Good afternoon all of you, sorry we are running little late, but very quickly I would like to give an idea about the course that is being planned, which is titled as solar photovoltaics fundamentals technologies and applications. And this course will run from 12th to 22nd December this year. So, as you can notice that the course on solar photovoltaics is not a normal course, which normally you will find in engineering colleges. So, why there is a need for such a course on photovoltaics. And so, I will quickly go through this few slides which will give an idea. First which I guess everybody knows about today's energy scenario particularly India's energy scenario that whatever conventional energy sources that we use I have limitation in terms of the availability. It has environmental pollution or the negative impact on the environmental and also there is a question of energy security. So, this all three factors which are kind of very well noticed in the last couple of decades. And therefore, lot of governments are putting efforts to bring a non-conventional energy sources in the portfolio and India is no exception. So, therefore, we need to find out alternative sources to the fossil fuel sources like coal, diesel, petrol etcetera. So, therefore, as I said Indian government is also putting lot of efforts to promote solar energy as one of the alternative source the wind energy biomass and the other possibilities. Solar energy is one of the major renewable energy resource that is available to our country particularly. And just to promote the solar energy, the government of India has launched this Jawaharlal Nehru National Solar Mission which was launched last year in January. And the target of this national solar mission is to actually implement about 20,000 megawatts of solar power which includes solar photovoltaics as well as solar thermal in the country by 2022. And it is a central government policy, but many state governments are also coming forward and making their own policies and promoting solar energy. So, for instance, Gujarat government has their own policy and they are very aggressive on that policy. So, there is a plan to have 500 megawatts installed by 2014 by Gujarat itself and there are many other states including Maharashtra, Andhra Pradesh were planning to bring such capacities. So, I feel that by year 2022, 20,000 megawatt will be 20,000 megawatt of solar power will be actually a small number and we will have much more installed capacity in the country. So, what this national solar mission is doing is trying to promote solar energy and various aspects related to solar energy which includes technologies, the feed-in tariff plan, there is the renewable energy per se obligations, the domestic production of the PV as well as thermal technology, the implementation side, the stability issues and how we can reach up to 20,000, what are the various applications, whether this PV technology is to be connected with the grid or it can also be upgrade. So, there are various issues that relates to the solar photovoltaic technology and the ecosystem surrounding it. So, one such issue which I have highlighted here in green is actually the manpower generation and it is estimated that if you want to achieve the targets of Jawaharlal Nehru national solar mission, we must have 100,000 trained people in the country in the coming years, but our own estimate suggests that this 100,000 is a smaller number and we would require something like 300,000 people. So, it clearly indicates that there is a huge shortage of manpower at various level in solar PV technology field and this level may be a technician, may be engineering or even at research managerial, at all levels we actually need lot of manpower and therefore, such a course, such a training program for the teachers is important because teachers then can train several students and we will have significant training done in the solar photovoltaics. Also, the government of India, the ministry of new and renewable energy has funded a big project to IIT Bombay. It is called national center for photovoltaic research and education and as you can see from this chart under this center, we are doing various activities which includes research and research in various areas, silicon cells, new materials, PV modules, systems, power electronics, but we are also focusing on in a characterization, modeling and simulation, but education is a important aspect of our national center and therefore, lot of activities are being done at IIT Bombay to educate and train people in the field of solar photovoltaics. So, here is the some of the programs that we are doing. For example, we are running several courses which is for our B take, M take, MSC and PhD students. We are also preparing the course material, so that other engineering college can make use of it, which will include not only the lecture materials, but also the laboratory creates and teachers training program about which I am talking to you right now and things like that. So, we are also working towards training people and students and teachers and scientists outside the IIT campus. We are also doing distance education program like one which will have on PV and training programs for the industry. So, we are also running lot of short term training programs about 4 to 5 programs we run every year on PV technology, various aspects of PV fundamentals, technologies, power electronics and things like that. So, this course which we will run in December this year will is mainly divided in two parts. The part one will cover the fundamentals and technologies. It will be about 20 hour lectures and it will cover the basics of semiconductor physics, theory of p-n junction, operation of p-n junction, parameters of cell, solar cell design, fabrication of solar cell particularly crystalline silicon solar cell which is the main workhorse of the PV industry, PV module and its variation output in output power is a function of temperature radiation. These are the very fundamental thing that people working in PV area must understand from the technology point of view, but in the second part we will also be looking at the applications for PV technology which will include various power electronics. For example, the grid connected and off grid systems, the charge controller DC to DC converter, inverters we will then look at the maximum power point tracking, energy storage options, solar radiation availability and eventually the design of the whole PV system and various costing related to it. For example, life cycle costing and eventually estimating what is the cost of electricity that you can generate using solar photo tech technology. So, this course is very comprehensive and basically covers fundamental aspects of the PV technology studying from materials, fundamentals, technologies and as well as applications. So, teaching faculty for this will be a professor B. G. Fernandez who is a professor at electrical engineering department and here is the web page you can have more details about his research work and his teaching activities. He will mainly be covering the electronics part of it and to cover the fundamentals and technologies part myself from Chaitan Singh Solanki from department of energy science and engineering. So, you can have more details about my work and my teaching activities from this web page. So, we will be doing two other thing one is we will be giving this book which I wrote about two years back which is titled as solar photovoltaics fundamental technologies and applications. So, we will be giving this book as a reference book, but some other material will also be given where you can use it for the reference purpose for this course. So, other than the lectures the very important thing that we must do is of course, the tutorials and the experiments typically solar field is something where you may have to do the practical experiment and a first hand experiment not done not just a simulation. So, other than delivering the lectures through the A view which you are receiving right now a similar to that we will also have the kind of contents provided for the tutorials, but most importantly we will give we are actually designing the laboratory kits for the people for the teachers and the coordinators centers. So, what we will do is we will actually assemble this laboratory kits at IIT Bombay and then we will be distributing this laboratory kits to those remote centers and the participants at the remote centers will be able to do experiments on their own. So, there will be about 10 experiments and this laboratory kits will be given to the remote centers. So, that will be unique feature that not only the lectures, but you will also be able to do the experiment on solar energy. So, about 8 to 10 experiments will be provided where you can do the experiment in the remote centers. Of course, a long term perspective is not only the and this actually experiment will be more on the characterization, but we will also want people to do the fabrication of solar cells. So, in a later maybe one year down the line we will also make laboratory kits where people can make solar cells themselves. So, but that will come as a later part of it. As it says we are doing this course in December, but we are starting the registration from 15th July itself. So, very soon from now, so that you can register here and take appropriate permissions from your college and the remote centers. All the list of the remote centers are mentioned here, but I think you can also find the details about the remote centers at the Eklavya web page, which you must be aware of already. So, thank you very much for your attention. And I hope that many of the interested faculty members do join this for this course and let others know who are interested in the energy and particularly in the solar energy. This course may be very useful for them and the students for the future. Thank you very much. If there are any questions related to this course, please ask your questions. So, there are some questions for example, what is the maximum efficiency we have achieved in solar cells? So, maximum efficiency is 43 percent, more than 43 percent for the multi-junction solar cells. Can we make DSSR, that is dye sensitized solar, I guess it is dye sensitized solar cell from the plant leaves. Actually, any material that absorbs the solar radiation and can conduct electricity having semiconductor property be able to use for any solar cell applications. So, it has to be absorbing the light, it must be having semiconductor properties and it must be electric conductor of electricity. My question is, I have not taken any energy management subject, but can I attend this workshop and what are the prerequisites for this course? So, this course, I mean energy management is not a prerequisite. Particularly, this course is open to many branches because we will be covering fundamentals, but it would be like preferable to have a kind of background in electrical, electronics, physics. So, basically this course on solar energy is not associated with one particular department. So, depending on the interest, it can be made open to other disciplines, but like more preferably to electrical, electronics, physics, sometimes many mechanical departments across the country and across the world actually run the solar energy related courses. Hello, I have one more question. I am the coordinator for this course in our college. We need teaching assistants familiar with the basics of the topics in this particular course and I am finding it very difficult to find such persons. So, how to handle this? Yeah, hello. Good to see you. I just remembered that you are there in the coordinator's workshop. So, what you are saying is correct that because it is not a common course being taught in the engineering colleges right now. So, it may be difficult, but I think as we can see there are two parts. One is on the semiconductor heavily on semiconductor, the part one and technologies and the part two is more on electronic side. So, you may choose a TA which may have some background in semiconductor and may be TA which may have some background in electronics or power electronic side. So, I guess a combination of the two type of TA would be able to do the job and you can not only student, but you can choose your faculty members to be a TA for this course. Yeah, I have a suggestion. If you can send your book early, then we can probably train some of our students as a teaching assistants. So, a book I just inquired before coming for this lecture. We will be getting book at IIT in about 10 days and then we can send the book to various coordinator's centers. And later we will make arrangement to send the books for the participants directly to the coordinator to the remote centers. Good afternoon sir. I want to know the recent research activities going on at IIT Bombay in the use of nanoparticles in solar cells. Definitely, I can answer that, but if there are questions related to course directly, it would be nice. So, there are a lot of work going on where people are working on like silicon nanocrystals, titanium dioxide, dye sensitize using semiconductors, sensitize and dye sensitize solar cell. Some people are beginning to look at the organic solar cells. So, there are many areas where like plasmonics also the metal nanoparticles as well. So, there are many aspects or various different PV technologies where the nanomaterials are being used. And you are most welcome to come and use the facilities. So, this facility we have in terms of this nano electronic center as well as national solar photovoltaic center. Both of them will have the facilities straight up the all facilities which anybody can come and use it. Thank you very much everybody. I can visit the web page, a club of web page where you will find more details about the course. Thank you.


With our favorite ice cream shops closed summer doesn't feel complete We have a solution Delicious ice cream made with household ingredients by the end not only will you have a yummy treat But you also have learned very interesting chemistry concepts The materials required are Two tablespoons of sugar one cup of half and half you could also use milk or heavy whipping cream instead of the half and half one fourth teaspoon of vanilla extract half cup of salt Any type of salt would work here At least four cups of ice cubes at least one small sealable bag At least one gallon size sealable bag a timer or a clock and you could also use a thermometer and some oven mitts or a small towel Now let's get started with this experiment. Put one tablespoon of sugar half cup of half and half you could also use milk or heavy cream instead of the half and half and One fourth teaspoon of vanilla extract and seal well Now add four cups of ice into one of the large gallon sized bags and Pour half cup of salt into the bag You could also place a thermometer inside the bag to measure the temperature To understand why salt causes the temperature to drop we must be well versed in the molecular structure of ice Water changes from solid which is ice to liquid which is flowing water because of the movement of the water molecules Energy or lack thereof causes the changes to the movement of the water molecules Energy in this case refers to the difference, which is the increase or the decrease in temperature The temperature at which water turns from solid to liquid is called the melting point When salt is put into a bag of ice the salt ions prevent the water molecules from getting into that rigid formation Making it hard for the water to turn back into ice cubes In fact, the salt water may not freeze until minus six Fahrenheit The more salt there is the colder the water will be In technical terms the salt causes something called freezing point depression that causes the bag to remain cold for a longer period speeding up the ice cream making process Now put a cream bag you prepared in step one into the large bag with ice cubes Make sure both the bags are sealed shut Shake the bag for five minutes or until the cream begins to certify in this scenario The cream energy is transferring to the water molecules If it weren't for the salt the ice would probably melt way too fast Which means that the cream wouldn't solidify as quickly Soon the cream would have solidified into yummy ice cream If you'd like to see what happens if salt is not added to the ice, please follow these instructions Now add four cups of ice into the other large gallon sized bag, but this time don't add any salt to it What do you think will happen without the salt? Put another small bag you prepare using the same steps as the steps in step one and Put this in the large bag. Make sure both bags are sealed shut Shake the bags for five minutes as you did before Did you get the same results? Now if you didn't get the results, there is a reason salt We will now answer this question as the ice cream freezes and forms ice crystals The shaking breaks up the crystals into smaller pieces and let's air into the mix So the ice cream becomes more smooth than icy if you did not shake the bag The ice cream would not become a solid In case the ice cream didn't turn out the way you wanted. It's completely okay. Science is all about trial and error Try putting the milk back in the large ice and salt bag and shake a little more to see what happens We've successfully made some amazing ice cream as you take a bite Be reminded that you're trying all three states of matter The solid ice crystals, the liquid milk and the gassy air bubbles within


So I just got done working out Excuse my hair. This is this is what my hair looks like when it's not all dolled up. This tripod is terrible Wait, you like this beard? You want the beard back? So Sarah wants me to bring back the beard. You want the beard back? Hope you guys have been joining the vlogs lately. I've been enjoying posting them And it's just been fun kind of documenting our lives up until the move to Missouri currently It's actually nine days, right? Yes, nine more days until we officially move out of Wisconsin Also for those of you wondering, this is what four months of spin-arts with the same drill looks like Looks pretty dope. I like it Now got to go ship this. Thank you Hey you too. All right, so let's go do some spinner It's always interesting when I bring out the camera to film and vlog people just are like, whoa, whoa huge huge camera Whoa, whoa, whoa, whoa, and I'm like, yeah, man I almost forgot to To change up my wristband on my Apple watch here. So it's not that it's not the newer one But it's the the paint the one boom gonna do some spinner. I got my list I've got about 25 people to make paintings for and so I'm just gonna go down the list Here's my sheet for reference. This first one's a 12 by 16 purples and pinks with a white and black We're gonna do we're gonna do just that to be like That's pretty dope. So up next we're doing one of my favorite combinations One of my favorite paintings to make and that's the Johnny Q galaxy So we're doing a 14 by 18 piece and then we're combining galaxy colors.  So let's get this that right There's a Johnny Q galaxy piece, but it's not done yet. So hold on what I do afterwards is this I grab some white paint That's kind of almost empty.  That's what I do pretty much every day Because business has just been exploding. It's been awesome Okay, thank you all for following the journey on tick tock on Instagram and now YouTube So I'm excited to be making more vlogs more episodes Now I got to go get all this paint off my skin It's gonna just the whole bathroom is gonna be a mess and it's gonna be a whole whole thing


Okay. So thank you, Stefan. So today we will have two hours of lectures, but I will, I mean, I will do one hour and then we take 10 minutes break and then do another hour. So when I was preparing actually that's like for this series of lectures, I was thinking that I will go faster than I'm doing now. So which means that I mean for today's lecture I might not go, I might not go beyond, I might not go to the end of the group of counting these closed geodesics. But I just want to make sure that like, at least what we're doing, like, it is understandable. I don't want to rush too much into getting to the final end of these counting the closed geodesic. Well, as you see last lecture, we were building a specific measure, which is a measure of maximum entropy. Do we have improved it yet? We will just finish the proof today and prove that it is a matter of maximum entropy. And then we will prove that this measure is unique, because you see the vibrational principle it gives you the supremum or you know all event measures so if you find one, the question you might ask whether it is unique. And the proof is, we will see that it's just general proof. I mean, it's an expansion of the general proof by form on using the specificity and shadowing or specification gives uniqueness. So today I will show you how to get specification and expansivity. And then I will, I want to actually discuss the proof of mixing that we did in this paper, which follows by some nice argument, which follows from some nice argument by Babilo for like a negative curvature for instance, when the structure and it uses a very, a very geometric function, which is the closed ratio function. So I want to say for some people who knows a bit about explanation mixing by like the proof of explanation mixing for contact flow by Daniel Leverani, he uses this temporal dispersion function and the fact that it is older, it implies some nice like explanation mixing for the contact and so forth. So today also we see a function that is an analog of that one in this area of geodesic flow, and then the continuity actually of that function will implies mixing. But here we just have continuity for that for that function so yeah, so today we are going to talk about four and five. We will be talking about MMI, like the measure of maximum entropy, and then we will talk about mixing for the MMI, and then I don't know if you will get to the counting, I don't know. Maybe MMI just say few words, or like how to use, because I spent the whole lecture last time to discuss the cross section of this back sense of measures which is key in proving mixing for instance and also in doing this counting. Maybe MMI just, I will just say few words about how the property of the measure is used to prove mixing. Okay, so let me just recall where we were, we have the energy. So, the results that I'm going to prove today like the uniqueness and the mixing, the whole, in our case we were able to prove it for any surface with genius at least two. Yeah, I told you that this genius at least two is just to answer that we have positive topology to entropy because on the tourist, if the, if you have a tourist with that point to get points flat, so do you don't have any entropy there. So, everything that I'm going to talk about today is for surface. Do we have a general results, but that uses some non-trivial assumption, okay. So surface is a point to get points. So if you remember this notation, I just replaced so X is minimal self cover, which is a copy of R2. And remember this notation, this was the idea boundary. It was defined by the equivalence classes of asymptomatic to this grade. Okay. And then, yeah, so what we did is we defined this family. That's, that's essentially one family. And they satisfy some nice properties like this one is going to vary and this family, they are absolutely consistent with respect to each other and the density is given by the good one function. And these measures are full support. And from this page, everybody wants to be fine. D. The new bar on on this space, which is the space of your design by this taking new bar is exponential of H beta P of the data, the new design. The new data. This is exciting. And from this measure, we build a measure now on the internet and longer by using this, the inverse so remember we had this map. So by the way, when I was watching again this lecture I noticed that I was talking about this map, you know this square minus the diagonal. D minus plus. I was talking about the activity and so the activity of this map but what I mean by as x here is this, you know, I portion is the geodesics. Okay, so you use here in relation by saying that B and W are if you want, I'm the same equivalence class, if they belong to the same this was really what I mean, I just take it before with like I'm saying is. Yeah. So, this is what the model is doing. You have to be, you're saying to this point, he might be minus E plus. But for this point also W they declined the same time so I'm pushing the, you know, with a set of geodesics. So, this map was not injective but next time I told you that there is a way to choose a measure of what was for this month. And last time somebody asked a question I promise I was going to think it was about it. So, so this is just from a very abstract result so what you do is you put define an equivalent relation saying that B is equivalent to W. Even only, you know, you have H, B is equivalent to W if and only if. So what I'm going to say is so I should write the picture and then you will see. So, if this happens, if you have two different geodesics so the picture is that we have the whole sphere. They do like this. So, B is equivalent to W if and only if B plus because W plus and B minus because W minus. We just saying that all these geodesics here, all these vectors here, they belong to the same equivalence class. And from there you can define the map Q tilde from Sx to, you know, Sx modulo this equivalence relation, and then you can project this map to the boundary to the to the manifold and then the boundary of the manifold Sx to Sx tilde. And now the, the theorem I did not write, I want to write wrongly the name for instance, like Kura Towsky in. Can you like a little bit bigger Khadim it's a bit. Okay. It's just a very abstract result that uses this. So it tells you that this map will have an impulse as long as the equivalence classes they are compact. And here we have compartments by most of them, like it's bounded they say compact. The equivalence classes are compact. So this theorem. The theorem of this is that there is an impulse, a measurable impulse, impulse for this map and then you can lift it and you will have an impulse of Q tilde. Yeah, so this is just something very abstract. So the way you could define, remember we define this measure me tilde by just taking like a intercept means the inverse of this map, Xi eta, D mu bar of Xi eta, and this is the x squared minus the diagonal. We have some nice properties. Okay, and you project this measure to the, to the money for binding this government and finite and compact sets, so it project down to a finite measure. So, yeah, there were these properties that I told you that this is what gives the fact that this measure has full entropy. You have new P of P R side remember this collection B X row. So raw is big enough. Okay, this is of the order of exponential of minus H distance between X and P. And so this is what we use to prove that to prove that this measure has full entropy. So I will add again that lemma. So new H new it was H top of you want. I mean, it's the measure when you project this. Okay. The measure you get so to prove. So to calculate the entropy of a measure what you do is you take a measurable partition. Let's say we have a, which is anyone. So let's suppose that the diameter of these elements are less than. It is small compared to the injectivity rates of the injectivity rates of this of the manifold. So, and you define this set again. You define this partition by using the dynamics to use your zero to n minus one of feet minus a and an element here alpha is in a and means that alpha is in the intersection of V minus J of the epsilon. So J equals zero to n minus one where V is some, some points in alpha some V. So in elements in this partition is just given by these bone balls like these are a set of vectors. You are in alpha, you stay at a distance epsilon up to time and until time and okay, at least until time and So the entropy here, the entropy of this measure for this system. It is just a limit when m goes to infinity of the entropy of this set, which is the sum of minus mu alpha. Love of mu alpha where alpha is in a and so proving that the entropy is H. So dilemma to prove dilemma it is enough. So let me maybe go to the side. So, to prove that this measure they have full entropy. We have to prove that you prove that new of alpha is less than some constant C exponential of minus H and where H is a topological entropy. Is that stop. Okay, because if you plan that here you see, you get there. Did I do this wrong. So, we just want to prove this plan here. Yes, so we pick an alpha. We want to prove this equality. So let's put everything in the universal car. I'm a big picture because it's going to be so I have the V is a vector says that alpha is in the intersection. So this is V and it has a geodesic. So this is V. And let's take this point. Let's call this point X, which is C V and so remember to know the major of the alpha, you should project it to the you should send these to the boundary and know the major you get here. You know, you should do this. And to do that, let's take another generic vector in alpha so big. W in alpha. That means generic so. W here. So V and W this day at a distance of time. Okay, so that's the definition. Okay, so that's the definition. And so this is Xi, which is that minus. This is that minus. There was a point of that. So let's see that. So yeah, this is that it stays at a distance. I want to draw now this geodesic. So what I want to do is I want to let me say whatever I want to say is if I do the geodesic connecting X, this point X to Xi. This point X to Xi, let's parameterize this geodesic so that this is a point zero. Okay, see Xi X zero is this point. Let's call it Q here. So the first claim is that Q is in a fixed range ball of V. This point is in a fixed range ball. This you can just do it by by by clang inequality. Okay, you can just add W and then, you know, the distance between Q and I of V like the food point of V. Let's say some are not, which is maybe twice the injectivity radius. Okay. So, or also the distance between Q and P is less than zero where P is a point where we did where we did remember when we construct this measure we have this family but we just fix one point and we do the construction so new P this guy. So P is might be different from the V but it's somewhere here. So what I'm saying is that Q is in a fixed range ball that doesn't depend on on end. So the distance between Q and V or key and P doesn't depend on end this by using clang inequality. Okay, so if I project this alpha. So it would be like, I'm projecting here. So, if I take this guy, and I send, if I saw this is what I want to say, let me write. So, this is what I'm saying is the alpha is a subset of the union. So, this is z in P, P. This is our zero z times. So remember this projection of Px. So to know all this all desire. Now it's enough to project X to this to the point in this ball like that set will contains the backward, the backward end of alpha. Okay, because key is in this ball like generally point W this Q that joins X in the geodesic journey exercise isn't isn't this ball. So this guy, they run from this set. And now to know that the other end points is that you just project project is this guy to X and then you have the other end point. So that's what you have here. The fact that this is in a fixed ball here doesn't depend on and is is crucial. So, yeah, once you have this, it's, it's basically, we are basically done. So see that. Now if you use the property, I will have new P, new P of P, B, P, Q, B, Q, B, X, epsilon. This guy is less than by the fact that these measures absolutely continue to inspect each other. This that's between P Q and new P of PR, PX. So he was this point here so if you market you take the other end you have, you know, the point from here to here is just you are changing the base points here. And then, and then this is less than by what we have seen before, this will be less or equals to a constant times exponential of minus H distance between Q and X by this property. And here it's easy to see again that the distance between P and X is like is is is a port of N distance between P and X again by Chang inequality is less than N plus is bigger than N. So if you want the other way around, it's bigger than N minus some, maybe two out of zero. Okay. And you would have that this is a port of N. So this is less than a constant times minus H and this guy is bounded and the measure has a product, product structure. So this is like that me of alpha is less than minus H and a constant.  So, this time. So now, for today, you will move to so this measure has full entropy. We will prove uniqueness here. So actually, uniqueness of enemy. It was used by Bowen that if you have what is called expensive, if the flow is expensive, and you have what is called specification, these two together imply uniqueness. I will say what that is, is imply uniqueness. So, I'm not in Thompson what they did is that they try to relax these properties, like the exclusivity and also the specification, and they will be able to find some nice conditions that you still uniqueness that is this, this argument. So before I let me just define that is expensive. So definition is expensive. Yeah. If this set of point, the set of W. The definition is for general flow, but I'm just giving it here in the, in the, in this case of geodesic flow distance between 50 B, 50 W is less than epsilon 40. So if this set is a subset of, so if this guy is a subset of the geodesic minus s. I'm just telling that if to, I mean, if two vectors, they are the bounded distance absolute. So it's the same. Sorry. Yeah, it's the same journey. Like, you cannot find, if V is expensive at scale, you cannot find another factor that stay in absolute distance over time. That's different from being the same. Expensive set at scale. And the action is set of non expensive vectors at scale. So the second factor that I'm expensive at scale. So for this, I think we could see just some some exercises from last lecture. I mean the second lecture probably. So they were an exercise in the second lecture where we say that if you have negative curvature, this imply that WSSV, we just said WUV is a single point. In particular, I don't think it's as to maybe to check for today would be that if you have this property, then there exists epsilon positive set and E epsilon is anti like the flow is expensive. Yeah, that's easy. And also, remark that the non invertibility of this map that I was talking about, it really is connected to the non-expansivity of the geodesic. Now, E, this map which was so SX under I take explanation between like the geodesics are different classes to minus the diagonal, not invertible. So this implies that you have an excellent zero you have an excellent policies is that this guy is not empty because see this condition. You might have a strict if you have to to two vectors that define the same end points but not belong to the same geodesic. If this map is not in depth in is not invertible. You have, you have that property. Yeah, so this is. Yeah, so what was used by Bowen is that, expansivity is that there exists an excellent zero says that this set is empty. So that's expensive. So, for instance, in negative cover to you have expansivity and not positive curvature. So in particular from this remark is not positive curvature does not imply, you know, expensive, and let alone expensive. Let alone no points. And there's a specification. So specification is just like a shadowing lemon for, you know, for other sort of a mechanism plus so definition. So that was expensive. So given delta positive.  You say F has specification at scale delta positive. And if every give me any pieces of orbit V1 T1. The end, the end, you give pieces of orbits. So these are in sx minus times. So there exists one orbit that shadows these pieces. There's W in sx and times. I will draw a picture for this. Tn positive says that you have 2j plus one minus 2j minus 2j isn't your delta zero. Oh, give me if there exists. I forgot here. There should be if there exists tau positive. So tau is a transition time. There is tau positive. So we will draw a picture for this is that feet. Tj of W is in is in this set of point that stay upon the distance until be in sx distance between us. This is one single orbit is less than the whole s for us from zero to this to the sum of time. So what does it say this is that you give pieces of orbit. So this is V1 and this is V1. This is V21 V1. I have another piece of orbit here. Another guy here. Another guy here. So this is V2 V2 V2 and so on. So Vn Vn and this is VTn Vn. So this is saying that you can find a single value that stay at the boundary distance here. This distance will be delta. This is W and spend some times outside. So this time is tau. This is tau. The time that is spent outside here is tau is the transition time. And then this is VT T capital T2 of W and again stays at a boundary distance to this geodesic and they state time tau and do this again and then come and share this guy. So this is specification. You have one single orbit that shadow this guy and spending a fixed amount of time outside like this transition time is fixed. This is a feature that you have in hyperbolicity like if it has negative curvature, there is a standard truth for this. Okay. It is using, it's just about shadowing. You have zero orbit you shadow in class transitivity to find these times you use transitivity. So this is something you always have in negative curvature, for instance, or the geodesic flow is transitive and another. But there is a problem like if you want to generalize this phone like expensive video and shadowing and specification implies uniqueness. But if you want to allow non expensive video to prove uniqueness, you should, you know, you should expect the non expensive set to have less entropy. So you should expect the non expensive set to have less entropy, because if the non expensive set has the same entropy as a topological entropy, then you can build a measure of maximum entropy there and a measure of maximum entropy in the expensive set, and then you will have to. So to help to get uniqueness, you need what is called an entropy gap, like there's a difference between the entropy in the non expensive part and the one for the food for the whole for the whole geodesic flow. So that brings this definition entropy of traction. Excellent is the supremum of the ventric entropies of measures, you know that I supported in the non expensive set was zero equals one. So the entropy of measures that are supported in the non expensive in the non expensive set. Now the term by term by the monologue and Thompson. This is 2016 is that you fix scale is absolutely positive. Because I mean they are very precise. So absolutely be very big compared to that. And having an entropy gap, like the non expensive set doesn't have more entropy than the full field has specification at scale. Then the measure of maximum entropy is unique and it is supported in the expensive set. So then there exists a unique. This is a very upset with us so they just do the recording the born coding is this type of specification and because usually the way to use specification is that you want specification at every scale, but if you don't have enough of the city like in our case, we cannot have specification up over scale. There's one scale that we stop and we can have specification there because of lack of capital city again, and. Yeah, so to prove uniqueness, we just need to check those conditions. So, for instance, we will first want to. So first of all, specification in this service is a little different. It's, it's kind of easy just rely on more so them. So specification is given by most of them. So you gave me pieces of geodesics like this. What I do, I found, I find that corresponding. There is a I'm in the service. For instance, it I had mentioned repeat this argument we would need to suppose that there is a background of which there's a background metric for the geodesic flow has specification, for instance, and negative curvature is a good one. So this is we know we have a program. So, this metric, this is the zero, this is the gene. This guy being a result, we can have a specification for this guy. I respect the one and then. So this is a geodesic. Now you go back to most to say that there is a geodesic that say at a boundary distance to this one. So you see the distance might not be small. It just depends on the constant you get from most level. So this is up really big. This delta is up really big. Okay, so this is how we get specification scale. But, so, but to ensure that we have epsilon much bigger than 40. That is something we don't have control on, but this absolutely can make it big how I just by just taking a finite cover. Which is large for the injectivity ranges is big. If you take a finite cover of your money for you can ensure that is big but so. So to ensure. So this one is bigger than 40 delta like to be able to choose such as to know what we do is because it actually is very high and your money quality is small. So, you won't have this like they're not expensive say will be everything so I mean I should start with that so. So, that's one is larger than the diameter of the money for is much bigger than the diameter of M. Then, of SM, then the non expensive set epsilon is is a whole thing. The non expensive is a whole thing is actually is much bigger than because everybody will stay at a distance. Okay, but what we can do is we can increase this game so we can take a finite cover of actually large injectivity ranges. So, so, that's actually something that people call the money for being relatively finite. So, so relatively finite it means that the fundamental group you have a, you know, you can do, you have a sequence of subgroup for the fundamental for which the corresponding injectivity ranges is going to infinity. So, basically we use about that and so, which is true in services actually. And it's true that for services and is finite. I mean, we can take a finite cover and is a finite cover of M of actually large, very large inductivity ranges uniqueness for the genetic flow and SN. You have uniqueness here, you will have uniqueness downstairs because this is just, you know, it's, it's a finite cover. It's a finite cover so you don't have to measure that gets multiplied because the orbits, if you leave them finally because one just to find it. Okay, so you don't go down for even though you don't go there's these measures. So, this is the fact we use to be able to make excellent because we are having this problem that if the manifold is small, which it can be, but this guy. The most delta this shadow this specification that you have from doesn't depend on the cover because this is a universal cover. So this is the universal cover. So, we can. So this allow to take epsilon, take epsilon bigger than 4g delta. Actually, now how to get this the gap. This is also two dimensional two dimensional argument in higher dimension. We don't know how to prove this entropy gap. So, to get this, if so if you have a positive entropy here. So, if, by the way, there is this remark that I forgot to say that. So, an excellent. So, see that sx minus and epsilon and e epsilon. This is roughly the set of expensive. Expensive vectors. This is a set of point for which this guy said of the says that that view. Maybe right. You can correspond to be to desire. P is the food point and I is the plus. That HP. I have to be plus. So the expensive set of these vectors. So, this is what I'm saying is that expensive vectors, the dual spheres intersect only at one point. So this is the. Expensive vectors in this picture that you have to join in this, you'll have the whole street intersect like this, and this is not expensive. This is a non expensive. So, if you're expensive, this is what you get. You can only expensive. This is what you get. So, but if, if this guy is positive. So, actually, we are, we are proving resources is we're proving that this has zero entropy is this is positive. So you have this person manifold, you have this person stable and unstable manifold and be in two dimension. I know that this money for the intersect transversely. So you will be in this picture. You have that transverse intersection between table and like between stable and unstable. These are passive manifolds. Okay, when you have a positive topological entropy, you can have this money for the entire dimension, you cannot have this intersection being transverse because you might have two surfaces that intersect just along the line. You might have two surfaces that intersect along line. So, but in this case, we know these are one dimensional. So the intersection is of this picture, which contradicts that this is supported on the set of vectors that are of this form, because if you remember the definition So this is a simple argument that it for services. Yeah, so what you basically have is that if you if you get at least things we have specification given by most. We have this quantity by going to a finite cover with the we make a month for big we have this. We certainly have this and then we have a unique him and me. So this realize so on the existence of a background metric of negative curvature. Other questions on this. Yeah, so, so I would start the proof of mixing actually saw that the geologic flow with this major is mixing. Mixing doesn't use every doesn't use you just use this every city, which you get from the next. The proof of mixing that I will do here is just using is just using the fact that it is everybody can then and then, and then the, the construction of the major by a sense of a mess. I should maybe call the 10 minutes break and then we stand meeting at the top of the second hour. Yeah, so we have a 10 minutes break now and then. Yeah, yeah, yeah, five minutes past the first the next hour we will start to prove off. Yeah, we started from mixing. Sorry, may I ask a question. Could you repeat why the condition on the expensive entropy implies the possibility of relations. So, so you know you have a major that realize this entropy here. Yes, you take a typical point points on that major so for almost so respect that major V in in any epsilon because this measure is supported in any epsilon. You have WSS, the patient manifold, and yeah, this is actually not quite trivial. This lock V and that us you you love V. And these guys, they intersect transversely. This is a patient. These are two curves. They intersect transversely. So what is left to show is that these basic manifold, they correspond with our definition of these are busy money for you, you just need to check that they correspond with the manifold that you get from the whole sphere like the manifold that they define. Clearly, you don't know where these they correspond to, you know, the set I defined as stable and unstable. But that's not very difficult to prove that this guy, that is law V, the stable set that you get from the whole sphere from who's my lunch. If this is the set that you get from who's my function. Then you know by this intersect transversely, because these are just curves. If they were higher dimensional, you cannot know whether they intersect transversely. These are curved and they are defined as a set of points that, you know, they are converging. Like, if you throw them, it's converting in forward time if you're here. But this is not very difficult to see because this is just I see what I'm saying here. More or less. So, so to see that this guy, this person manifold, so this person manifold law of V is just a set of W such that D 50 V 50 W is less than, okay, some cheetah. And then, yeah, so these are the guys that say for all to positive. It's not quite to go to see that these are the manifold you get from who's my function. It kind of uses the divergence property here again, which was key. So, so see that. So, so, yeah, I want to say that the if you take so V is the point that I'm looking at this is being going in like that. So, if, if I take another back to W here. W, let's say W is in this set. W is in W SS lock V. So I have the geodesic that is, you know, staying at the center by definition. But what happened is that, at this point also, there is another geodesic that is going to end point of V. So this is the plus this point out new. There's another point that's going to do the geodesic to do to V plus. Okay. So, so what I'm saying that these two geodesics are the same by the agents property. There's another geodesic that is, and that is the geodesic that you get from the, like the Boozman function, okay, the whole of spheres like. So, this geodesic that is going to this V plus they are bound to distance to this one. And this is also at a boundary distance to this one because this guy does. So, by divergence property, you have that these two are the same. So this uses that business and let me just. So this is basically telling me that if you have that business property, what are called the Boozman assented are the only assented that's that's how they call it was my assented it mean that the geodesic you get from the Boozman function is just this geodesic. That is asymptotic to this one. So, yes, I understand. Yeah, this is a Boozman. And that is property tells you that Boozman assented have only assented like at this point you're going to have another one that converge to that is going to be plus and, and that's different from this one by that. The Boozman assented are the only assented so that's the only assented this is this is actually something don't you know that uses the divergence. Like any geodesic that stay at a boundary distance to another one is given by the boozman is given by the boozman assented these are called boozman assented so boozman assented. So this is a geodesic that I've given by this back to fit gradient of PV and gradient of PV, like if you follow this vector field, you get a geodesic so these are called boozman assented so this back to this geodesic follows this guy is a boozman assented and diverge property that these are the only The only thing that you need to check is that if you used to dimension you have the transfer of the basic money for now you need just to check that the basic money for are exactly the money for even by boozman. And that uses that and take maybe two minutes and. Thank you very much. So, let's get back. So for mixing. So, I mean, this is a definition of mixing that I'm doing. I mean, it's a user definition. It says that, you know, he is a major and is mixing. So that's the property that. The component is. Is converging to zero for all means. I mean zero functions. Yeah, this is equivalent to the user definition of mixing that says that the enemy that says that the manager of 15 a intersect B is converging to. 50 minutes of a meal. So, um, yeah, so I'm going to do you this this function so I will use this version. I'll use this version of mixing. So as I said, the proof it uses a general I mean argument due to Babylon that was done in negative curvature. And it relies highly on the fact that the measure projected to the boundary is a product measure. The measure projected to the body is a product measure which we have from Paterson's 11 measures. And, yeah, and then you have some happy policy. Yeah. So, yeah, you have some happy policy. The one thing I forgot to say actually, in our proof of uniqueness is that the measure we build the measure that the measure of maximum entropy will be supported on the expensive set on the set of points where the stable and unstable whole sphere intersect at exactly one point. So, and if you have the measure is supported on that expensive set, you have some type of hyperbolicity, even though it's not like uniform or exponential, like you can have this whole sphere contracting exponentially. Like for a time, but you know that the limit is going to zero if you have the, if you have a back to the expensive sets. So this allows us the fact that the measure is supported on that kind of good set where you have a certain hyperbolicity. It allows us to, to kind of do the arguments of Babylon that I'm going to say here. So, there is this abstract lemma. This is very abstract that is used here. So, see that this is just, this is just saying that this function is converging in the weak style to. So, this is abstract lemma is that it says that if you have this function fee. So, that's not converging. So, that's not converging. That's not converging with L2 in the, in the week L2. This is the conveyance week out to then, then there exists a non constant function, constant function. In L2, this is that fee composed with fee Tn is converging to xi in fee composed with fee minus Tn. This is also converging to xi. This is L2 convergence and for some subsequent Tn is a different sequence. So, this is very abstract, this is what we are going to use to prove mixing basically. So, see that if this convergence was point wise, you would expect this function to be constant on stable leaves and so constant on unstable leaves because if this convergence was almost surely. So, you have, if you take two points in the same stable, you would, if you use the fact that stable is contracting. So, that is this part actually. That's that there exists. A bar full measure, full measure, me is a measure we construct. This is that if subsequent W isn't W, SSB, you have the distance between Tn and B as in constant. So, let's just say that if I'm in the in the expensive set, the set where you have, where you have, you know, this model spirit intersect like this. You can choose a sequence of time for you to have this and this is a trigger for if the coverage is negative for instance in the annals okay we have this always and it goes like this. So, I wrote this fact because I was saying something that was going to use this fact that if this convergence was almost surely surely, and this sequence coincide with this one, you would, you would get that he will be constant. And that will be constant in a stable set and also be constant on unstable set. And that's something. So to get that's something useful and to get the almost convergence you use a kind of the chase algorithmic so you, you, if you have this convergence out to you can define this, you know, you can see that this function. You can see that this function converge to that almost surely and plus or minus actually plus or minus almost surely. So, this implies that for instance, if I am in this a, so this is why that is the W belongs to W SS. Then, this is what I said so, and then a similar thing called if you are also in the same unstable so it's done. They won't be used for the one day minus one. Yeah, so, so see that me to do this group we do it by contradiction, we support that. It doesn't converge we have to. So, there's this lemma by a popular that is this function that is not constant, the fact that it's not constant is beautiful. I will say why, you know, it's a non-constant function, which is constant in a full set like if the V, or I forgot to say, if the. For this is true for me almost everything. So, for being a, okay. For all being a something this whole for all being a. You know, there are some technicalities with this subsequence, but here the fact that we can have the subsequent that works for all be that's something good. But you want the subsequent sequence. The subsequent sequence you get from barbillos here, but that's also something you can do because that actually the demo tells you that for any subsequent you have a further subsequent for any sequence actually you have a further subsequent where you do this so there's a way to this, this, this sequence is I'm having but just to, to not be very technical, I'm assuming that I have the same, the same sequence, okay. But there's some small technicalities that that can be fixed that. So, now what we do is we take a lift. So, like, so we suppose that we need to feed. Tell that I write it here again be a lift the universal cover. And I consider this map here. The multi. This is something crucial that gives feet. So I like it as a saying at feet TV. So, okay, I should, I should do something. I should be very explicit here. So, so if you give given to me. So, I'm, I'm now being in this full measure set where I have kind of this type of policy given being a, I can define this map. This is a fact. So now I do this. So this is defined from a to the subgroup. It might be brutal but we'll see what I'm saying. The subgroup that takes you give me V. It gives periods of, of this mark of, of this mark of T is xi equals this TV. Let me spend some, some minutes in this month. So if you take B, you can look at the periods of this month. T gives this map. Okay. You look at the periods of this month. It's a subgroup of half. So you have this map that's defined from a. So think of, okay, this is just a negative curvature. You would take the full unit and a bundle, but I'm just, I just, I just want this type of policy. So I have to reduce to be say A. This is a subgroup of R. V of V is a subgroup of R. The set of periods because it is added and it's a subgroup of R. There is some technicalities that I'm avoiding here that, you know, this map might not be nice in the flow direction. It might not be continuous in the flow direction because this will just help to function. But up to, you know, making it nice in the flow direction by integrating, you know, you could replace, you could replace. You could replace V by this guy, the integral of some epsilon F. And this will be nice in the flow direction just to make this subgroup work nicely. So yeah, so you suppose that this map is nice in the flow direction. So few of these are subgroup of R. Okay. So this map is invariant by the geodesic flow, obviously, so she is easy to see that V of V, it was free of TV, because it belong to the same word, so they have the same previous. V. So he's invariant in the geodesic flow. This is invariant. So by a good city, this is the point where we use a good city of the major meal. So by a good city fees constant for almost every or me almost every week. So me. So basically I did not talk about it, but it follows from uniqueness. Okay. Me being everybody in life. He is constant for me almost every day. So there is one subgroup that works for all fee for almost everything. So in particular, you can write fee at V equals easy. Okay. At the questions about these markets. And maybe skipping from start. Now this is not the time we will introduce the to get the contradiction we need to use this course ratio function that I was talking about in the beginning, which is similar to what was used by Caravaggio, liberally improving exponential mixing to get a contradiction from that. So, I mean, I'm defining this function inside the proof, but so I would say so, so close ratio function. We'll come back to the proof I've made you find this close ratio. So this is a function that is defined in the boundary. So you take four points in the boundary gives a gives a number. So what is it doing, you take this. There are there are the definition of the process of function, but it is equivalent to this one that in the same.  Eat up. So this guy is joining this to this is doing this to this is here and one here. So what is the cross ratio. So you start with. So there is a, I mean, I will say a problem, but let's suppose that side. These are chosen in the set of expensive factors. Okay, like they are. You don't have two geodesics joining these guys. Like, these are the only two days. It's not necessarily, but just for the future. Let's do that. So if you take this whole sphere, the whole spirit is take start with any point here.  And you are here, you take now the whole sphere, going through this point. And you're here, you're taking a whole sphere through that point. And this distance you get is the process of function. This is V. If this is V, this is. Fee of the cross ratio. So this distance here. Is. So this is just a lot of if you're familiar with the temporal distortion function, which you go stable unstable stable unstable and you measure the gap you have in the fluid direction. Exactly what this is doing. So I was stable, unstable, stable, unstable, and then I measure the gap. So, so I live this for this part now so I said that these geodesics they belong to the set a, okay, this geodesic they belong to the expensive saves. And then, so the function, if I take V here. But I'm just going to say is that the cross ratio, this guy is a, is a time. Is a period of this map of TV, where V is this guy. Because of this. So, the function five is the same from this vector to this one. So let's say this is V, this is V one. And you go this is V two.   So, by this thing here, the fact that Xi is constant. So I have side. So why, why is it that is I, V one V, because I, if you want, because V and V one they are in the same. Okay, this is unstable. I say I was saying, but this is unstable. So V and V one, they belong to the same unstable. So you use us to say that is the same. And then it is the same or so in V two, because they belong to the same stable. So this is Xi and V two. And it's the same at the three or so. Which is the same as Xi before, but before is what before is just the procession. I'm writing this guys.  And, and yeah, so this guy is continuous that this function is continuous. Why is it continuous because I said last time that these forest fields, they depend continuously on Xi. So this guy is continuous. It's continuous. Because I said if I spend this function be, be, be, be, or if you want Xi gives the team that is continuous. You use the continuity of the forest fields, which in a group of two we know the stable for the Asian is holder. So here we just have continuity and that's enough. So how do we use that is that now if I take Xi Xi, this guy is zero. The procession of this guy is zero, but I can change this a little bit, change this guy a little bit this guy a little bit, or just change this to a little bit and I create something that is absolutely a really small. If you take. So this is why that, you know, you can find that this is again using the fact that the major is a product major in the major is not a product major. You cannot, you cannot be changing these to, you know, you can be changing any video. So you find that I'm actually small by, you know, this. So which content is this guy because there's a period. Again, oh, sorry, I forgot to say that a is not zero. Actually, a is not zero. Why, because this was not this map was not constant. Actually, yeah, this was crucial. Why we need this much money is a is not zero. Okay. And it is different from zero because the from the bubble is that my side is not constant. So I create a period which is actually small. And this is the previous artist. So that's a contradiction. So it's, it's just using the continuity of, of this cross relation function and the fact that you have certain have a policy. This guy is certain have a policy and also the fact that me is a product manager to be able to to this guy and if you do. So these are the people but it's all. So these are if you want to remember what is used here is that me is product. This is used to be able to put up this guy individually. So have a policy. I mean, some kind of happy policy is very happy policy on the support of me, which is the set of expensive factors. And then continually, and then continuity of this function, continuity of this function, this question. And every city of course, yeah, because every city was used to say that is exactly one. And the questions about this. So, so I'm going to show you like some steps at least to do certain counting of closed geodesic, some steps you need the steps where you use, where you use this guys like the product measure and mixing. Yeah. So, let's go. So this is about counting. I have an hour which is given very sketch sketchy proof of this. So we are in this energy is a surface. Remember drama was a group of eyes on this. As a matter is, yes, which is it. To do this counting. So, let's say what we want to do is we want to count the number of so problem number to this counting we're going to concentrate the problem, the problem of, you know, we want to identify close geodesics. We use elements of the fundamental group, like elements of it as a metric group. So there is this lemma which is not actually difficult to prove that for every gamma gamma. So this is the geodesic C gamma C equals C, like this is left in vain by the azimuth element by gamma. And in particular, in particular, you have that if you project C C project to a close or to a projects to a close geodesic to a close geodesic. The proof of this, I can just give you the sketch. It's not very complicated. So proof of this is, you know, given gamma, you can define this quantity L equals the infinitum of distance between P and gamma P, where P is in X. It's not easy to prove that this supremum is achieved. So there exists P0 since that L is the distance between P0 and gamma P0. So this you can write as an exercise. So I don't do everything. So as an exercise, this is the fact that this portion is a compact portion. You can choose elements of the fundamental group to bring everything, every point that is going to escape and bring it to a compact set and combine anything on there. So the geodesic joining the geodesic, let's call it C P0 gamma P0, the geodesic joining P0 to gamma P0 is invaded by gamma satisfies gamma C P0 gamma P0 equals C. P0 gamma P0. Why is this fact is just the fact that it just uses the fact that it was an argument. So I have P0 and I have gamma at P0. Okay. So if this geodesic, so this is C, so let's go and see this. So this claim tells you that gamma marks this geodesic in the same geodesic. If that wasn't true, let's say gamma marks this geodesic one as a geodesic. So this is gamma C. If you take a generic point here, T, C, T, and a generic point here, gamma C, T, like the image of that point by gamma. You can see that the distance between C and gamma C T by just using the quality you can see that. And the fact that so this is strictly less than the distance between C T and gamma P0 plus the distance between gamma P0 and gamma C T. Yeah, this is great. And this guy, this one here is L minus, this is L, this is L minus two zero, L minus two. Okay, and this one here, because gamma is an isometric. So this guy is just, it's not the distance between C T and so this is T plus T. And this guy is strictly less than, strictly less than L, which is contrary to the fact that L isn't true. So the picture is that C is marked GZ, is that right. So already given any elements of the isometric group, there is a gamma, there is an X, an X for this element. And so, yes, and then there is this quantity, actually L, the size of gamma is defined as that L is a multiple, you can see that is a multiple of, is a multiple of the period of the length of the exposure, is a multiple, the length. So finding, okay, but this is just general. Now, how we do is now we, we fix a set, but now that we have actually to come, first of all, we define a set, a set, you know, a nice set with some property. I'm sorry, I always forget to look at this camera. There's another camera there, but this is the one I should be looking at. So, we build a set, which now we are going to come the number of closed geodesics in this set, in going through this set and actually after that there is another element to, you know, to come to total geodesics on the, on the, on the whole unit in Bangal. So, there is this set that we define like this. So there is. So, this is Mark, which is called the, the hub mark, which is defined from V from SX, sorry, to the boundary squared minus the diagonal. So, I'm just going to build a nice set. This is using the argument of the Margules argument in counting, which is, he has this detector, this nice set where you have, you know, created by stable and unstable nicely and then you can count in that you can do the counting in that set. But here, we, we, we build that set just using, using this, this map. So, to R, you give me V, it gives V minus V plus and V, V. So, P was fixed, okay. P is fixed, okay. P is fixed, BP. So, this is just, you take a vector. So, this is, you take a vector, you send it to these guys. So, this is V. I have P somewhere here. P is here. At V minus, I know P was going here. So, this is V. This is, this is V minus and this quantity is just this guy here. So, this guy is BP. Now, you can build. So, you have this, you build this guy, V for some theta is, so before that. So, remember, I pick, I pick P. So, I also pick V zero in SX. So, this guy to be an extensive factor. That's a technicality. We will, we will do that and then see that we can define the set, the R, the theta. Oh, sorry, before that. So, you define the set F theta, which is a set of W classes that the angle between V zero and W is less than theta. The integral feature for this set and PT is the set of W minus less than theta. So, this is what, so I have here V zero. I take an angle theta in a project then. So, this set is F theta. So, this is theta. This is F theta and I do the same and this is V theta. Now, the set I want to build, I want to build a set around this vector. So, V theta is just defined to be, I'm just, I'm just trying to think about how much details I should say. Now, and then some epsilon here. This is different. Theta and theta and zero. So, so this is a set that is around here. These are vectors that they send to this set and they project to this set at the boundary and then you take epsilon along the flow direction. Okay, it's just like you have something here, and you do, you do excellent thinking of these guys. This is epsilon. Okay, this is what this is doing. And now what you do is you look at, so being closed. It's, you know, if theta, if C such that C is geodesic, geodesic, such that C at zero is in V theta epsilon. And this is like that there is an element. So it's a close. So there is an element such that is that if you do this, if you do phi minus gamma applied to gamma star of C prime zero. This guy is again so by this is again C prime zero isn't the data epsilon. What I'm trying to say is that if you're capturing close geodesic that I'm going through be, it's like doing taking elements of the isometric for which this set is, is, is kind of non empty. So I'm just trying to motivate the definition I'm going to write now, which is this gamma T. So gamma in gamma says that P minus T intersect P minus T. So be theta excellent, just like P minus T gamma star. Be theta excellent. This guy is not empty.  So these are the said that if you, if you take the, this is the epsilon. You apply gamma, it gives again is gamma star B. And then you pull it back by the dynamics is going to be something like this. So it's kind of a recurrence like this orbit will be kind of recurrence to this to this set, you will have an orbit that is recurrent to this set. So the one step is so I should really maybe not go very deep into all the technologies here, but one step is, is that this set is in bijection. This is the set of so the T is a one to one correspondent means the set of closed geodesic will be will be be the epsilon of length. I mean, this T plus or minus some maximum. Okay. And what I was just saying is one way like this part is just what I was trying to say here, but the other part is a closing lemma. So to prove this by this injection in this set. This is the index to this set is injection is a closing lemma that is not very difficult to prove that I did not talk about the group. And yeah, and once you have this, this bijection, you, you, if you want to count the close geodesic through this set, you just count this number of guys. And you can see that the major meal of the data, I'm not writing all the details intersect with. So, because this set actually he will write, you can write it as a subset of the union of gamma and gamma of the data intersect. So the measure of this set is like the cardinality time measure of one of these guys. And it's not difficult to say that, like, this guy they have roughly the same measure so the measure of this guy. The part of the measure. Excellent. Excellent. This guy is roughly the cardinality of gamma T times the measure of these guys. And then you, you can use mixing to prove that this guy is converging so this is mixing. This guy is going to measure of the two types, you know, squared. Okay, roughly. And here, that's another thing of the team. Here you use the construction of the man, but that is given by this, but some new measures in particular the product structure to say that this measure is roughly exponential of minus ht times the measure of the two types. So, this part is using products factor like that. Yes. And from here you can, you can see that this cardinality now will go like I'm sorry I'm being very, you know, very precise here but there's no time I just wanted to tell you the two points, the two points mixing. But central one measures they are using these two steps in counting here. And then this is like the ht times the measure of the data. Yeah, and then there is a way to generalize so if I want to summarize everything that I have been saying here we prove we prove. This is two terms in two different papers. So these are joining work is. Clemenaga. You can have people in myself that says that you have energy surface that could get points of Jesus at least two. And the geodesic flow is has a unique MME unique MME meal and that the flow is mixing. Actually, this is better only actually. Yeah. You can prove that it's barely. I mean, the very property from from the construction of from the from the Clemenaga Thompson result, and that is more like we have this and new is fully supported. But you can see it easily by the fact that the person to measure definitely supported on the idea of boundary. That's what we've been like. Well, that's right. And then I saw that me is obtained, can be obtained. And the long clause long clause long clause geodesic. This is not very clear, but I'm just going to write the full term here. And the other result is that in another paper. These are two different. Same. Is that you have in the same setting surfaces like the HD over HD where PT is a set for the classes. And geodesic geodesic because we know it's one geodesic. Thanks. The close to geodesic. At most. So this is known as the, the margulis estimate my duty for typically mixing and also flaws and then there are many results like it's again proved by it's also proved. In the case of one cat zero spaces by. And also there's results for actually flows for instance by. But and, and, and very for actually flow. So there have been many results on this, but yeah. So there are different type of estimate that was studied in the literature, not exactly this one, but this is the sharpest. I think you can get in this counting and yeah I'm sorry this last part was very very sketchy. I'm dressing to read the paper. These are the two things where we need mixing in the passenger when construction. So, I'm going to, I'm going to end here so I'm open for questions. Or yeah, we have to answer questions. So, thank you for. Thank you very much had him. Thank you to everyone who's followed the course.


Well, hi there, I'm Sandy Olnok, and I'm going to show you how to make your own bird feeder. There's so many bird stamps, you're going to want to learn how to do this one. This is a new release stamp set from Ellen Hudson, and it's got some really great birds in it. They're really simple to color, which some of you might really love because I know lots of people love to color super simple types of stamps. These have nice big, heavy lines, especially when you stamp in VersaFine, which is what I'm using on some letterpress paper because I'm going to use my Zig markers with it, and VersaFine for me just stamps much darker, like thicker lines, so you can also stamp these with thinner lines with an ink that you can use for your Copics. I'm going to show you how to make a bird feeder of your very own.  You can do a round lid, you can do something that looks more like a roof, whatever you want, and then cut out the glass portion of it.  I'm using a fingertip knife. I find that I get really good control with that because I can stick my finger into the little handle and it feels more comfortable in trying to hold a big exacto knife of some kind. And then I'm going to trim around the bird heads. And the reason I did the bird heads in front of this rather than just making it kind of a standalone rectangle shape, which would have been easier to cut is so that it looks like the birds are interacting with it. So the little chickadee at the bottom is standing on the footer and the other one is flying in front of it. If you wanted, you could make your birds outside of that area and make just a rectangle cut, or you can use the dies to cut the birds out, that sort of thing. To finish it off, just take a pen that is compatible with it, whatever medium you're going to color with. So I'm using a water-based or waterproof pen, since I'm using water-based markers, my zigs for this, and draw around the edges of the pencil lines that you drew. And Bada Boom, Bada Bing, it's ready to go. And it looks like I had a bird feeder stamp in my stamp set, but I don't. I made my own. So these are the zig colors that I'm going to use. And that picture is going to be on the blog if you forget what those colors were and you want to use them. But for the most part, just use whatever colors you want. I recently did a post and there is no video for it. But if you're interested in customizing your birds, see how I did it with a different stamp set. If you look on my blog under Customizing Bird Stamps. And if I remember, I will add that link to the blog post so you can see more of that because I did a whole bunch of different bird breeds so that I could use my same stamp in different ways and use different tree branches for each one, et cetera, to make them all feel like they were very different. But for this one, I'm just going to keep it nice and simple using my zig markers. They work really well on letter pressed paper or a bunch of different kinds of paper. But I like letter pressed. Letter pressed is also a little heavier. So when I did that detailed cutting on the little birds beaks around the bird feeder, I had a little more stability there because it was a thicker paper. It's a hundred and ten pound water. Water based letterpress. Yeah, that's the word I'm looking for. Boy, my mouth is not working as fast as my brain today. So the chickadee has just a little bit of gray and a little brown in him. And then the hummingbird can be all different kinds of colors. You can Google hummingbirds and I want to do some other stuff with this hummingbird sometime and make a whole bunch of hummingbirds on a car. I think that'd be really fun to figure out how to do that. And they're all different colors. So I made mine kind of rainbow-ish with yellow, red and green and then blended the colors using yellow in between all of them to make him look like he's a shimmery little little birdie. The coloring in general, I keep really simple when I'm doing my zig clean color markers, and that's one reason why I don't use them very often, because I love to do scenes and fancy things and the zigs don't lend themselves to that really well for the kinds of things that I do. I know some people like to use them a lot. And I am not as big a fan as I was when they first came out because I've gotten so much into scenes that's kind of become my gig. And that's what I do on my my teaching site a lot is I teach you how to make backgrounds for things and there's a recent class that was just launched in in three different mediums over on the art dash classes site. If you hadn't heard about it yet about how to color rain in Copic markers, watercolor and colored pencils. So that might be something of interest to you if you like to color. But those are of course a lot more complex because you have to have color all over the page when you make a scene and these zigs, they just don't lend themselves to that as easily. If I'm going to do a whole scene background thing, I would use a brush and water and if I'm going to do that, I might as well use water colors. As I have more color available to me when I mix colors than I do in my Zigg markers, so I have a little more availability as well as control. So now I'm going to make my shaker part. It's a really basic shaker, nothing exciting about this one, other than how I made it to go around the birds there. So I just took a little scrap of packaging from something and cut a rectangle that's going to fit over top of my square window or my rectangular window.  And this stuff, they changed the size last year, the width on it. And I still cut it down really skinny because you don't need a big wide chunk of it for the most part and I get double use out of the whole roll of it since these rolls are not inexpensive and I just slice the whole thing down the middle. That's why mine looks really skinny and it also looks uneven because I just whack at it with a pair of scissors. And then I just put a couple pieces all the way around it so that that panel sits up the same distance off the background paper. So I've got a couple of different kinds of shaker contents that I can use for seeds, just going to hold them up to the window to see which one I like. And the third one seemed to do the best in terms of looking like the kinds of seeds I wanted. Now there's not seeds, there's butterflies and all different kinds of flowers and things in here, but it's still going to work because it's going to have all the multicolors that I was picturing in my shaker. You could also do one with little beads in it that would look very realistic or you can use just wacky color ones. All different kinds of fun stuff would make a great shaker. And you could just stick this straight to the card base, but I thought I'd be safer turning it over if I actually just put the little piece of paper over top of it before adhering it onto my card base. And then I'll stick it on. The full length of the panel is the full length of the card. And I've left a little bit of blue showing on the left and right. So that gives a little color to it without a whole lot. And then I decided I wanted my sentiment. I was going to stamp it right on there. And I thought, no, I want to make a little sign hanging off of the bird feeder that I'll draw a little extra attention to it as well as just making it fun, adding a little tiny bit more dimension to it.  So that was a little fun treat to add on there. And I'm just leaving the pencil line on there, that little V shape to make it look like it's a string. It'll make it a little different than the string that's holding the shaker or the bird feeder up to the tree that it's hanging on. And it's really fun. Look at that cute little bird feeder. It's adorable. You could make a giant one on your card. You could make a little one all different kinds of ways you can lay out a bird feeder on a card like this and stamp all kinds of beautiful birds on it. So there you go. There's my card for today. The whole new release is out and linked up below and more information is over on the blog. And if you're interested in seeing those other bird cards, I will have a link to those on the blog as well. I will see you guys later. Take care and have an awesome, awesome day. Bye bye.


Welcome back to the videos on correlation. Today we're going to continue to talk about correlation, but we're going to get into that statistical inference side. And in particular, I'm going to introduce to you how to conduct a hypothesis test for testing the significance of correlation. So let's jump in. So here we are in Google Colab. I've already loaded in the data. We're still working with that Marcellus well shell or Marcellus well and water data that we used up here to make this conclusion. But we're going to do a bit more of a quantitative analysis and focus on hypothesis testing today. And so the first step in any hypothesis test is to state the hypotheses. And so our null hypothesis is row, the Greek letter equals zero. And it's always going to be this correlation because the significance test for correlation is whether or not the correlation is significantly different from zero. And so we have row equals zero. And then our alternative here is that row is greater than zero. Now this could be less than, this could be not equal to, but for this case, I'm going to go greater than zero. And so once those are stated, then we need to conduct the randomization procedure. And I've outlined these steps here so that you can have them later when thinking about the randomization procedure. But we start with the data. And now in the past, we've shuffled data around, we've done reallocation, randomization. In this case, we're going to do a reallocation of sorts, but we're only gonna do it with one variable. So say we start with this data here, X and Y. What we're then going to have to do is choose one variable to shuffle. Here I'm doing Y and we sample without replacement. So reallocating the data for that one variable. And then we're going to recalculate the correlation coefficient and repeat that n number of times. And really the goal with this reallocation is to break the pairs. So we don't wanna keep them paired together. We wanna break them and then recalculate the correlation because if the correlation is equal to zero, then it shouldn't matter how much we shuffle the data. It should always be equal to zero. And so by shuffling, we can sort of get at that significance level. And so we can go ahead and get started with the code. So the first thing we need to do is define our sample correlation coefficient. And I'll call it SAMP Core. And this is just how we calculated the correlation before. Total basewatervolume.core with total yes. So this is our sample correlation. And then before we get into our randomization loop, we need to initialize the variables. So I'm going to make a copy because we don't want Python to accidentally override our original data frame. So I'm gonna call it SIM. We need capital N for 1,000 iterations. We need lowercase n, which is our sample size, which I'm just going to set to be the number of rows in our simulated or copied data set. And then we need to create an empty array called SIMCore, which we'll use to create our simulated correlation. So I'll go ahead and run that. We come down here and check out SAMP Core. We can see that it's the same value that we got up here, just it hasn't been rounded. So then we need to start the randomization loop. And so we started like we have all of our randomization loops for i in range n. And then we say SIM, total guess. So again, we're selecting one variable. So this could be water, but I'm gonna do total guess. It doesn't matter which one you select, as long as it's the same throughout all of your code. So I'm going to take SIM, total guess, and I'm going to shuffle it. So I'm gonna say np.randomChoice. And then I'm going to randomly draw from, again, total guess. So making sure it's the same variable. Our size is going to be n, and then replace equals false. So there are two things that are a little bit different here from what we've done in the past. We're sampling without replacement. So you need to make sure that you set replace equals to false. We've done that before, but the main change is that we're just gonna simply, we're just going to randomly sample from one variable, shuffle it, and now we're going to recalculate our correlation. So this is total base water volume dot core and total guess. Making sure that you're using the simulated code here so that you are taking the correlation with our new reshuffled dataset. So we can run this, and if we come down here, we can print simcore and see that we've got a lot of different correlation values, occasionally even some negative values. So one of our reallocations led to a negative relationship between total gas and total base water volume. So that is the randomization. And like we have done before, a lot of times it's easier to then visualize as the next step. So before we can do that, we need to create a data frame for simcore because right now it is a NumPy array and ggplot doesn't like to work with those. So we can say simcore, the columns, I'm just going to write as simcore. So we'll keep the same columns. And then we can say ggplot simcore, df, and we can do a histogram. And so our x value is the only one we need, just simcore. And then I'm going to add a vertical line where the x intercept equals samcore, get our sample dataset. The color is blue, and we will do line type equals dashed. And I'm going to enter that down so it's a little easier and wrap this in parentheses. So Python knows it's all one line. And then we can go ahead and run that. And so here we can see this is our simulated correlation values. So they're centered around zero. And this is our sample one. And so depending on how we framed our alternative, this value is based either going to be zero, if it's however much is greater than that, or one, however much is less than that. But we have done a greater than test. And so we can say print the p value, and it is just the length of simcore where simcore is greater than or equal to samcore divided by n. So we can run this and see we get zero, which we expected there's no data above our sample line, which leads us to reject the null hypothesis. And in particular, we can say that the p value is less than the significance level of 0.05 and give exactly what the p value is. So we reject the null hypothesis in favor of the alternative that the correlation between these variables is significantly greater than zero. And so that is how you can do a randomization procedure to test the significance of correlation.