, the uncertainty on initial condition. And then you have certain tolerance. So, you accept tolerance, you can decide that you accept tolerance of 10 percent. Kako je tarko, zelo to se pomega, zelo to postavljati. Pozivajte, da ga jazem zelo tudi, da posahajte delta, delta-zero. Tukaj je tukaj preditabilima. Tukaj preditabilima je 1 over lambda, log delta, capital delta. And after this time, your position is not good enough. So for example, in the Lawrence model, you discussed the Lawrence model with, i guess, you discussed the Lawrence model with Fabio Ciccone, you have something like that, you disincrease, and then desaturate. So imagine that you fix here delta, you have delta zero, this is the result. Begin je. Vse je neorrlandovano, zato nezavršen. Vsame jaz ima nezavršen. Vsaman. V tem vse moraj formula zrita da je v tem, evoč da nisi tukaj odsleda načitelje ljuda stremičen desperately in vse je. Zelo da poddaj sem tukaj izorite recognizing. Chvaš to lego. Pozor dela ta oveč, ok? In mest več, da je taj injere, da je taj injere izgledaj z vsega rač. Zelo je, da je ta injera, da je ta rač. Tako, da to je taj injere in taj da se karatterist, in danes prav. Vse sem vse, da je to vse, da je to vse, da je to vse, da je to vse, da je to vse, da je to vse, da je to vse, da je to vse, in v temosteriju sistemu, tudi je zelo vzelo, in vzelo, da je vzelo, da je za začetek milijonov milijonov, to je zelo vzelo. Vzelo, da je to, da je vzelo. Vzelo, da je začetek milijonov, ma in izgleda. Vzb. Vzb. Vzb. Tako. Vzb. Vzb. Poz. So. in vseč je to vseč, da ne moguš pošličiti svojo system, z kajim vseč, in zato je naredil, da se pošliči pošliči. Pošliči pošliči, da se pošliči pošliči. To je pošlič, da se pošliči pošliči, the terministic point of view, differential equation and so on. The practical approach is the following here. To consider not only these, but also these stuff. OK, so this means, if this is different from zero, these means that you… practically you have a probability distribution at the initial time. OK, so you can imagine it, for example, this is the point that is at this X0 and you have a certain uncertainty in this in this bowl. We have that this is 0 in this region. So I am constant in this one over the the the volume of this volume, z vsem z vsezina vojstve, z razljime za tazir, na practicesite, in tudi imaš to, in u tijliši sveti, potrebno je, da vidimo, da evolutišlji iz vsezina vdin zomoza in tudi vstaj, ako vdin zelo izvaleno, ne zdaj, da je vstaj a so, da in v strategijskih situaciju. Zelo vzostaj, ki je mene, ki je vzostaj, je Maxwell. Maxwell v zelo na vsej leksu, v napostelju KLAB, vzostaj, da se ideje od neko odličenih, da se je neko situacije, je metafizijski idej. je metafizika, ker je in poslutno, da se repejte, občasno, nekaj več. Čekaj, da jih je zelo metafizika, ideja v maxi, to je unika, pravdučne opročovanie. Protočno, preformulajte to v matematike, občasno. Čekaj, to je ideja, z vseho statične opročovanie v termisji systemu, zelo, da je bilo vsef počka. En je to, da so se počka. Tečno, Maxwell ne nekaj. Počka, in na nasljah. Však vene vno več, že je to in stabilitiva. In druga je vsef počka, imač je, da je vsef počka. Veselo, da je vsef vsef počka. Nič, da je to začunil ki je to počkaj. Sdaj si je zelo se na vrlo zelo. Zatim, da je tudi zelo se očil, da je to začunil. Zato se je tudi začunil, da se mi se prideli nekaj. Zato se je tudi začunil. Tako in zelo se je, da je počkaj zelo se predaj, da je tudi začunil, da je tudi začunil. je to veliko, veliko in veliko in so, ali, na vsema, na matematika, je to zelo vsezvalo v nekaj, nekaj, nekaj, nekaj sistem. Zelo ne je nekaj zelo. Zelo, da se prišli, ideje Bolzma, zelo, da je zelo, da je vsezvalo, da je dobro vsezvalo. The bolzema, you have an amytonian system where Q is some partner to something. The same. And you have the amytonian equation. Let me call x the point in the phase, the representing point in the phase space. You have the amytonian equation. So the amytonian equation, you have an evolution, an evolution law, an evolution rule. chocolate. Sit, tMP, tMP, tMP, tMP, tMP. TMPP. TMP. TMP, tMP, tMP, tMP, tMP, možda, je mikro-canonica, ne? Mikro-canonica, je to. In so, and you know that the micro-canonica measure is invariant under time evolution, in the sense that if I take a region A, and I, on a certain region, and then I, this is the region, then I take the evolution of this region, is let me call S-T-A, and I look at the probability to say here, the probability to say here is the same, and so on. So, yeah, what is the ingredient of the problem? You see the mathematical terminology, and here you have that the system is on the region, let me call omega. Omega are the X, such that H of X is equal to energy. Okay, so the structure of the mathematical term, of course this is not the language of both, this is the language of the mathematician Birko von Neumann. And so the language is, you have a phase space where the system lives. Then you have an evolution, this is the phase space, this is the evolution, and then you have the invariant measure, so here the invariant measure is nothing but the integral on the region of the level. Okay, invariant measure means that I have this property. Or the mathematician prefer to write in this way that usually the evolution is not invertible. So this means that these are the points such that you evolve after time t, you arrive here. Okay, so you have this situation. Okay, this is the prototype, so if you take a mathematical book on dynamical system, this is the phase space, the evolution, and the measure. In this case, the meaning, this is nothing but the hypersurface, constantane, this is the Newtonian equation, and this is the microorganism equation. And then so you see here, this you have a mixture of determinism, and probably, and then the godic hypothesis is the following, that when you perform, you can wonder, so you know that from the elementary course of statistical mechanics that you have a micro-canonical, and so on, and from this you perform some average of some quantity. Okay, and then you can wonder, but okay, I have some observable, and from these observable, I compute an average, for example, micro-canonical. Okay, this is an exercise, but then you can wonder what is the relation of this quantity with the real world. We are physicists. And so even in the theoretical field that you can, you had to consider the problem, what is the physical relevance of your computation? How this stuff is related to the real world? The connection, at least, according to a certain way to consider statistical mechanics, is in term of the ergodic hypothesis. Boltzmann, I don't remember, more or less. Okay, let me reformulate in modern term. In modern term, the hypothesis is the following. You have an observable, and you have an instrument, and when you measure something, for example, you measure the pressure, you measure the pressure with the manometer, there appear on the display a certain number, how this number is obtained from the instrument. So the instrument perform a time average, in the case of the manometer, and you measure the transfer of inputs on the more region. So you have your time evolution. You start from a point x, the system evolves, and the instrument perform an average from zero to t, and then divide by t. This is the time average, which is performed by the instrument. And this is what, in some sense, this is the real fact in what the physicist measure on the table in experiment. And of course, if you do this, you realize that this is practically impossible in analytical computation. Why? Because you need the knowledge of initial states. This is the first problem. How to measure the phase space is enormous. Then you have to find the solution of the equation, which is billions of billions, this is the second difficulty. The third difficulty is that you perform the integral. So this is the real stuff, but immediately you realize that this is impossible. But this is what you want. So the remark of Balsam is the following. So if this quantity in principle is a function of x, which is the initial time, let me write x0, just to remind, the initial condition, and the duration of the measurement. In principle, actually this depends on these two stuff. If this is the case, if it really depends on initial condition, it is a disaster, it is not easy to understand the initial condition. So you want, at least if t is large enough, this doesn't depend on t. So you have a long integral. This is the first problem. And then the second problem, if there is a dependence on x0. And what do you open? The open, that t, when t go to 0, this is the average with the invariant probability. And this open is the ergodic hypothesis. The ergodic hypothesis is that the real stuff, the real quantity is the time average. But nobody is able to compute. On the contrary, the average with the canonical ensemble are relatively simple. And if this is true, if this is true, we are happy and we solve the problem. The question is, is this true? Okay, this is the, okay, then this will be for our Newtonian system. But you can repeat all the problems for any dynamical system. You can repeat this problem in any dynamical system in which you have a phase space and evolution and invariant measure. So this is the sort of problem in the measure theory. The problem started in physics. But then can be generalized. And then you have the problem. And now the start many, many years ago. And now the mathematician arrives and to fix the, to put order in the question. And there are two important results around 1930, due to Birkhoff and for Neumann. Oh, sorry, there are the theorem one, the theorem one is very important from mathematical point of view, from physical point of view is with due respect of these two joints completely irrelevant. The theorem one is under set hypothesis, blah, blah, blah, you can imagine the limit one over t in the other stuff exists. This quantity, the limit t to infinity fx0 t is a certain quantity, g of x0. Okay, this is clear. So the limit that exists is not surprising for us, is not surprising, this is what you expect. But remain the problem of the dependence of x0. Okay, so the first theorem solves the problem for t go to infinity. The limit that exists. But if it is true or not, it is not. The second theorem is about the tough point. Okay, this is okay. Okay, there exists the limit. The second point is about the dependence of x0. And this is a tough point. And this is called ergodic theorem. But it's not the solution of ergodic hypothesis. The ergodic hypothesis is still there. Okay, the second theorem is the following. There is a necessary and sufficient condition for ergodicity, I mentioned now, but in some sense it's just a shift of the problem. The system is ergodic if something happens. But then you have to prove that this happens. And this is not easy. Okay. I don't discuss the proof of the theorem. The second theorem is ergodic if and only if it is impossible to find b in omega, d in omega, such that the evolution of b is b. And so, this is omega, this is b, and if I start from b, I remain in b. In variant, this b is invariant under evolution. If I consider the evolution of b, it's still b. Such that, and the measure of b must be different from 1 or 0. Of course, if I take b, the system is okay. If I take b, a point is not true. So this means that if it is impossible to divide the system in two parts, and each of these parts is invariant, and are non-trivial, this means that these are non-trivial, then the system is ergodic. So you see that, practically, you shift the problem to this. So this theorem can be useful to prove that the system is not ergodic, because you realize immediately, for example, if there exists in a direction, the proof is trivial, imagine that there exists a situation like that. In this case, for sure, the system is not ergodic. Because, for example, I consider the observable, which is 1 if x is in b, and 0 if it's not in b. Then I consider x0 in b. Of course, xt will be in b. And so a of xt is always 1. OK, then you perform the average. Oh, sorry, let me. This average usually just, the average respect to time, which I introduced with an overbar, the average is, for sure, is 1. If you start from here, but if you start a point which is not b, you have 0. So you see the time average depends on where you start. So if you start from here, you say 1, you start from here, you say 0. And that's it. So this is the proof of the theorem in this direction. If you are able to find a partition of such that, then the system, for sure, is not ergodic. OK, and this is the end of the story. In the other direction, the proof is not trivial. But this theorem are very important, very important, but, however, this just shifts the problem to understand if the system is ergodic or not. OK, so the problem to understand if the system ergodic or not is not trivial. Absolutely is, absolutely is, is not trivial. So what is the connection, the ergodic hypothesis in some sense is the fact that the system is ergodic is, you are assuming that ergodic hypothesis corresponded to assume the validity of the law of the large number in an interior situation. Let me discuss this point. Ergodicity, which sounds very exotic argument, is not exotic at all, because it is the extension in the deterministic situation of the law of the large number. So you know the law of the large number, so I consider the case continuous time and also discrete time is the same. It is not important in the case of discrete time instead of the interior of the sum. So you know that the law of the large number, you remember you have x1, xt, where these are independent, the anti-distributed variable with finite variance. Remember that if you consider sum of xt, if you go to 1, 2t, 1 over t, these tend in some sense to the average of x, integral of x. You remember this in more formal way. Now you can assume that this is the elementary statistical and elementary probability. Now imagine that you have a dynamical system, xt plus 1 equals certain f. So this is a dynamical system, and maybe you have an invariant probability distribution, which is non-trivial, another point. In the mathematician starts assuming the invariant probability distribution, but the physical level, the invariant probability distribution is already a non-trivial problem. But imagine that you solve the invariant, you already know the invariant probability distribution, let me call it rho. Sorry, the mathematicians speak about measure, the physicists speak about density. Of course, the mathematician writes, and we are a bit rough, because in general you say that the invariant, the measure, typically we physicists write this, but this is not as precise, because in general it's not true. In general it doesn't exist a rho, which is an acceptable factor, but for the moment let us assume that this is possible. Ok, then I show you that in general it's not true, but a problem, not all the problem together. So imagine, this is the evolution though, then you have this rho, then how to determine rho is another problem, and then you can wonder, if I start from x0, and then I obtain x1, this way, and so on, x2 and so on, and then I consider the limit, t go to infinite, 1 over t sum of xt, this is equal to this, this is the ergodi problem, the ergodi problem in a map. The ergodi problem in a map, ok. And so, but here you have also the trouble that you don't know this stuff. So we have to reconsider the problem from the very beginning. Ok, but just now assuming a very empirical approach, just you have a computer, you can play in a simple way. For example, I know that you are very busy, but if you have time, just an exercise in homework, for example, let us consider this stuff, let us consider this stuff, exercise, take this stuff, take this, then consider x0, reset an x0, I don't know, point, a number between 0 and 1, a number between 0 and 1, and then from this you take, you fold your stuff, and then you compute, you can compute x, the limit equal to infinite, 1 over t sum x t. And then you can check if this depend or not of x0. You can try to check empirically if the system is ergodi or not, ok. Then the problem is that, but then you compute, if this doesn't depend on x0, you are tempted to say this is ergodi, ok. Ergodi, in matematica, the word ergodi is ergodi, respect to a certain measure, not ergodi, in the, in the Hamiltonian system is obvious that the invariant measure is the micro canonical, but in this point, in this problem is that, respect to what, what is this row? How are they termed in row? For example, in this situation, what is the invariant measure of this row? So at empirical level, before you can give an answer at empirical level. At empirical level, you can consider the following, you can consider a set of points, x1, xm, let me put here, 0, no. You can consider a initial probability distribution at the practical level, you take 10,000 points in a certain region, uniformly, for example. And then numerically, you can look at the evolution of this. What? You take 100 points, informally distributed between 0.3 and 0.4, then you look at the evolution, perform an Instagram, and you can compute numerically stuff. And then you look what happened when t go to infinity. If this go to something or not, this go to some row invariant. OK. Oh, believe me, this happened. This happened. And you can, OK, this is at empirical level. Now, in a more mathematical way, how I can construct this stuff. So this problem is the correspond exactly to the part in statistical mechanics. So in statistical mechanics, you have an Hamiltonian question and you write and you read the question. Right? So do you read the question? Just to show you. OK. The analogy. So yeah, Hamiltonian question. The Hamiltonian question, you have x dot equal f of x. I write in a formal way. OK. Then you have the you read the question. This is your equation. OK. And you know that your equation of the stationery is micro-canon. And then you can repeat the game here. The game for the map. One D map. One D map, you have x t plus one is equal f of x t. OK. Now you have to find the analogues of this equation. So the problem is the following. Now the time is discrete. The time is deep. So the problem is the following. The probability distribution at time t and you need the rule to construct the probability distribution at time t plus one. So if you find this rule, this is actually the analog of this. OK. And OK, if you write this equation, then you have an evolution law and you can wonder a lot of problem, a lot of question about this stuff. And this question is exactly the same question. I don't know. You wonder when you study, for example, Markov chain. Is exactly the same problem. Now more difficult because we are in a functional space, but part of the mathematical state conceptory is exactly the same stuff. So, OK. Now this is possible. OK, this is possible. How is possible? For example, let me consider just a simple case and then the gerengestion is obvious. Let me consider the 10 map. 10 map, the F. This F. And I want to write the evolution equation for the probability density in the 10 map. Now let me consider the probability that x at time t plus 1 is between x and x plus delta. This is if delta is small enough this is let me write in a physical way, it's clear. This is just raw t plus 1 x delta. OK. So this is x and this is x plus delta. This stuff is the probability to stay here. But the probability, there are only two possibilities that at the previous time you start from here or from here. So this point here, this point this is x over 2 because the slope is 2. This point here this is 1 minus x over 2. So this the length of this that we call delta prime delta prime is delta over 2 because f prime is equal to. So then you have this is nothing but the probability that x at time t is between x alpha x alpha plus delta prime plus the other probability x t is between 1 minus x 1 minus x alpha plus delta prime. OK. Then OK, this quantity, what is this quantity? This quantity is just rho time t compute in x alpha time delta prime but delta prime is delta alpha. In the other side the same. So now you simplify delta simplify delta and what you have, you have the evolution law which is very horrible but but at the least you have the equation in the following rho time t plus 1 in x is 1 alpha rho time t compute in x alpha plus rho time t minus which apparently is very horrible actually is horrible you have the law but then if you follow the reasoning you understand how to generalize this maybe after the stop the generalization is the following I write then I discuss later so if you have a the evolution law is the following where x k are the so-called pre-images of x so are the points such that f of x k is equal to x so just to what I mean so if f is 1 for example if x is here I am only to pre only to that if I am here I am 4 so this is in a compact way that rho t plus 1 is equal to a certain operator rho t for sure rather horrible because it is not local and this is at conceptual level this is the analogous of the of the equation which give you the evolution of the rho now you have this problem and you can wonder a lot of problem how be it is at a large time if there exists an invariant probability distribution of probability and blah blah respect to a certain specific rho I get this time for the stop for the stop I don't know do you prefer do a question now or after at the end? Questions I don't know say a great loss when you from the first equation to the last I show you this the next class I prove this equation could you please repeat quickly the process which start from x and x plus delta and then you get to the second step and how you relate this to the equation you wrote about the probability do you agree about this? so the probability to stay here at 90 plus 1 is possible in two way the previous time you are here or here because if you avoid this point you are right here that's it this is what the any other question stop for ten minutes take a break until 12 ok ok, so let me prove this this relation this operator is called Perron from Benius Perron from Benius operator or from Benius Perron and ok, I discuss this in 1D just for simplicity of the notation but it's not difficult to repeat the treatment in any in any dimension map of any dimension is an exercise for you to generalize to generalize to large to two, three dimension ok let me show this formula let me take something like that ok this is f this is x let me assume that x is always between 0 and 1 ok, now let me consider this x this delta this x plus delta I repeat the game of before if I repeat the game of before let me call this x1 and then we call this x2 the two preimages and here here I have delta here I have said that we call this delta1 the size of this what is this so I have that delta is f prime of x1 delta1 if delta is more do you agree that delta1 is delta over f prime of x1 if I repeat the same game here the same story delta2 is delta f prime x2 but here you have to put the modulus because what is important is not important the slope but it is a model of slope so let me line this way delta1 is delta over f prime compute in x1 then I repeat the same story of before and I have that just repeating exactly the same argument before if delta up to here in this region in this region the rule is the following the rule is some from delta2 and rho x f prime fj in this region if I consider this region this part I repeat the same story but now I have four point I have this this and this so the same story in this one where k can depend of x this region is 2, here is 4 and here is again 2 so they are apparently strange but this is nothing but the conservation of probability you are saying that in order to stay here you have to consider all the possibility at the previous time there are two, here you have four just to remind the formula there is a very simple way to remind the formula if you remember the simple way you remind the elementary property of delta here it is it is very simple to remind and then it is also very intuitive so the probability in x at time t plus 1 is given by all the possible combination to stay at time t ok these are equivalent and this is a way to remind and that's it so if you repeat this game in more than one dimension it is not difficult if you have something like that x t plus 1 is equal to a certain f x t where x is in r2, 3, whatever you repeat the game and you have something like that x is equal to all the preenegies here instead of f prime you have the Jacobian of the transformation so you have let me write d d of x where d is the determinant where d is the determinant of matrix A and the element of the matrix is obvious the argument is the same so in any dimension you can repeat the but we consider only the 1d is enough to suffer now let me write x t plus 1 is equal f of x t and I have rho t plus 1 is equal a certain operator rho of t in a compact way this is in a compact way now this situation is very similar it is completely parallel to the problem you have in Markov chain let me open here a short parenthesis of Markov chain in Markov chain Markov chain are much simpler because in Markov chain you have discrete time even here the tiny discrete and you have a discrete state discrete states 1, 2 and here the state is not discrete it is continuous this is the difference so now you have the probability let me call p of t the probability to stay in the state 1 or 2 and so on you remember that if you know the probability at time t you have the probability at time t plus 1 and you take this using the transition matrix so you have something like that this is the translation of the formula p to stay in p at time t plus 1 is the sum of the probability to stay in j at time t for the probability to jump from j to i so you see you have this structure and this structure formally is very practically is the same structure of this one but this situation is much more difficult in mathematical terms because here you are in a functional space and there you are in this discrete state but now you know that in the Markov chain everything is even here everything is even in this transition matrix in p the probability to jump from 2 states looking on this transition matrix you understand all the possible things you can wonder about the Markov chain so in the Markov chain for example you have the idea of invariant in mathematical vision you determine invariant stationary probability the p stationary or invariant to this equation stationary means that if you start from there you remain there if you start from the stationary this remains there and you can wonder if this is a unique or not for example and this depend of the I don't know if you are familiar with Markov chain you are familiar with the classification periodic, periodic periodic, irreducible then the invariant the invariant probability is unique and then not only is unique but also attractive so it means that in the everybody are familiar with this stuff for good Markov chain where good means periodic, irreducible you have that if you start from certain then you consider the evolution and then you consider the limit t go to infinite this approach to the stationary probability not only but you approach also exponentially fast you have this stuff now you are and in addition the system is gone in the sense discussed before this is the idea of Monte Carlo in the Monte Carlo in the Monte Carlo is there to compute the average as an average of the probability distribution perform a random walk among the states in the certain rule now you can it is quite natural to repeat the same problem there so now the invariant invariant measure variant distribution let me use not formal the invariant distribution is something like that invariant means that which satisfied is a wish I apply the I apply the operator then you can wonder if the row invariant is ergodic and if row invariant is attractive attractive means I start with a certain root 0 I compute the evolution and I look at the limit of this you have in the case of the map then you can translate also you can repeat in 2D and if you want you can repeat in dynamical system you are something different but the concept are the same problem now let me just show that ergodicity is a nice property but not enough sometimes not enough so let me just the first day consider an example very simple example the rotation on the circle easy there the rotation on the circle so you have a circle and you have x0 and x1 is just x0 plus something omega ok, since in the circle you take modulus 1 modulus 2 pi but just for simplicity the circle of length the rule is the following xt plus 1 is equal xt plus omega modulus 1 and the question try to reply so what is the invariant distribution is the invariant distribution ergodic is the invariant distribution attractive ok, this is an exercise in this case we can do all the computation explicitly but I guess is one of the few cases just an exercise ok, first point the invariant probability invariant probability is very simple the rule I discussed before because the f the f is this way this f is x this is the rule this is omega this is the rule so I have to write the evolution equation rho t plus 1 x is equal so in this case I have only a pre-enegis only this ok so I have only x the pre-enegis x prime is x minus omega modulus 1 you see that I have only one and the slope is one so just that rho at time t plus 1 is nothing but rho at time t compute in x minus omega consider on the circle ok, and that's it this is the pro Frobenius equation in this case which is absolute trivial because now if you impose the rho invariant x is equal rho invariant x minus omega immediately you have the solution is rho invariant equal 1 this is the reason why I take modulus 1 this is constant this is constant which sure is invariant ok, so the rho invariant is constant fine then you can wonder if it is ergodic or not no? rho invariant rotation no, you can wonder if this is ergodic or not so the answer is that is ergodic is ergodic is omega is irrational is not ergodic is omega is rational how you can prove this you can prove with brute force computation brute force explicit computation is not exactly a grid force in explicit computation so I take in observable I consider just for as mathematician say for generosity I assume that h in observable which is analytic so it is analytic so I can perform Fourier series s n differ from 0 a n a e 2 pi n x ok then I take this observable of course if I perform the average according to the invariant x in 1 so this is 1 so the the integral of this this permit give you a 0 and the other give 0 give 0 because it is integral of this so the the average on invariant distribution is only h0 now I have to compute I have to compute the probability the the average I have to perform the evolution and then the average so I have to compute 1 over t sum of t from 0 to 1 to t 0 to t minus 1 here I have a a x t so 1 over t t minus 1 I have this stuff 0 plus sum n different from 0 a n here i2pi n x t but x t x t is x 0 plus t omega modulus 1 if I iterate so I can put here x 0 i2pi omega t so you see that modulus 1 is free ok, so now this is real this term is h0 now the second term we have sum of n different from 0 a i2pi n x 0 and then I have 1 over t sum t from 0 to t minus 1 this stuff, but this is I can write it this way ok, so you see I have a sum of something like t from 0, t minus 1 of alpha to t this alpha so, what happened? so there are two possibilities alpha is different from 1 or alpha is equal to 1 let me consider the case alpha different from 1 alpha different from 1 is when omega is irrational because in order to have this 1 means that this number must be integer so if omega is irrational this is impossible so you see that this is true if omega is irrational in this case just to apply the elementary geometric series here you have 1 minus alpha t 1 minus alpha and that's it in this case you have a0 plus sum bla bla bla and here you have 1 over t and you have this quantity take the limit t, go to infinite this go to 0 remain on this if omega is irrational this is ergodic if omega is not irrational what means? means that omega must be omega is something like k over m if I put here k over m I have repeat the consideration so if n is not multiple of m I repeat again in all the case where this number is is integer I had to maintain this quantity is 1 ok, so at the end of the story I have that what remains a0 plus the sum no all the n such that nk to m integer and remain this is 1 this is 1, remain this one remains am pi mx0 so you see that in the case this is rational you can perform the average but the important result that now the this is the time average that exists because of the theorem but remain is a function of x0 this is the problem is a function of x0 and so this system is not a godic but if it is not a godic another way to see that it is not a godic you just to note that the system is omega is rational, the system is periodic this is periodic why if it is rational the system is quasi periodic this means if you plot if you follow the system on the circle in the case in rational at the end you cover uniformly the circle why if it is in rational you are just a finite number point so in this case when you see that you have that the system is a godic is a godic but then you can find is a godic but is attractive or not no is not attractive in fact is not attractive do you realize immediately I delete the question let me write again the question for the evolution the question of evolution is this one what is the meaning of this equation the meaning of this equation we are just a rigid translation so this means that you take and rho1 is just this just shift of omega that's it this is 0,1 you start from this is a rho0 this is a rho0 rho1 is simply this shift by omega so if you start with someone no uniform distribution you just remain for the rest of the the life just this move if this triangle remains triangle if this is semicircle remains semicircle and that's it so there is no way there is no way that this approach to the invariant apart the case and you start with the wrong variant so of course if 0 is different from the wrong variant then rho t cannot approach to the invariant just remain trivially the stuff so you see that you can have you can have ergodic system but but somehow somehow trivial ok now this is about the this is about the now apart the game let me show something on this something on this there is one you discuss already with Fabio Cerconi the logistic map and I don't know if Fabio discuss the fact that the logistic map you can perform a change of variables such that you have the 10 map you discuss this stuff ok but don't worry so let me just use this last 10 minutes ok consider the 10 map rho t plus 1 f x t f is this situation and you can wonder what in this case what is the invariant distribution sure it is uniform but it is a coincidence it is not true in general that this is we just repeat the just look in your notes beyond the question in this one ok this is evolution question and then you see immediately that rho invariant x is equal to 1 half rho invariant x half plus rho invariant 1 minus and you see that you have a solution a solution not a solution a solution that rho invariant is equal to 1 you see that it is satisfied rho invariant is equal to 1 is satisfied that if this is attractive or not yes it is attractive there is approval but it is not completely trivial so you can check for example numerically you start with so what is the the deep reason that this is attractive the reason is the fact that f prime modulus is larger than 1 this is which is an effect of instabil so for example if you start from probability distribution here in this pattern probability distribution this shape so no let me here so what you have that these remain something like the size the height the altitude is half of the after iteration the area is conserved but this length decrease of factor 2 why the size increase factor 2 because of the spreading and then you of course when you went in the other regime and followed the argument but this is the basic argument you can do numerically so actually what you have that if you look at the probability the evolution this actually go to this go to the invariant and the the deep reason is because you have you have f prime larger than 1 and of course then you have the the system must be bounded and these repeat the folding ok but why this interesting so and then you can repeat the game you can repeat the game in just a stupid exercise if instead to consider a you consider an asymmetric an asymmetric tent something like that is always the same so in this case in this case is independent is perfectly symmetric and this is interesting because because there is the following trick or technique what is the basic trick in mathematics to change the variable to change the variable in such a way that is the simple ok to find the smart change of variable so the Fourier transform Fourier transform Fourier series something which is difficult in the original space Fourier transform is simple ok in the mathematician use topological conjugation topological conjugation invertible change of variable so imagine that you have imagine that you have a system like this no not this imagine that now we have this one ok no this is but this in this case you can repeat you can try to write this is not impossible you can write you can write but and then you can try to understand what is the invariant the invariant probability and you have some horrible stuff you don't understand nothing now the idea is the following imagine that I consider another variable x which is a function of x invertible invertible ok so in some way I don't there is no missing of information then of course this we evolve in some way we evolve like g of x t plus 1 so this means that this is g of g of f of x t is g minus 1 y I have something g f g minus 1 so I have something h is this y it is a rule ok it is invertible j prime it is always different from z so I take the this equation and you have another question this is for free but now I do this only if there is an advantage so the idea is that I start with this this equation is not easy to treat and if I write here something much simpler the idea is that then from here imagine that I something simpler and I am able to compute the invariant probability of y let me put here y if I am able to do this then I just simply to remind the elementary property of the change of variable probability and then from this I can come back to the invariant problem to the invariant probability of the original problem ok so this is the strategy this is the strategy is the change of variable such that the new the new the new system which always the same system is the system observed in different different way is simple enough in order to compute invariant probability and then come back so in pratics for example imagine that I am able to find a change of variable such that for this equation I have this then I have the invariant probability of the solution then I come back so there is a change of variable then I I will discuss tomorrow discuss tomorrow and it is possible for example this idea to show that the invariant probability of this stuff is an expression which not obvious I obtain this using this trick change the variable in such a way the new variable is the 10 map and in fact the 10 map is 3 then I come back so you see that the probability distribution is the fact that in this case is constant is just a coincidence in general is not constant at all so this is a probability solution with the version something like that absolutely not obvious it is obvious that if you know the trick to perform that's it ok I guess we can stop here for the first day I guess is enough question on the on the from on the web somebody asks about the you already answered up to now I have no specific hypothesis about which kind of dynamics I have now I consider only one dimension a case I can consider tomorrow I can consider larger larger dimension and but it's better to maintain a separate system with a uv theorem a system without conservative and dissipus are completely different so in the case conservative system in this case the invariant measure is obvious is the is the big measure is absolutely so imagine you realize immediately in this case the situation is trivial because since the system is invertible you have an equation completely trivial and from there immediately you realize that the invariant probability is just uniform and that's it but this is not true if the system is not conservative if the system is not conservative in more than the one dimension we will have a horrible situation in which we touch tomorrow or the day after because up to now I discuss about invariant about density assuming that density there exists but this is not true absolutely is not true just for the first election just to start in a smooth way so in just considering I guess Fabio Shekone introduce you the Inom map which have very innocent aspect is a monster from the point of view of the measure is a monster monster means that it doesn't exist any density not only but it is also singular respect to fractal geometry so even extremely simple system you can write in two lines on the blackboard in almost very rich structure but we will see you tomorrow maybe I should watch the video again but I don't understand how you go from here and you obtain one the invariant so the invariant is when I assume that the invariant means that at any time I assume that this is raw invariant because invariant means that when they perform the evolution there is no evolution so I just translate this repeating invariant here, invariant here that's it now the one I don't know how you obtain one look because one is equal one half plus one half so if I assume that this is one this is satisfied you see okay so in fact I say this is a solution non de solution then you can wonder why you don't consider all the possible solution and anticipating this question how many possible solutions there are infinite so it's better to avoid the solution this is a subtle point I discuss tomorrow in some sense I have 20 seconds just to introduce the problem this is a very fascinating problem I like a lot not only me the people are working in dynamical system the problem is the following you can wonder there are other invariant and the answer is yes this is f if I consider f2 so just ff which means the evolution I have something like that if I consider f4 I have f4 is enough then let me consider this point this this this and this so this point are fixed point of the the map f4 so these are periodic point so if I immediately I realize that if I consider rho if I consider the slope of the stuff are the same if I consider some of normalization delta x minus this point xk this invariant this invariant because you saw one point moving the other so this means that if I consider fk many times all the stuff which correspond to unstable periodic orbit and I consider the measure concentrate to the unstable periodic orbit of any period any period all the period you like this is invariant that's it, this is invariant, this is remark and then you can wonder why this is a problem so mathematically this is a problem this is horrible so the first the first we just and of course but you have to be an authorization to remove this stuff and so you have at the least some not say prove but some argument to remove tomorrow I will take so the idea is that this a solution not the solution the problem is that but in some sense this is the solution the solution because this solution are unphysical unphysical and can be removed with a very physical argument ok so thank you very much so we meet again at 4pm here