 Okay, so let me explain the essence of the Oselege theorem, just the idea. Let me consider the trivial case, but which is the essence then the trivial case is the case multiplication of random number. We have a sequence of random number a1 a2 a n where this number these are positive and distributed according to a certain probability distribution, smooth enough. Okay. And then you consider this problem, you take X, Z, Z, and equal the product of this, this number. You know the probability distribution this time. Very independent and you wonder what happened when and go to infinite. Okay, this is the difference. What happened so if you, if you have to guess the behavior of this quantity when n is very large. One temptation is a this is the average of a to n. This is wrong. This is not true. Okay, the behavior, the most probable behavior, the typical behavior, which happened with probability one is that these go like exponential and the log, the average or log of a white is more formal way is a the limit of one over n Z and go to this. Go to this with the probability one. Why it's real because you just take this. Perform the logarithm, you have the logarithm of Z and is some of log of a and. Then, since these are independent, even log is dependent. And so, and if you divide by n, you invoke the law of the larger number and you have this approach to log of a. And that's it. Okay, this is just trivial, just a trivial application of just trivial remark. This is the consequence of the law of general larger now. Now, the Oceli deck theorem is the same kind of result, but I know trivial situation. The solution is the following you have now instead to have a number, you have matrix, let me call a one, a two, and so on. Okay. And let me define the be and as the product in order away. And then let me introduce the, the matrix is transposal matrix, which is a one. Doug a two. And you wonder, they said it is about this, you take this matrix, you multiply this by this in such a way the matrix is, is, is a symmetric. And then you wonder what happened when these go to infinite. Okay, now, just to do in a more precise way. Take this quantity. Okay. Which is a formally, this is the same, I think that is correspond to that and formally that and is the same. One over two and these go to log of a. Okay. Okay. Now, I consider this, this now this is a matrix. Now it is a matrix. And you wonder what happened when I go to infinite. So, in this case, this quantity go to this with probability one. It's absolutely not trivial but in the same spirit. The statement is when all go to infinite approach to a certain metrics, let me call the approach a certain metrics. Of course, this metrics that you can expect that the metrics depend of the exact sequence you have. Okay. But you see that in analogy with this, you added here the this quantity. Almost all the sequence approach to the same value. Even here you're the same all these things approach to the same, the same probably to the same metrics, which is independent from the specific sequence, this is the theorem. Which is in some sense is a generalization of the this is trivial. Okay, this is not trivial at all. Okay, so from this factor, if you translate this in term of the of the of the tangent vector, you have the reference point. Okay, why, because so you have this and this is certain certain certain certain certain metrics and this metric is some some eigenvalue, let me call alpha one alpha two and so on. And these are positive, of course, because it is a real and then you're also some some eigenvalue, some eigenvector, let me call the one the two and so on. The fact that there exists that this is this is asymptotic metrics. You can translate this stuff for with for the Lyapunus for the opponent. Now you have that with the notation yesterday ZN is equal to be N at zero. Okay, now you take ZN model square. And this is ZN. This is B and that zero. You got this. Okay, now you see that appear appear this magic. Now. Now, you want to, you want to, you want to, you want to understand one over N log of this stuff. And this is, this is one over two N. This quote. Okay, so now you see now you understand the why isn't that it's important that this is some some limit. Okay, because this means that, of course, this depend on that that zero. Okay, and so now depend on the, if that zero is perpendicular or not to the eigen eigenvalue eigenvector, all these metrics. So, and you have the Lyapunus point. Okay, this is the, this is the, there exists the Lyapunus point and the meaning, the meaning. Okay. This is the mathematical part without any proof just just just to mention the. The, the method by Benetton et al. Carlo Benetton is very wonderful mathematician, very kind person in Padua, he went in pension just few years ago. He's a great gentleman, not only a great scientist, but also a great gentleman. Benetton et al. Actually the paper, the first paper has been published on physical review, but is the, the first paper was say, not, not, not very rigorous. It was in the style of physical thesis, what there is a, some years later it was a rigorous paper published in a very obscure journal, and make the name is mechanical. And, in spite of the fact that this journal is completely unknown almost to everybody. It's very important. Okay. And the, the, the method is the following. So, I, I discussed yesterday, discussed for, for maps for the, okay, for maps or differential equation, there is no big difference. So the case on map, okay, like map and ordinary difference. So in the case of a map, we have something like that x. At time t plus one equal g of, of x time t differential equation we have x dot x dot equal f the, the, the, the, the, the engine vector is written this way. We are a is a j here is a set dot equal a, the same. Then the method is the following you, you, you have, okay, you have this equation for, for the evolution of the system, and then you have the equation for the evolution tangent vector. Imagine that you want to call the system is the dimension and n, n is the dimension of the system. So for example, in the case of a no market to in the case of the, the, the, the lower system is three and so on. Okay, now the, the, the, the, the, the algorithm is the following. At equal zero, you start with the zeta one to n zero, and these are orthonormal vectors. Okay. These are i zero j are mutually orthonormal. Then each of these evolve, each of these evolve according to these equations. Up to up to a certain time t equal tau, okay, this evolve this way. And after this evolution. I consider, let me call alpha one equal alpha one tau equal the modules of that one. And let me call alpha two tau, the following quantity. So this is that one is that two. And I look at this area. And this, let me call this up to down, let me call alpha, alpha three, the volume when they tell me by zeta one, zeta two, zeta three, and so on. Okay. Then after this. After this, I, I perform a orthonormal, you know, a grass meat orthonormalization. Okay, so I rescale this. I take this, I rescale. Let me call W. W one. Z is just just Z one. Normalize the divide by alpha one for for W two is a vector which is perpendicular to this one. W three is perpendicular to W one and W two and so on the orthonormalization. Okay, and then I reconsider this and restart the game, a repeat again. Okay. So, I look at the evolution after a certain time tau, I perform a grass meat orthonormalization I compute this quantity. And then, and then go on. So, then I perform this, iterate this a long time. And what happened that from this quantity alpha one are computed time tau to tau and so on. So I have that today. So one is the limit and no N is M. M go to infinite one over M tau, some of log of alpha one alpha one. Jerry from M. Jay. So if you want to lambda two, I have that lambda one plus lambda two is the same, but here I have alpha two and so on. So if you want to lambda one plus lambda two plus lambda k. This is the same stuff with here I have just alpha k. And this means that they can compute all all all the all the characteristics on it. There is no problem. So it just it just long because the computer time increase if they want to one, practically a to double the time if you want to get factor three, and so on. And you see that it's not possible to compute the second without the computation on the first and you cannot compute the third without the computation on the first and the second and so on. And this is, this is the method. So this method was in the history of the dynamic assistant was extremely important was really a turning point of the history of dynamic assistant, because before this method. What means chaotic is really very generic is something which is irregular looking at the spectrum, looking at the correlation function. So, this is this is the borderline between between the pre history and the history of the dynamic assistant before this dynamic assistant was a matter for only for mathematician. After this it was possible even for us physicists entering the game, studying in a not just some very abstract T or M just looking at what type just look at some, it's a number. Okay. Okay, this is the for the part of the app. Okay, of the of the Japanese point. Now, the other aspect I mentioned yesterday is the characterization of the tractor. And I said yesterday that a way to characterize the tractor is so before before to go on before to go on. There are a list of the list of the property of the leopon exponent which I give you without any, any proof. Okay. So, the conservative system, the sum of the leopon exponent is zero. And in simplicity system, I mean Tony I'm sorry, it's in practice, in simplicity system. We have that the, the, the last one, let me call lambda to N is minus lambda one. And this is a, this is a specular situation. Okay. And in the C but a system. Okay, this is generally the sum of the leopon exponent is equal to the divergence. So it's negative. In the law in the Lawrence model, so in the Lawrence model in the, in the, okay, this is for the, sorry, this is for the differential equation otherwise is the determinant of the matrix. For example, in the, in the case of the another nice property, which in the case of ordinary differential equation, a part of the stupid case with fixed point. Okay, if the system is just before this if you have a stable fixed point, you have that the, all the leopon exponent are negative. This is real. Okay. If you have a stable stable limit cycle, you have that one leopon exponent is one leopon exponent is zero. And the, and the, and the other and the other are negative. And the entire situation is when at least one is positive. So in the case of ordinary differential equation at least a part of the trivial case of stable fixed point at least at least one leopon exponent is exactly zero. Okay, this help in the computation, for example, with this result, when you have to, you can use this result for the pain in the end on map, you have to the opponent, the first and the second, but the second is fixed by this condition. In the, in the law and system, you have three leopon exponent, but only one is independent, because the first one, because the second must be zero. And the third is fixed by this condition. Okay, so there's a lot in long, in long dimensional system, this, this kind means that practically just one leopon exponent, because the other are truly determined by, by this general Okay, now, the, the, the, the structure of the track before to go on. Let me, yesterday I discussed one of you in a private form, an important, important fact that why, what is the, the, the intuition why you expect that in the dissipative in the dissipative quality system you have some strange things that are, the, the, the, the reason is the following. These are to say to the stretching a max, stretching, stretching, stretching, stretching, no, stretching and folding mechanism. What is the stretcher for the image in that the just to fix the idea the enormous. Okay, so you have a disability you have a disability system, which is chaotic. and live in a bounded domain. Okay, then you start with the set of initial condition like this. Okay, so whatever. So because it is dissipative, these evolve and this area must decrease. Okay, into this area volume agent must decrease. Okay, but but can decrease in different way for example one one possibility just decrease in a trivial way just decrease. But this is not possible. No, this is possible way to decrease, but this is possible only in stable situation because in a stable situation in counting situation this is not possible because if you take two points here. These two point the distance must increase. So this is not possible. So what is what happens that this must become something like that. So if you take here some some point the point and so on the distance must must must increase. So this means that you are you are something like that. Okay, so because the system is dissipative, the area must decrease but because it's chaotic, the shape is necessary some location. Okay, and then this mechanism must go on. So since there must must be bounded in a finite region, in a finite domain, the unique way, the unique possibility at that not only you have this stretching but there's also a folding you have to fold this stuff in order to remain inside this. So it's necessary something like that. Okay. It must be repeated all the time you want. So at the end, so you start from something like that. And then the end that you is necessary to have something like that. Okay. This infinite time. So this means that this cannot be as much stuff can be something so you can hope that this is self similar for example you can imagine that if I take this and I perform a zoom. I obtained the same. This is not the proof of the course but so you can, you can expect something like this and there is always this this mechanism. So it means that putting together the fact that this is put together the fact this is secret is chaotic this is bounded these three things together. You have the intuition that you expect something like that. Okay. So this is the reason why you expect factor. This is not the proof. Nobody has been able to prove that some attractor is really fragile. Maybe there are some specific case but there's some. But it is what it is what you expect and you find that in my guess so I guess you are familiar with fractal is not necessary to define the fractal dimension. I get to find fractal dimension. Yes or not. I have to. Okay, so what what what means factor. What is fractal, which, if I say fractal, you expect to say that the father of fractal was Mandelbrot wrong. The father of mandel fractal was not Mandelbrot but was another gentleman. I suspect that you don't know the name of this, this guy. You know what this guy. You know, so every day you see the weather forecasting program. This way, this guy, this gentleman invented the modern modern methodology to perform the weather forecasting among the many among the other things. And so and he had the first was the idea that the famous question how long is the cost of Great Britain is from him, not from Mandelbrot. Okay, apart of the historical remark. What the means factor. The real factor is the general dimension. Yeah, measurement. What they mean dimension. So usually I say why why you say that these is dimension one, you say this dimension one because a point here is determined by one number. The longitudinal curvilinear coordinate. This L1, this L2, and that's it. So if you have a sphere on the sphere, a point is identified by the number latitude and longitude. So this is, you have, you say this is dimension one. This is dimension two. The volume is dimension three because you need the X, Y, Z. Okay, and so on. So you associate the idea of dimension with discrete number. In terms of the, how many, how many coordinate I need to identify a point. Okay, this is an idea. Another idea, which is a empirical is due to Richard's is the following. You have a line. I have a line. And you have a compass. Okay. There is a second. You have an instrument. Okay, you have an instrument. Okay, you have an instrument with a certain precision. Let me call the delta, delta X, okay, and you take your your instrument. This and this. So you approximate the, you approximate this as something with a segment of size delta X. Okay. And then you count how many, how many, how many, how many segment I need to cover the stuff. Okay, I count this and then I multiply by delta X. And I take the limit delta X go to zero. And this is what you call length. Right. This is called what you call length, the length is one. So if you repeat the same game with the, so with a sphere. So you count, you count how many, how many square. You have the edge delta X, you need to cover the answer this stuff this and this approach to what you call surface. Okay, in the case of volume, you have n delta X delta X cube, and this go to the volume. Okay, this is another way to define. So you see that here you have a data. In this case, you have delta X go like delta X minus one. In this case delta X minus two. Here you have delta X to minus three. This is another way to define dimension because instead to say how many variable I need blah blah just just from empirical point of view. I count how many, how many boxes quantities, how many elements as necessary to approximate the stuff in the limit of delta X go to zero. I have that delta n delta X go goes like delta X to minus D. I define the dimension in this way. I found I have to count and counting looking at the scaling. Okay, fine. But, okay, this way I obtained what I know 123 up to now nothing tell us now the next question is, is it possible to have an object such that this quantity is not the one or two three. This is, this was the smart idea of Richard. There was also some in the prehistory of the stuff that I guess the richer son is the father but the grandfather probably was by a stress. He was in his work on, on, on Fourier series and on differential. So, apart the, the, the historical aspect, so you can go on the if there exists, if there exists the situation where this number is something for that between one or two or two or three and so. So, so what is important is that not only is possible to show some, some mathematical logic, such that this D is not integer. And these, maybe you can say this is a mathematical curiosity, okay. Now, the next step is this that exists in nature, or not this important from physical point of view. So, the first step in mathematics, and this is easy. In 20 seconds, two minutes. The, the, the, the, the following step is that there is this stuff in a trivial way, for example, in the electric attractor. Because a chaotic attractor is not very physical, it's just mathematics, okay, you can wonder if you look outside the window, I observe the stuff. Yes, in turbulence, for example, in turbulence, in turbulence, the, the, the, there is the energy distribution distribution of turbulence is an audible object. Which is not a factor, but it's more complicated. It's called multi factor, but don't worry. So, so now, now, let me just convince show you a situation where this is not, is not, is not the one. Oh, so in the, in the, in the paper. How, how is along the cost of a great beaten the gentleman take perform this performer explicitly this exercise. Okay, we are on the maps of course not not not working along the, the cost and the plotting log of N log of delta X, put the number. You said that this law, but you said that the was not the one was not the one but was something like I don't remember 1.2 1.3 so but something for sure different from one. So, why, so this means that this means that this means that the in some sense that the, the, the, the, the curve is not differentiable because it's differentiable this is one. At least it's not differentiable at a certain scale. Okay. No, now, let me just show an example of, of fractal. The phone cock. So the phone cock is the following is very simple, you can construct in an iterative way. So, at the, at the steps, zero, you have a segment of size one. Okay. At the step one, you, you divide this segment in three parts. You remove the, the internal part and you substitute with two other. Okay, this is one, this is one third. And then you repeat the game. Repeat the game is enough. This is one third. This is one. One third square. And so on. Okay, you see that here, yeah, delta, okay, yeah, delta x here is one is one third is one. And the N, and here is one. Yes for here is 16. Okay, if you go on, if you go on the scale one to N, here you have four to N. Okay, so now if you want to manage to compute the so you have a no differentiable cure course. And so, and then if you apply this formula, what you have. You have that N, N of one third to N is equal to four to N. Okay, then you apply this formula. And you obtain that D is in this, this is log of four log three, which is something between two and one. Okay, is a number. Okay, so you see this is this is an example. Okay, you can say, oh, okay, this is the pathology is artificial example. Yes. You are all the right to say that this is just a joke. This is just a game. Okay. But if, if the factor remain on this level are not important. Fortunately, they're appearing in a trigger way. Okay. So, okay, this is the way to compute the dimension. And the problem is that, okay, but if I, for example, if I have, if I have an attack, if I have empirically an attract, imagine that now you have your computer, your computer you iterate the system for example, the, the, the, the, the new map or, or, or, or Lauren system. And then you want to assume that you assume because you expect you hope that the, the attractor is the fractal. The first question is, how is how you computed the fractal dimension, how you compute the fractal dimension. Okay, these things was very popular when I have your age in the 80. And the first approach, the first approach was very, very, very nice. What just the first approach is just to apply the definition. Of course, okay. The first approach is what is called both counting. So just to repeat this. So the idea is the following. So this is the, the first method of course the first method is not efficient as you always the first method was counting. You follow me. Imagine that you have your system, and you generate the X one X to X, and so this is the, the, the point of evolving time. Okay. Imagine that the time is this great. This is continuous you take it at a certain time is not important. This point, for example, in the, in the, and then you have your, your system. And you perform a, a partition of your system of size, so it's a epsilon. So you have, you have this point. Okay, this point, well, and it is very large of course and, and, and must be very large. And at a given, at a given epsilon, you count how many non empty boxes you have. So you count how many non, non empty box you have that you call this end of epsilon. Okay, then if this is a fractal. If this is a fractal means that this is scaling this way. Okay, so now you take this you put here log epsilon. You repeat the game with the different epsilon. Small, your point. So if this is fractal this point will be on the line. If it's on the line. Do a fit from this fit you have a diffusion. Okay, this is the method this method is pretty stupid because it's just a definition just application definition. The question is the method is efficient. No, of course, no, it's not efficient. So he's efficient for the normal map. For the map is efficient because for if you do this for the normal map for, for example, the normal map with the B equals zero three a equal one point four which is a typical the typical value just to play the dimension is is around one point. Okay. Also, and if you, if you, if you look at you, you perform a zoom or the tractor you realize that actually there is a sense in the structure. Okay, but this is this is simple. It's okay if you zoom the stuff. If you zoom here you see that you are more. If you do this is normal map. Okay. So if you do the same for the Lawrence system. Are equal 28. Here you have the D is around two point zero five. This is the case of low dimensional system the method is okay but so you see the method is very stupid because they put for example in in today. This is not empty. This is an empty box, but the, the total number of boxes much larger is is epsilon two minus two. Okay, so it means when you do this. The dimensions of the box are empty. So you have an enormous memory consuming for nothing. Okay. And so it's clear that this method cannot be applied to large system. Even even something embedded in the same system with the dimension with the five, five, five variable. The method is not efficient. Okay, and so the, the, the, the, the, the, the, the work to solve the problem at least not completely but at least. This is the first step is the grass burger Procaccia. If I remember correctly was 83. Okay, so the progress burger progression method is to avoid the, the, the, to have a method, which is, which is efficient and is not limited by the fact that the, the, you have to use an enormous number of boxes and these boxes are empty. Okay. And the idea is the following. Actually, in this way, actually, the method is not the method doesn't have to compute the fractal dimension, but another dimension, which apparently is disturbing because you want to compute the fractal dimension but what actually what you compute is something different. And you can want you can you can complain on this but it's not true. This is the point to for a break. Oh, okay. They are the computation of another dimension. And then, after this, the people realize that the fractal dimension is just one dimension is not the dimension. So it's possible to use the many, many, many, infinite dimension. And, and the fact that dimension in some sense is not the most relevant. Even the dimension introduced by Gersberg approach is not the most relevant, but in any case, better than nothing. So, and the general generalizing this is the first step. And this is the correlation dimension. And then, so, generalize Gersberg progression method. We have as possible to compute the generalized dimension. And there exists a thermodynamic interpretation of this dimension is a thermodynamic. And some strange thermodynamics thermodynamics what we thermodynamics for the annual map is something ridiculous. No, but I show you that. Okay, so this is the, you see when you repeat you, you appear this, this. Okay, this is not the proof that is self familiar but convincing that you suspect that this is really said similar. Okay. And actually it's not exactly self similar. So in fact that the, the fact that the, the, the, the, when you try to generalize the idea you realize that there is no unique dimension and this dimension has been interpreted in terms of statistical mechanics formula. So apparently say, so you know the usual statistical thermodynamic formula, the free energy and so on. Partition function. Partition function and so on. What is the relation partition function free energy this stuff. No, there is a connection. There is a connection. I show you in the next, next, next hour. I stop you. Okay, we take a break. Questions. Oh, okay. Okay, thank you very much. Okay. The, the idea was very prokaccia is very, very simple but as usual when you know it's simple. Here is the following. So you started with your X one X and okay and very large. Okay. So this is the point to which, when you run on your computer you remove the first transient and take this. Okay, and these are representative of the attract. And you want to understand something like a dimension. The idea is the following. I take one point X. And I take a ball of size epsilon. And I count how many of these stuff are inside this. So, and then I perform an average on all the time. This means this I think so, in order to count out I can count I take the data function epsilon minus X i minus X j. So this is zero. The distance is larger than epsilon. And these one is the distance is smaller than epsilon so discount how many couple as distance less than epsilon. Then I take the average I and j and divide by n square n square and my n minus one over two but that is okay. And so, and now you're then I take epsilon smaller and the statement is this when it's you know go to his mall, this is epsilon to something, and let me call something the two. And so you expect that this the two is some relation with the fractal dimension. Actually, it's not exactly, but why imagine that the way you expect this and why you expect this is the dimension. Imagine that you have that the pointer are on a line. If the pointer on a line and you perform this is clear that what you can obtain epsilon. And if the pointer are on a two dimensional surface, you are epsilon square and if you are on volume. Okay, so for the same reasoning, you say this is an easy to make is a dimension. Okay, and you expect that this dimension some relation with the fractal dimension. But actually, it's possible to prove that the to is a lower bound is the dimension is surely smaller or equal to before to go on. Why this method is much a part that this is not exactly this but almost, for example, if you compute this this in for the, for the, for the animal map, instead of 1.26 say you take 1.24. It's almost impossible to distinguish but this is just a coincidence. Well, apart this, so this method is much powerful than the box counting method because it's not necessary to introduce a partition. You don't you don't introduce any partition and what is important at the end of the story that this dimension is not too large, but the original dimensions that we call x, I, in R to let me call the small D. So, this D here can be also larger, maybe five or six okay, what is important that this number here is not too big. Okay, so if this is large is larger, the bottom the boss county method is practically impossible. Okay, this is just at the computational level, so the computational this was a great advantage for the method, and then the people start to say that it's so important that this is smaller than what the means that is smaller, what is the relevance of the difference between the two and the and so on. Okay. Okay, and this is the story of the next step. The introduction of of the range dimension. And what is called multi fractal description, the multi fractal description of this object has been introduced in around 1984 in Rome and Chicago, and I present the, the Chicago version, just to be kind, because it was in the other group, whatever, and, okay, the, the, the group, the two leader was one was Parisi in Rome and in Chicago was kind of, and the, the, the idea is just posteriori so the approach is the following. In the mathematical term, you have the problem of the characterization of a singular measure. Okay, so mathematically you have this problem you have a measure. Let me go new day new bags, which is singular, singular means that this is, is, is not possible to write in this way, not possible to write in this way where this is a smooth function. And so you want to, you want to characterize how this is singular in which sense is single and so on. Okay, so the area is very simple. No, it's very simple, the area is the following. And you have your system, and you, I present a simplified version so it's possible to do something more sophisticated but so it's not very important. You introduce a partition epsilon regular or you can also also know regular partition but you introduce an epsilon partition, and you have this cell, you numerate the cell. And so this is, for example, the cell I, and you introduce, so you have a resolution epsilon. And you, let me call PI epsilon, the probability to say here. So the integral of this stuff on the cell. So this is lambda lambda I epsilon is the cell of size epsilon center in this part. So if the system is not is is regular. In the case of a regular, if it is irregular in the case regular regular means that you have the new equal raw days. If it is irregular this quantity is just just epsilon to the dimension to the dimension of your system, apart the pre factor 234 and that's it. Maybe there is a pre factor he had you do the same but the scaling is epsilon. So this is regular. Okay, is regular and you have always the same day, and this is the treaty. Now, the problem is that to, to put in evidence that they, the regular, the singularity, and the idea is the following and to perform a sort of a partition function. The terminology is actually the statistical mechanics argument. It's not a coincidence that the Parisian codon of some knowledge of this. And so the, the z, z of z epsilon q. Okay. They find this way, if I had some, I, of P, I, epsilon q, where q is not necessary integer. So, and then you look at the scaling of this prop scaling of this quantity for small laptop. And this is the definition of this epsilon go to q minus one dq you define, you define this number, you define this number. And then you look at the range dimension range dimension is no no Alfred Rennie was a great Hungarian mathematician. I guess he never introduced this stuff about so in honor of all these are cool. He introduced something similar for information to you but the idea is the same. Now, these are the, the, the, these general. It's, it's, it's simple to see that if I take q equals zero. If I take q equal zero. Okay, this quantity is zero is this empty or one is not empty. So if I take q equal zero, I have that the deal zero is the fractal dimension. Okay, then okay it's possible to prove that don't worry so it's possible that they take q equal to. The same to two is the grass burger dimension, the grass burger procachio dimension. Okay. And so it's possible to prove that this quantity this quantity must be a non increasing quantity. This is just a standard inequality in probability of it's a convexity argument. Okay, this the q is a monotonic. And so, now, what, what, what happened that the, the shape of the q is important to understand the, how, how this structure, how this fractal is trivial or not. Okay. Imagine that you have a, a, a, a, a simple fractal simple means that is the, the, the, the, the density of the point are homogeneous on the factor. Okay. So in the, in the homogeneous factor. In the homogeneous factor you have the view q is equal is equal to the fractal dimension. So this is a flat stuff. Okay. Something like a uniformly distributed in some sense, the fact that the if this is not true this means that you have a hierarchical you have a homogeneity on the, on the, on the, on the measure. So you see that we start to define a fractal as an object, but in mathematical term the fractal is not an object is something related to a measure. Okay, so in order to characterize in an interesting way you have to look in terms of the measure. And then you can wonder with this definition you, you are forced to use the, the partition, don't worry there is a trick to avoid the partition at numerical level is possible to avoid the, the, this is just a definition but so it's possible to, to, to, to introduce a trick to bring the idea of the grass burger Procaccia for generic, for a generic, for a generic. Oh, let me write the trick, the trick is the following so just, just for to be complete. So the idea in generalization of the grass burger Procaccia method you define the, this quantity you, you can sort of define this, this local density. So this way you avoid the, the, you avoid the box counting, and then you perform the, if you compute the moment of the stuff, if you compute the one over N sum of the, and I have, you know, to Q is not difficult to prove that this is K as, as what is Q, the. So, this is the definition and this is the, the practical computational method. Okay, so now you can wonder, so you can wonder, but there are all these dimension are all this dimension at the same level are. conceptually, technically, the same relevance. No, there is one dimension which is the somehow the, the dimension, the most important, the most important dimension is not the two is not the zero is the one. Why is this one and fact that this one, this is called information dimension. It's associated to the entropy. Okay. Why. So the statement, the important dimension is the one is this one, and this is this is called information dimension. Why. Okay, from this stuff. Let me take Q equal one plus epsilon. Absolutely. Absolutely. I use epsilon for the action to the small. Okay, and then from this, I, I take this stuff. And I, I have that Q minus one, your Q is limit epsilon go to zero, a lot of some PI epsilon to Q, divide by Q. No, the right dog. Okay, this is there. Okay, then let me take Q equal one plus absolutely there. And so I have that PI epsilon to Q is equal to PI epsilon PI epsilon to the. Let me remind the trick. It's a simple trick, but I have to remind the trick. Okay, okay. So, using the, the, the standard trick in statistical mechanics, let me hide this as exponential epsilon to the log of P, which is sure is correct. Apparently stupid, but it's not stupid. And then let me see I use the fact that this is smaller. So you have PI one plus epsilon to the log. Okay, so. So you have that sum of PI Q is equal to some stuff, some PI plus epsilon to the sum PI log PI. This is one definition. One plus epsilon to the sum PI PI log PI. So we are something interesting. Okay, now, I have to take the logarithm of the stuff. The logarithm of this, I have to take the logarithm of this, but the logarithm of this by definition, so the logarithm one plus something small. Okay, so you have that the log of some PI one plus epsilon tilde is equal to one. And then I have to add to this stuff to epsilon tilde. Some of PI log PI and I can go on and so if you want something more, but at this step at this level is enough. So I have this. And then, and then I am. I am, I am okay, because now, now I have here. This is equal to one plus epsilon tilde. Okay, so this, and so this is epsilon tilde. The one plus epsilon tilde equal to what to limit it to zero. Oh, yeah, I have. I have epsilon tilde. Some I PI log PI over log epsilon. Okay, and I take the limit epsilon till they go to zero. And they are the two one, I have the definition, a way to define the one is necessary to do this because otherwise. This is defined this way at zero as the sum I PI log PI epsilon log. Okay. Now, if you put here, minus. This is a sort of entropy. This is an entropy. This is an entropy of the, let me put minus here. I have that minus sum of PI epsilon log PI epsilon is minus is minus the one log of epsilon. No, up now. You don't, you know that to see the, the, the, let me write another way. This is the one log one over epsilon or log one over epsilon. The one, let me write this way. So if you like this way, you can appreciate so this is what is this this is this is the entropy. This is the solution epsilon. Yeah, the epsilon. The solution is equal to log one over epsilon to the one. So now you understand why this is the dimension. No, because you have a system informally distributed. You have that this is the log of the volume and the volume is the size to the dimension. Okay, so this is one. This is in the scary unit. Okay. So you see that the one is associated to to an entropy. Okay, and this is possible to prove possible to prove believe me that if you pick a point on the attractor at random with the major natural measure. If you look, look at the ball around this point and you wonder how, how, what is the probability to say they are outskate the probability. You realize that the probability escape as epsilon to the one. So this is the most, the most probable singularity to have in free singularity but at the end of the story this is most. This is the most. So, and this is can be computed because you can can be computed, you know, using this formula here you repeat the game and you take the logarithm blah, blah, blah. So, using the. Okay, with the same idea you perform the average of the log of an you can compute this stuff. So, so this can be computed. It's a bit longer. Not only can be computed but if you go on here in the if you go on in the computation, considering also the next order, what, what you have. If you go on, you have that the of q or q around one is equal to the one minus something, let me call sigma, sigma square, minus one. So I am considering the correction to this one. And this sigma one is really a fluctuation of something. Okay. And so you see that is true that the view q is something like that because he's an on decreasing functions on something like that. Okay, but what is really important is, is this value. And this look, so this look, which is given by this sigma must be negative because this will give you an idea how you know more genius is the factor. So if this is more you are in. If this is in almost almost homogeneous we have something like that. Something like this is very, very flat. Okay, so this is, so now I promise you that there is, there is a thermodynamic interpretation of all this stuff. The one question so so there are other ways in which you could define a dimension say for example by looking, you could define a dimension. Okay, so for example there is a spectral dimension. So if you look at the spectrum dimension is another story the spectrum. No, no, no, no, no, the spectrum dimension is another story because this is this is about the static stuff. The fractal dimension is associated with the diffusion property is a concept is different because all the stuff you have to work in a finite domain. Why the fractal dimension is associated to the diffusion property which is an open space and open domain. No, there is no, in general, there is no, no, no, no, no, as far as I know there is no connection. Yeah, it's true that also in different case that are different dimension but here the infinite dimension. Okay, it continues better. So now they, I promise you they interpret the thermodynamic interpretation of this stuff. The potential interpretation is a. I never, I never discussed with the, with the Kaldanov and the Parisi, how they arrive to this, now it's impossible to discuss with Kaldanov because he is no more with us, but also with Parisi is not simple to speak. And I saw this is my personal reconstruction of the origin of the idea is personal reconstruction. I don't know if they had this idea or not my personal reconstruction is the following. Imagine that you have a day standard. You compute the system with n particle. You put the partition function is something like that. You have to compute this. Okay. So, and the standard trick is instead to compute this stuff you connect with the micro canonical property. So you define omega. Okay, you have n go back very large. So, you, so you, you, you expect that energy, you write the energy as n times the energy for particle, and you write that the and the micro canonical entropy right as n time the micro canonical entropy for particle something like that. Okay. And so you write this, you write this way. This is just a change of variable introducing the density of states. Okay, so you remember this is, this is exponential of, okay, forget the, forget the Boltzmann constant just just for simplicity. Okay, so this is n s. Yeah, this stuff. Okay, so you, so this is exponential and s, x, and here you have minor beta. Okay, this stuff. Okay, but this is just a shift shift the problem. Okay, so of course it is not easy to compute but imagine that you. So if you do this, then, so then, okay, this is the like exponential of minus beta and F. This is the free energy for particle. Okay. Immediately realize that, no, let me put it here. Let me put here. Okay, so. Check. So you see that the these you compare this with this. So if you are interested in the limit, very large n, you can perform a Laplace methods and plus the last computation, and what you have you have that that you have that the beta F of t is the minimum on a of beta. Minus minus. Okay, so you see that you can, you can write the, the, the, the free energy in terms of religion transform of. Microphone. Now. Now know what is important here is the entropy, the entropy is the counter counter how many states you have such. Now repeat the game. Yeah. Now, if you were to repeat the game, using this district. So in this week, so you put together all the, the part of the face space such that you have a certain energy. You count how many you have. Now the idea is the following I, I look at the, at the point such that you have that this go as absolute alpha. This is some sense is similar that you look at the region where this you have a certain energy. And the analogous of this, you have to count how many boxes you have, you have such that you have alpha. So I count how many box or size x, you know, present the scaling. So, and these. This is water. So here this is an entropy. The, this is, is associated. So if I count how many box such that this I am introducing a fractal dimension. This means that they have epsilon to mine something. This something is called f of alpha. Okay, it's not this effort. Sorry. Chicago they call it. Okay. And so I introduce. So this is not in some sense you see that alpha is some some our analogous of energy. And the f of alpha is the analogous of entropy. And the, the fact that and go to infinite is the analogous of epsilon to zero, which are low to performance at the point. Okay, this is the analogy. The exotic correspond to small epsilon and so with this stuff. So you have that now you repeat the game. Yeah, this is integral of what of epsilon to q alpha minus f of alpha. The alpha. Okay. A person decoration. Okay, so now you repeat the game. So you have that you are interested in the limit of small epsilon. So in the little small laps, you know, this means that the dominant is given up by the minimum. So you have this epsilon go to minimum on alpha q alpha minus f of alpha. But this by definition, this by definition is q minus one. You have this the q minus the q is equal to one q minus one, the minimum on alpha alpha q minus f of alpha, which is in a general form. You see that this is the perfect parallelism of this. So this is the reason why the mathematician called this thermodynamic formalism. It's the thermodynamic form because at the formal level. So you have that q is some sense in analogous of the temperature of beta. No, and, and so on. So you see that here, when you, you change temperature, you, you perform an inspection of the phase space, if beta is very large you don't see nothing so when when you decrease when you take this, you know, if beta is smaller, you don't see nothing. When you change the temperature at the value in the temperature you you explore different. So here is the same. So at the value in q, you are able to explore the singularity. So if you want to see the very strong singularity, which correspond to small value of q to small value of alpha, you have you have to, you have to look at that very large q, very large is q give you an idea of the terrible event. Okay, this is the idea. So, of course, this is f of alpha cannot be a completely arbitrary function as as the entropy is not an arbitrary function. So you know the entry must be convex for some reason, somewhere, even here for some very good reason, F or must be convex, the derivative must be negative. And so down, let me just, so the F of alpha must have some, some property, F alpha must be, okay, let me, if this is F or must be something like that. Okay. F of alpha, F of alpha must be surely smaller or equal than alpha, and that there exists a point. Alpha star. There exists a point alpha star such that F of alpha star is equal F star, and this is, this is the, the, the, the important dimension. And around, around this part, F of alpha, F of alpha, behave like F of alpha star plus minus minus minus alpha minus alpha star square over two sigma square, which is the, the, the, the, the sigma square I introduced before. So, okay, you have a, you have a, a, a parabolic shape, a parabolic shape of the Gaussian approximation, translating the proper way. And, okay, you have all these must be convex, must be convex, F, second must be negative. Okay, of course, the maximum, the maximum, the maximum is given by the fractal dimension. The maximum is given by, by the fractal dimension. So this stuff can be measured. No, so simple, but true. And then. Okay, so your, your, your, your, your, your, your attractor and you perform all the stuff. So then this for the, this for, for the, for, for the attractor. So then you can wonder, okay, but this, the attractor of the law and system. We are physicists who care or the attractor or the system. So it's possible to have some property using this kind of formula in real physics. The answer is yes. So in turbulence, of course, I have no time to introduce you the problem of turbulence but believe me in turbulence you have in turbulence. What is the singular, what is the singular measure of turbulence in turbulence what is important for, for some good reason. It's called energy dissipation. So apart some decoration so you, you look at the, you look at the epsilon, the, the, the, I am not sure of the, the, the, but something like that. Some more. ij derivative ij plus square. So this is, this is by definition is a positive quantity. Let me call this epsilon. No. Okay. This energy dissipation. And especially in the point x, of course, must be a non-negative function. And so you can, you can, you can take, you can take as, as, as, as, as measure, this is the measure. Okay, this is the, this is the measure. This is the new, new of x, a part, a part, a constant to normalize this one. Okay. And so what you have that the structure of this is this stuff, which is why there are some very good reason to study this. Andrei Nicolai should come over to explain us that this is important stuff. And if you look at the, the stuff you, you, you, you will serve, you will serve. This story is called multi-fractal. Multi-fractal, why? Because it's not, because there are different fractals embedded in super-posted in, for, for, to explain that. So I have a question. So really interpret this construction in this way, in the sense that it's like you have, say, different fractals with different singular, so the different singularity are really placed. So in the sense that if you take a point. No. Then you ask yourself with, in some sense, but not, I can give you a trivial situation, but the, actually this is much more complicated. Okay. So the trigger solution is, as you are saying, okay, but for example, the simple situation is that what you call defractal. So imagine that they have that the tractor is a line, the point that distributed on a line, and then are also distributed on, on, on, on, on the surface. Okay. Then if I take here, I observe that the p scale like epsilon. If I take here, I have that p scale like epsilon square. Okay, so if I perform this, this computation, what you said by yourself, this or this depends on the value of, of, I have something like that and the, the, the IQ in the latest way, which is a bit stupid. Okay. So this is just to keep to have an intuition. So you have to imagine we have a super position infinite this stuff, such a way you have, you have this molecule. I don't know if this was fine. So this is just to develop intuition. So this is a big factor. So yeah, this and this. So the question is, is the converse now, whenever you find the fractal multi fractal structure, can you always interpret it as local singular. Yeah, in some sense, yes, but not not not not not too much because this is considered this as a probabilistic interpretation. So because if you look, if you wonder what happened if you look at a certain point with probability one you obtained the one. That's it. No, sorry. If you try to obtain something different from one is possible because zero mission. You have to consider in this way just perform me an average and so on if you just try to take a point and consider the quantity or the point you have in the in a certain ball. There is a general theory and say this you take the one that's it. This is the reason why the one is considered a day dimension, the dimension because if you take a point at random, at random in the sense of you take the one. You take something different because you perform an average, and you take the average and the average. You are waiting at the point in different way because there are these two, this is the reason. So in some sense that the correct that the correct average is when Q is close to one which correspond to entropy. So this is for me, my point is very nice because it is a perfect correspondence between the statistical mechanics formalities and the stuff. I understand and say, is there a way to tilt the probability such in such a way that you can take that you can sample rare points, whichever. You sample their point this way you sample their point this way so you know that the performing the study with different you do realize that there are these rare points my rare point you cannot serve a directive because if you try to serve a rare point systematically you take a typical point. This is the trick. Okay, I guess we can stop and so tomorrow I finish the. Tomorrow is the last day not part. Yes, tomorrow is the last lecture. Okay. So no, okay. I have to discuss before before to pass to change argument. I have to discuss the connection between the dimension and the opponent because apparently say, I have the opponent from the side and dimension on the other and apparently are to this disconnected chapter of the book. And no, it's not true. There is a connection. This is very important. Let me finish the problem is the following. If you. If you try to compute the dimension. Don't worry which dimension and because that's better dimension care. Of course, the method as a limit. But when the dimension is large, it's not possible to use them at large means the 45 or six. Why, because you are trying to, to feel a dimensional space. If you try to feel a dimensional space you need an astonishing number of points. Okay, so imagine that you want to feel a segment segment with 100 points enough. And then you need 10 to four in three to 10 to six and so on. If you want to feel a system with dimension 10 with the same precision use you need the 10 to 20, which is out of your possibility. So you can wonder how it's possible to compute the fractal dimension of an attractor in a system with the dimension is around 20 is not possible to do with this method. It's clear it's not possible to realize immediately, maybe to 10 is possible but 20 for sure it's not possible. So but there are a lot of system with the whose dimension is large is not two or three maybe is 100. So how I can compute the fractal dimension of the system with the one on the first point is why why you want to compute the fractal dimension of the system which is 100 why why is important for example in two bullets. So in a atmospheric science. So the reason is the following the dimension, a part of the integer part, the dimension attractor give you an idea of how many relevant variable you need. In principle, so imagine that you are a system with the 10 to five variable, but the dimension is only 22. Okay, then the conclusion is that they may be in the future some bart and mathematician will give us the proper 22 variable between 23. Okay, no, this is the idea. And this is the reason why so defective dimension is given by the fractal dimension attractor not by the dimension of the, the nominal dimension, but how you can compute not with this method with this method is impossible. Unfortunately, there is a formula that actually there is no proof that is only conjecture that put the relation between the upon exponent and the fractal dimension. The fact that the upon exponent that there is no problem to computational level if you want to compute the fact that the upon exponent of a system with 2000 variable, you can do. When I, when I was young also with the Stefan roof, we compute the system like system with the hundred and under the variable, and this was almost 40 years ago. It is, it is absolutely straightforward. So the upon exponent are not difficult to compute all the upon exponent you want, even some 1000. Now the point is, knowing the upon exponent is possible to understand something about the dimension at the level of theorem. No, but at the level of acceptable argument. There is a conjecture is called Capra and York conjecture which put relation between these, these two fields, and this is the reason why the upon exponent are important, not who carries the upon exponent is 1.2 or 1.3. Okay, it's important to have this one or 10. Okay, what is the relevance of the other the second and so on. Okay, the mathematically at this and this out close the area and so on. Okay, I would say that the physical level, what important is the first one, but then what's important, all the opponents opponent together to try to understand the, to try to understand the fractal dimension of the, of the attract. I don't approve of this but I guess everybody believe that this is the essence is true. In any case, it is not possible in the case where you can check the independently, it works. And so you hope that is correct. But in any case, this is the only possibility. In fact, the people invent an extra name called couple of York dimension. Assuming that this conjecture is true. Which is tautological. Okay, but everybody believe even great. Even very sophisticated mathematics, even see now I believe that is true. Okay. It's in I believe this rule, must be true. Okay, thank you very much. So we meet the