If you're a permable and you believe in the whole bull market theory, the bulls need to stay above these levels. It's crucial. OK, it's absolutely crucial. Welcome to Access a Trader, the number one community for those who are committed to taking control of their trading in order to achieve success, profitability, and longevity. Thank you for joining us. Here's Dan Shapiro to help you find your edge, master your process, and own your future. Hey, guys. Happy Hump Day, everybody. Welcome to another edition of the Access a Trader dot com. That the wrap up show, hope everybody had a good trading day, hope everybody is happy and healthy and alive and hearing this broadcast. So let's get to the market. So no deal done, right? No deal done. The deadline, which really wasn't a deadline, which everybody knew it wasn't a deadline, it came at Wednesdays Wednesday, right? The deadline was yesterday. The problem is, it's not that the market is not doing anything. We all know the market is doing things. And we'll talk about what the market is possibly telling us what's happening next. But if you trade the technology sector, and that's kind of what we do, you've been seeing something that's been happening now for two, three, four days. And not even two, three, four days of selling, but two, three, four days of the market kind of basing right on this big, big level of support. Now again, if you're joining us for the broadcast the first time, you understand why this 282, 283 level is really, really big. Because this is where the whole market took off, right? This is the range that started all the way back from September the 11th, finally broke out. And now we're just nestling. The problem is the longer we continue to sit below the five day moving average and build and build and build, eventually something has to give. And you can see here is one closed below the five, second closed below the five, third closed below the five, fourth day below the five. Now again, the bulls are doing a good job hanging on, right? G.I. Joe with the Kung Fu grip, they're hanging on for dear life, right? They're seeing this whole area here get defended, but the problem is how long can it last? And if you do trade the technology sector and you pay attention to the high fliers of the world that had their really, really big runs, the JK Solars of the world, right? The JK Solars of the world and the docus of the world and the ZMs of the world that had their really, really aggressive moves up, really big moves up, investors did great, traders did great. But as we've been talking about for a very, very long time now, you get, you cannot look at the stocks that already made their big moves. That's right for weeks and weeks and weeks, we've been concentrating on stocks that were coming out of the channels, the bottom channels, the middle of the channels, because the first thing that gets always pulled are the high fliers. And as much as the bulls are defending these levels, you can clearly see the money is aggressively coming out of these big, high fliers. And the most important part, and again, I know people love their stocks. Again, I still don't understand the world-win romance you have to your individual stock, but I get it. Okay, some of you guys are new traders, new investors. I get it, you have some sort of bond. In time, you'll realize your stocks don't love you. But again, for the rest of you guys who've been trading for a very long time or even starting your trading journey, again, technical analysis, again, is not subjective, okay? It's not a discussion if the market is gonna go or it's not gonna go. Usually when you get multiple points of data, eventually that data is really gonna get breached or it's gonna reclaim high levels. So again, here we are, three days in a row, right? Mainly four below the five-day moving average. We have the line in the sand now for three days in a row, something has to give. And if you do your research tonight, okay, and you go from stock after stock after stock, they were basically the biggest high flyers even in the last month, month and a half, you'll notice they're all on the chopping block, either got already destroyed in the last two days or ready to hit that panic button. And again, you don't need to have a lot of clues, guys, all you need to look, right? We talked about JK solar, massive reversal. Again, and you can turn around and say, well, people were talking about this top for a while. Again, our job is not talking about the top, our job is talking about when stocks break technical levels. As soon as it broke the five-day moving average, that's where again, it's moonshocking back to the 10. If you look at, again, the names that had the biggest runs, they're all setting up. Look at Amazon, right? Look at Amazon, literally, right? 3160 is low here, 3160 is low here. Again, what do you think is gonna happen if the bears take control of that 3160 level? Again, this is, you can make an argument, well, this already came down from 3500. It's 3160, the stock is oversold. Maybe it is, maybe it isn't. But again, this line in the sand is real. Peloton that had a monstrous run, really huge run. You know, rallied three months in a row, put in a high of 137. Again, people tell them, well, this is just a pullback. Maybe it is, maybe it isn't. By the way, Goldman Sachs downgraded them after the close. Again, now it looks again, first close below the 10 day moving average. Now it looks like it wants 115, 110. Zoomed that we had a phenomenal move on, right? Phenomenal move, yesterday was that first pivot. We talked about yesterday, 562. We'll talk about the pivots tonight. There was another pivot this morning at 533. The stock now, again, took out the five, took out the 10. This thing looks like a moonshot going up to 500. So again, if you do your research tonight and you really see where the next possible players are at the downside, it's pretty obvious. Amazon really needs to defend this level or this thing's gonna get clipped. Nvidia really needs to get defend this level or it's gonna get clipped. BYND that I traded yesterday, held on today for one day, right? Stopped at exactly the same level but if it loses this level tomorrow, it's gonna get clipped as well. And again, just go trade by trade, set up by set up. Again, these stocks are not hiding, folks. They're not. There's some names that actually look pretty good. Baidu, right? Baidu had a pretty aggressive option buying today in the November, November and December, 135s, 140s, 150s, we even saw 160s. So this thing is actually setting up very, very well. But again, if you look at the majority of the market, they're looking like they're about to get hit and looking like they're about to get hit incredibly aggressive. Again, just ask the players what happened today in square, right?  Again, they pulled the plug on square and that kind of brings us to what happened today. And if you notice, again, I share my morning notes, especially for the people who are not in the live webinar, they do not hear the kind of the play by play during the day. So I kind of give a little bit of a micro analogy of what exactly, excuse me, kind of what's going on and what I'm thinking throughout the day. And the most important part is again, we were ready for the downside pivots. We saw the lower highs being put in for the last couple of days. And now we're just looking for rotation. Stocks, they just haven't got hit yet. They're on the chopping block. And if they do get confirmed tomorrow, you should see more selling. And again, guys, that line in the sand, again, is not a subjective point of view. It's helped three days in a row and if the Q's lose that third, the fourth day's a charm, you're gonna see a really, really aggressive mass selling tomorrow. So again, big, big day coming for tomorrow. Again, if there is no stimulus deal done and it doesn't look like they're close, doesn't sound like they're far, but it doesn't look like they're close. Maybe you get something done by the weekend, maybe by Friday, but for now, again, you can't trade on the hopes of a stimulus package saving anybody. Again, you have to take technical analysis for what it's worth. And after all the cards are dealt, then you can see how it kind of plays out to your individual process. So let's talk about today's session. Baidu, 134, 135 big spots upgraded this morning. I thought Baidu was gonna be, I thought Baidu was gonna be much more than it was. I really thought Baidu was gonna be much more than it was. It had a nice run. Don't get me wrong, it was actually very, very strong. It traded to that 36 area, okay? But again, I think tomorrow, if it starts confirming, if it really, really starts confirming tomorrow's action, then you have 139 really, really quickly on deck and that's very, very important. So we're looking for a big, big push here. Again, if the options market continues, again, this is at least one of the stronger names that could benefit if the market doesn't completely tank. So I still like Baidu for tomorrow. This is the trade that kind of, I was very puzzled about this trade here. NIO, so I got one NIO and it kind of stole that, which was very, very odd considering yesterday there was a lot of call volume. It took out the previous day. So I wound up losing about 18 cents on the trade, not horrible obviously, but the point is I'm still trying to figure out why the stock, why the stock didn't work out considering that Tesla was coming out with earnings today. Earnings look pretty decent. Again, for Tesla to really get stronger, it really needs to get above like that one, that 442 area, 443 area to confirm. But again, we'll get to that for tomorrow. Uber never confirmed. And again, here's a perfect example of a stock that was incredibly strong, had a big run and then it got destroyed, right? Tullo 308 307, if it builds below, can flush. Here was Tullo, right? Here was Tullo, here is took out the 308 307 level, went all the way down to 297. Again, this thing confirms as much more, much more room to the downside. ZM got destroyed as well. Again, second day in a row, second pivot in a row. 533, if it builds below, can flush more. I thought it was gonna stop. Personally, I thought it was gonna stop at 521. It traded to 521 and traded all the way down to 513. Again, I get it, it was a big move, but if this thing confirms today's prices, this thing has a shot to get to 500 tomorrow, so we're gonna keep an eye on that. Netflix actually had a pretty decent trade on Netflix. It was tons of upgrades, they got down 5%. And obviously again, this was for experienced traders only a dollar max paying. So I got along Netflix and it went right up to 507. I was making sales along the way and I knew 507 was gonna be supply. My last sale, I was literally offering out, right here, you see right here? I was literally offering out 507, traded 506.85. So I wound up, didn't get my sale, my last sale, and wound up scratching on my last piece, but again, it was a fine trade. Beyond, I still like, still held this level, didn't reach it today, but I still like it tomorrow. Tesla, nice little spike ahead of earnings today. 526, if it builds, can spike for cashflow ahead of earnings tonight. So here was Tesla, right? Here was Tesla, excuse me, let me just show you on this channel here. So here was Tesla right here in the morning. Here was the whole 526 area, stock traded to like five, almost 531, obviously. And this is kind of where I'm talking about that 442 level for tomorrow's session. So Tesla was fine as well. Square got destroyed, 181 if it builds below can flush more, again held that level twice so big support, so here was Square, right? So here was Square, here was the 581. This whole channel here, 581 went all the way down to 175, just got completely destroyed there as well. Yeah, take on the way up. Again, you gotta be in scout mode. Guys, Sennler, for all you guys who are trading these smaller names, Sennler for INO, I really like this setup here. It's not really my thing, but I found it today and I kinda like this setup here. Look at this INO, just for all you guys who trade smaller names, look at this INO setup. If this thing clears out this whole channel right here, you see that guys? If it starts clearing out this whole channel roughly around, let's just call it 10 and a quarter, so if it starts clearing out 10 and a quarter, this thing could really, really get aggressive to the downside. So keep an eye on that, obviously still valid, destroyed, new lows coming up. But yeah, I thought 527, new lows, and I thought it was gonna get down to 521, it got really ripped off. So again, I think tomorrow is going to be another big day for the indexes. I think it's incredibly important if you're a permeable and you believe in the whole bull market theory, the bulls need to stay above these levels. It's crucial, okay? It's absolutely crucial. They stay above these levels. Matter of fact, they really do need to reclaim roughly 288 on a close for this whole ugly downtrend to kind of wake up. And if they don't, any close below 283, again, you could see what's gonna happen in this future because it's gonna go right back into the channel that it literally took a month and a half to get out of. So it's going to be an important day. Guys, God bless, have a great night everybody. Hope everybody stays healthy and I'll see you all in the field tomorrow.


Thank you very much. So it's a great pleasure to be in Trieste again and I'd like to thank the organizers for the opportunity to be here and for the invitation to speak and So I want to talk about something which is very easy It's about some properties related to a fairly straightforward measure on a very very simple space But in fact the measure is the so-called Kazulka measure and so the hardest thing About proving results is actually knowing what it is and so I will give a definition In a few slides, but I want to kind of creep up on it Just by way of audience participation does anybody recognize where the location is of this picture excellent It is of course in Trieste It's near the canal and and the person with the the the worst pallor is me But the other person is James Joyce or at least a statue of him. I assume that he's probably buried somewhere Okay, so I want to talk about The Kazulka measure and in fact when I define it, I'll actually define it on a Sub-shifter finite type in fact a full shift on three symbols But its origins actually come from Fractals and so in particular I want to just creep up on it by talking a little about Fractals and in particular the siropinski triangle mainly because it has nice pictures no other particular reason and To define the measures as I said it's easier to code the siropinski triangle using sequences and Then define the measure in that context So the game is I'm going to talk about this measure its origins are as a measure sitting on the siropinski triangle In the plane, but I'm going to view it in terms of a space of As a measure sitting on a space of sequences just Infinite sequences of one twos and threes kind of easy The problem is that Usually if you look at measures on sequence spaces you like them to have nice properties you might want them to be Gibbs or equilibrium states or something like that and It's usually usually easier to prove things in that particular context But this measure when I get around to defining it is not a Gibbs measure of that type It's a invariant measure for the shift, but it's not a Gibbs measure as we know it And so this leads to some complications if you want to prove stuff about it But if you are more of an optimist then you could say it's an interesting class of measures to which previous techniques don't apply and so consequently it's interesting to study for that viewpoint and Also, if you prove stuff about this measure, it's supposed to have some applications to the original context where the Kazalka measure was introduced in the context of measures on on siropinski triangles and The good news is that although this measure is not Gibbs and In particular, we can't apply the usual techniques. We can move stuff around We can modify the ideas and there's analogous things that we can prove And indeed we can establish the sort of things you'd expect to be true for a Gibbs measure in a hyperbolic system Just we can't follow the traditional pattern. We have to do something a bit different, but that's okay. I guess So that's what I just said. So here's some context for no particular reason so in the first talk of this conference Keith Burns was talking about measures on For for geodesic flows and in his case the geodesic flow had the property that it was not sort of uniformly hyperbolic But the measures were quite nice because they were from a particular nice potential a holder potential and In fact, this also appeared In the talk of Lima on Tuesday where he was also talking about non hyperbolic systems flows in that case and and Nice regular potentials giving you the measures and so I'm doing the opposite in some sense I'm going to look at a nice dynamical system, but the potential which defines the measure will not be so nice So the measures so I'll be looking at the simplest possible hyperbolic system Such of the finite type in fact a full shift on three symbols and for this very simple Dynamical system, which is as hyperbolic as you could want It's we'll be looking at a measure, which is associated to a non holder potential and non holder in a very Bad way. It's not even continuous in any reasonable sense, okay, so that's that's the game so instead of looking at More general systems, but regular potentials We're going to look at reasonable systems very nice systems But with less familiar Potentials because that's the way it comes and as I said these measures originally appeared in the context of Fractals in particular things like the siapinsky triangle gaskets things like that So let me just whittle away some time by saying some words about that. So the usual construction of The siapinsky triangle as you just take a triangle in the plane and then you simply remove the middle triangle So that's the picture there and then you remove the the middle triangle of each of these pieces and you keep on going and so it's just a two-dimensional analogue of the The usual construction the middle of the kentel set for example But we're in two dimensions where you're taking out triangles rather than the middle of the kentel set where you take our intervals and So that's the construction and at the end you get some fractally looking picture, which looks like that and You can Do stuff with it compute its housed or to mention easy of course And other stuff So this is just a traditional construction So siapinsky is is now mainly associated with the triangle But in fact he was a very distinguished number theorist and that which is sometimes now forgotten But particularly by people who do fractal things and so here is a picture of the great man Which I just mentioned in passing But I won't be looking so much at the siapinsky triangle because to define measures on dynamical systems It's usually sometimes easier to actually look at sequence spaces So we can code the the siapinsky Triangle just by sequences one two and three and these just correspond to the three triangles That are left after we take out the the previous one So at the top we take the big triangle and then there's three pieces here And there's three pieces here and that's what's going to give us the coding so the usual coding is Given in a certain way so I look at the space of all sequences of ones twos and threes and I can turn it into a metric space Anyway, I feel like doing it, but in particular I can define a metric which may or may not look something a bit like that So basically sequences are close if they agree for a long time In the first few digits and this is a very simple metric to define on the space And then we can associate to every infinite sequence a point in the siapinsky Triangle just by the coding of it which basically means that hopefully you take an infinite sequence and you write down some Expansion like this where you take the right vectors these may be the right vectors These are my guesses at what the right vectors are and it's pretty much the same as if you take the middle third Cantor set There you just have two branches So you take a dyadic expansion of points in the unit interval and you just skip anything that had a one in the expansion You just take zeros and sorry Yeah zeros and twos in that case here I'm just taking this and I'm expanding points in this way so every infinite sequence in here corresponds to a point in the triangle and Most points in the triangle Correspond to some sequence upstairs. There could be some lack of Bijection, but it doesn't matter so much in what happens and of course There's also some dynamics, which is more apparent or more familiar for the shift Then it is for the the siapinsky Triangle and the dynamics is just the usual left shift map. You just take an infinite sequence index here from zero to infinity and throw away the first term shift everything to the left and it's the usual Shift map and if you were to write it down in terms of the coding of the siapinsky Triangle, it would just mean that you blow up each of the smaller triangles by a factor of two until they overlap Okay, but basically we're just interested in this familiar object space of sequences and the shift map Very easy to to deal with Okay, and so if you want to define the Czorka measure, it's an example of an invariant measure on the space of sequences for the shift map and Usually the easiest way to define a measure on spaces of sequences is to define what the measure is of a cylinder set so just a subset of the Space of sequences So it has a nice topology coming from the metric you look at the Borel sigma algebra and Within the Borel sigma algebra you have these open sets which are simply the cylinder set index by a bunch of numbers and The way that they are Defined is it's all the sequences which happen to take these values for the first end places So it's a standard definition of a cylinder It's simply the guys which have the property, but they look like x0 up to xn minus 1 in the first end places after that you can do what you like and that's what gives you the cylinder set it's a whole set and These things are enough to to generate the it's a sub basis of topology so they generate everything and Consequently if we know what the measure of these guys are it's it defines the measure of on the entire space basic basic stuff and so in particular if it's invariant it means if we shift backwards and We look at the set we get we want its measure to be the same as the first set So hopefully it's the same as this condition So it's a shift it will interested in shift invariant measures which we can define just by saying what all these sets look like these cylinder sets Okay, and so the most obvious example. I like to do obvious examples It always gives me more confidence when I get on to more complicated stuff So the obvious example is you take some Bernoulli measure you can take the one-third one-third or one-third Bernoulli measure so in that case The measure of each of these cylinders is just one over three to the length of the cylinder And this gives you a measure which is well-defined. It's invariant and somehow it looks like the natural measure to look at on on at least on the siapin ski triangle when you code it down, but it's the easiest measure you can think of on the space of sequences and You can generalize this definition and you can look Gibbs measures and so by a Gibbs measure I mean that you take The measure of you see have a measure mu and it's going to be a Gibbs measure if If you have the property that if you look at the measure of one of these cylinders And you look at the measure of the shifted cylinder. So this is one way around all the other Then you take the limit as n tends to infinity and if it exists for every sequence x x is one of these infinite sequences x 0 x n off to infinity So if this limit exists for every point x in the space and it's holder continuous Then you say that this measure is a Gibbs measure. So it's a general class of kind of useful measures to look at on a sub-shift of finite type and if you take The obvious example, it's an example of a Gibbs measure Simply because when we look at the previous definition and apply it to this measure or the measure of this cylinder would just be one over three to the power n plus one I guess and the one on the bottom will be one over three to the power n and you take the limit and You get that You get that the measure is But the ratio is going to give you a third and then you take the log and And it's minus log of three. So this is a Gibbs measure for a constant function. It's not the most exciting of Examples if you look at the next most obvious example, you could take a Bernoulli measure and In the case of the Bernoulli measure we can choose different weights for the three symbols one two and three and So I'll take the weights to be simply p1 p2 and p3 And so we define the measure of a cylinder which therefore once we know the measure of every cylinder It uniquely defines the measure on the space of sequences and we define it simply to be a product of the corresponding weights P whatever p whatever p whatever where the whatever's are simply the terms of defining the cylinder So it's a standard definition and if you carry out this analysis correctly before Then the associated potential is just something which takes different values Depending only on the first coordinate and these values are something like log of p1 log of p2 and log of p3 So in the case that they're all the same p1 is equal to p2 is equal to one third then we should get the same value as above and this defines this explains why Bernoulli measures are Examples of Gibbs measures with the previous definition. It's because this is a continuous function Defined on the space of sequences. It's a locally constant. It only depends on the first coordinate Because of the way that the metric was defined This is going to be continuous. Okay, so this is very easy stuff easy ish stuff and As I said, I mean generally one talks about Gibbs measures or equilibrium states The words tend to be interchangeable nowadays. Well, they have a slightly different meaning In the context that these potentials are holder continuous because then you can apply a rich theory Which dates back to such people as Sinai and Ruel to prove all sorts of things But in the case of a Kazulka measure it does not have this property It's a measure which is defined on the space of sequences So in particular, we can actually say what the measure is of each of the cylinders That's how we define it But when we try to work out the potential it turns out that it's not a holder continuous potential which Causes some problems. So am I going to tell you what the Kazulka measure is? Yes So here again, I'm just saying that in order to define the measure on the space of sequences Ones, twos and threes infinite sequences I need to specify what the measure is of one of these sets one of these cylinder sets So for every choice of I zero to I n minus one where n could be arbitrarily large I want to say what the measure is of this particular cylinder set and In the case of Gibbs measures usually there's some function hanging around Like in the case of these Bernoulli measures, but in this case it's actually defined using matrices And so instead of giving the general definition I'll start off by talking about the classical definition, the classical example And the game there is that if I want to specify the measure of this particular cylinder So each of these I zeros up to I n minus one is just a number between one and three It's one two or three So there's going to be presumably three to the end different choices of cylinders of length n And the way I do it is I look at the indices the things that define it So each of these guys is an I zero Each of these I zero I one up to I n minus one is either one two or three I look at the corresponding one of these three matrices up there and I take the products of the matrices So in the case of Bernoulli measures you multiply numbers together Here I'm multiplying matrices together And then I do this twice, actually it's the same matrices So I look at the product matrix, I take the transpose of it And then I sandwich in between it this extra matrix So I'm defining my Kozolka measure on the space of sequences of ones, twos and threes The full shift on three symbols And the way I'm defining it is by taking a product of matrices Corresponding to the indices of the cylinder And the hope is that this measure should be well defined And also that it turns out that it's shifting invariant In this case you can just do it by some sort of computation In the more general case, which I haven't said yet It follows from the more general definition But it gives rise to a measure which is a probability measure defined on the space It's also a sigma invariant So if you're an ergodic theorist you might ask the question like Is this measure ergodic? And the answer is yes, it was proved in 1989 By Kozolka that the measure is ergodic 1989 incidentally was the last time I was in Trieste Prior to this week I'm not sure if that's fate or not And I haven't actually said much about why this measure was introduced Or who cares about it Well the people that care about it most are people that study Laplacians But these are not people who study Laplacians on Romanian manifolds These are people who study Laplacians on fractals So for example there is a theory of defining a Laplace operator A Laplacian on, for example, the Sierpinski triangle And the way it's constructed and the way it's defined Works best with a certain class of measures Of the so-called Kozolka measures So up until yesterday I had three slides following this Telling you how the Kozolka measure appears A bit of its history and something about harmonic measures and functions On the Sierpinski triangle But I decided it was better to shift them to the end of a talk In the hope that I wouldn't get there And talk more about the ergodic properties of this measure So let's for the moment just say it's a funny measure With some connection to a different area Which is a ergodic shift-invariant measure On a well-known friendly dynamical system A trivial dynamical system And we can start asking questions about it And so if we were given this definition The first thing we might ask is this measure a Gibbs measure That is is there a nice function psi With the property that when we look for the potential function For it Does this potential function have nice properties So if it was holder continuous Then we'd have a Gibbs measure For the potential it was holder continuous For this Kozolka measure Then the good news is that we could just roll out The classical theory And we could prove all sorts of things about Central limit theorems, large deviation theorems Stuff like that And it would be easy But the bad news is that the thing isn't continuous In fact there's a result from last year Which shows that in fact this function has A dense set of discontinuities So not only is it not holder continuous It's not very continuous And so this is my attempt to evoke the fact That it has a dense set of discontinuities It's simply something I plotted in Mathematica Which conveys nothing in particular Except it's meant to look like something Not very continuous The limit exists almost everywhere Yeah it exists almost everywhere The limit but it's not With respect to itself Which is not overly helpful really But it doesn't have any good topological properties In the sense it's not holder It's not continuous It's not whatever you'd ask for after that It's a simply defined measure Which has bad properties in terms of this potential Okay so therefore One wants to still prove things about it And one has to develop some other approach So our first result Or maybe our main result even Is that not only is it agotic But of course it's also strongly mixing And so strongly mixing simply means That if you take any two functions F1 and F2 And you shift one function by the shift map Which is the dynamics here Multiply it by the other function And then compare it with the The integrals of the first two functions So this is a correlation function Between the shift of the function And the second function Then this thing is strong mixing So in particular this quantity tends to zero So the dynamics kind of spreads out In some nice way And moreover it's actually Exponentially mixing in the sense That if we choose these functions to be more regular And in this case we can choose the functions To be Lipschitz say Still doesn't say anything about the potential Because the measure is given to us But when we test it with these Lipschitz functions It transpires that it has Exponential decay of correlations A fact which of course is very easy to prove In the context of Gibbs measures But it's not so easy to prove in this context And so the rate of decay is given by this alpha Which is a number between zero and one And in this particular case It's anything very close to five over seven This is just an explicit calculation Because it's a very explicit measure Okay so it's a result It's a result that says That not only do we have a periodicity But we have strong mixing And in particular we have mixing Which is exponential fast In fact once we have strong mixing We have things like exactness And stuff like that But let me concentrate on that This is a picture of my co-authors Anders Johansson and Anders Uberg This is Anders Uberg I think it's at the Uppsala train station Which is now some sort of restaurant And the picture on the right Is a picture of Anders Johansson Who does not have many photographs But you can track down There's a picture of him explaining something Very patiently to me Which usually he has to do very patiently Because I have trouble following what he's saying It's usually complicated and he's a smart guy Anyway so these are my two co-authors And this is the result And of course once you can prove Things like exponential mixing The same method or at least a direct application Of the result gives you a host Of other kinds of results So let me just mention Three kind of things That one would want to do And so once we know That the measure is a godic Then of course it means That you can apply the A book of the godic theorem And the book of the godic theorem Simply says that You have the dynamics for shift map You have the measure mu Which is this fancy A Kazulka measure Defined in terms of these funny matrices And then the book of the godic theorem Tells us that the temporal averages Are the same as the spatial averages That is you average along an orbit And you get the integral Of the function over the space For almost all points x It's what it always says basically So here's a picture of Corino Who actually gave the book of the godic theorem As the definition of ergodicity Which is even easier And once you have the godic theorem Which is just a beautiful Classical theorem You can ask how you can improve it And the classical improvements Are you assume more about the function f If you keep assuming the function is f1 You can prove very little more generally But if you assume that the function is more regular Maybe lipchits Then you can ask if you can prove things like Central limit theorems Which are different kinds of averages Or large deviation results Or if you're particularly attached To the book of the godic theorem You can ask if there's an error term In this convergence So these are three kind of classical things People tend to ask At least in a godic theory Or a smoother godic theory Once you've established ergodicity So if you know that the measure is a godic Can you prove these results as well And the answer to all three of these is yes There are analogs of these results In the case of classical Gibbs measures The newly measures Gibbs measures with all the potential These things again are very classical And easy to prove nowadays But in the context of Kazulka measure It's a bit more difficult So here again is just a statement Of the book of the godic theorem So we're taking the shift A full shift on three symbols An old friend We're taking the Kazulka measure A new friend Which we know is a godic So we know that this theorem applies And then we want to prove A central limit theorem So central limit theorems work on the principle That somehow you want to replace The 1 over n By a 1 over square root of n For example So you want to look at how things Fluctuate where the scale is changed By from n to 1 over n So here is the same summation n equals 0, n minus 1 So the summation along the first n points In the orbit for a typical point x But instead of dividing by 1 over n We divide by square root of n And then we ask well What proportion What measure of points Has a property But this particular average Is within this range Alpha to beta Of the integral So this is what central limit theorems look like And possibly the normal distribution Looks like that I always write this down from memory And usually get it wrong But anyway it should converge To the normal distribution As n tends to infinity And indeed In the case of the Kazulka measure This is still true So the central limit theorem Which you can think of as an extension Of the work of the godic theorem In some sense It's also true In the context of these things And the other theorem I mentioned The second theorem I mentioned Was to do with large deviations And so again It's a variant on what happens With the book of the godic theorem So the book of the godic theorem Sells as I said That for typical points You have the averages Converge to the integral Classical result And in the case of large deviation results You ask well What proportion of the sequences What proportion of the space Has the property that this difference Is greater than some fixed bound From where you're going So you know that These averages are converging To the integral of fd mu You look at the difference Of these things And you ask well How long do I have to wait So that the proportion of the space For which this is bigger than epsilon Is smaller than some particular value And these large deviation results Typically say that the measure Goes down exponentially fast So our bound here is fixed The epsilon is fixed Bigger than zero But the proportion of the space That haven't quite made it yet Is going down at some exponential rate So it's another kind of statistical result N capital which one That one You are perfectly correct It should be a capital N So let's not go back to that Yes So all the ends that should be capital ends Should be capital ends And all the ones that shouldn't be Shouldn't be So yesterday or maybe the day before I read through the slides And there were endless t's That should have been sigma So sometimes my transformations were t's So there are some It's like when I try and speak Italian When I only speak Portuguese There's some loss in translation Between these two settings But James Joyce certainly helps I can't understand him in any language Okay, so again The large deviation result could be thought of The kind of generalization A more sophisticated version In some sense of the Book of Egotics theorem And it also holds in the case Of the Kazulka measure And this is what the statement would say More or less Level two, type two Large deviation results Involving measures About which I will say nothing So that's two kinds of generalizations If you like of the Book of Egotics theorem There's the central limit theorem There's a large deviation result But the third thing I mentioned Was error terms for The Book of Egotics theorem So yes again, third time Maybe even the fourth time Here's a statement of the Book of Egotics theorem We always have this result But by ergodicity If we average some nice function L1 is enough in this case Along the orbit The average is converged to the integral But you might ask Well, for typical points Can we say how fast it converges? Well, if we assume that the function Is more regular And in shift spacers Lipchitz tends to be A good notion of regularity Then the answer is yes And so in fact You can prove the following Your function here is Lipchitz On the space So it's got some nice Regularity property Then in fact The difference between these two guys Tends to zero at some rate And in this case I've got the N correct It's uppercase N So it goes something like this Which has got a large number Of confusing logs on the top But if you ignore those It basically says That the thing goes to zero At least as fast as One over square root of N Minus something So it's an error term In the egodic theorem And of course Since it's an almost everywhere result The O here depends on your point There's a constant implied in this Which of course depends on the X At which you're applying it So you can write This is true for almost all points X And in fact The proof of these results The proof of the last result The error term is extremely easy And it's an immediate consequence Of stuff to do Just with the exponential mixing In fact some reasonable rate of mixing And a bit of spectral theory Stuff that doesn't appear in courses So much anymore But it's kind of straightforward to prove So these are just three applications Of the ideas to do with Exponential mixing And how it percolates through To egodic theory and egodic averages And because we have these results On exponential mixing Or at least some machinery That does that Then these three applications Apply in the case of the Xorca measure Which unfortunately is not a measure Which is a classical Gibbs measure But at least has these sorts of properties So let me say something about The strategy of the proof I always hate talking about Proofs in talks I always have this phobia That someone will suddenly point out a mistake But that hasn't happened yet It's happened to other people In talks I've been to But not in mine So what happens here is that We'd like to define We'd like it to be a Gibbs measure But it isn't But we can still define This candidate for the potential We just take the ratio of these guys And take the limit It's basically a Jacobian As it used to be called in the old days And it's not as good as we want it to be But if it was a nice measure Then we'd know what to do We could apply classical Ruelle perintrabenius techniques Which sometimes are now called Transfer operator techniques And basically we take the shift map And what we do is We just average over the pre-images With this potential And then God would be kind to us And this operator would have a spectral gap And this spectral gap would imply mixing So this is what everyone does In courses nowadays And then just using the usual duality Between the transfer operator And the shift map You get the result So that's what we would do If it was a Gibbs measure And the game here Is that we can't use Lipchitz functions Or holder continuous functions On the space But we'd like to follow the same strategy But we'd like to introduce A different space of functions One that works, for example And we'd also like to prove That the corresponding operator Still has a spectral gap Because then we will be able to do Just what we did before Okay, and so What is the space of functions Which I'm going to call b I think b is meant to invoke Banach space for some reason But that's okay So just by way of notation I'll let a of n It's the finite sigma algebra Corresponding to cylinders of length n So basically it's just Some collection of sets Which is nested As n gets bigger These things get smaller And then I want to associate The expectation operator Associated to these finite sigma algebras And the word expectation operator Here sounds extremely pretentious All it means is That it's the linear operator Which takes a function Which is defined on this space So it's just defined on the entire space With respect to the Braille sigma algebra And it gives you a Locally constant function And the locally constant function Is meant to be one Which only depends on The first n coordinates And how does that work Well you look at the first n coordinates Of wherever you are And you just average the function Over that And that's what you get So it's just an approximation To your given function Which is given in this way And classically it's just An expectation operator And the Banach space Well it's going to be Something like Holder continuous functions Except it's got to accommodate These discontinuities And so the way it works Is that we take some function f And we look at these approximations to it With respect to the expectation operator And we look at successive approximations And so the idea is These things are getting smaller And we ask that they get smaller At some sort of exponential rate So if I was going to say Something vaguely pretentious I'd say it was something like Moving from smooth functions To a sobble of space Where now you're using integration Rather than supremums Or something like that I'd use some words like that But I won't I would just say That this is some way of defining a space Which accommodates these rather bizarre potentials And the potentials psi Which I defined before Associated to the gazolka measure The thing that would have been a potential If it was Holder continuous Sits inside this space Which is kind of reassuring And it has some sort of norm Which is good And it's just saying Something about regularity Let me not say anything else about it This space, of course, contains The Lipschitz functions In fact, all Holder continuous functions If you choose suitable theta Just because it's measuring how things Approximate in an L2 sense And if they're Lipschitz They actually approximate faster than that So it contains Lipschitz functions So we can apply it to Lipschitz functions If we want to prove stuff And it has the property That the operator preserves the space Which is good Otherwise it's a bit useless introducing it So we have our gazolka measure We associate this hopeful potential Which is not Holder continuous But it's relatively nice Because it lies in this space B The old familiar transfer operator When we try to do it The analog preserves the space It has to fix the constants Because that's always defined And in particular if you quotient out By the constants Then you get that the spectral radius Of what's left is smaller than one Or if you like it's got a spectral gap So the original operator acting on This funny Banach space has a property It has a maximal eigenvalue at one If I do it that way around At one And then the rest of the spectrum Is in some ball Which is strictly smaller I is spectral gap And that's exactly what you use To prove all these results You might hope that the way to do this Will be to follow a traditional path Which is to use things like the Sotter York inequalities and things But that doesn't work too well either And so in fact the method of proof Is not actually to work directly With this Banach space But instead to look at a space Of matrix valued functions And that's simply because the measures Are defined using matrices And it turns out to be easier to do it that way But philosophically these results Are like the setting where you look At more general functions These things defined using expectation operators But rather than proving things directly With them you actually have to at the moment To go through this more indirect method So now I can't remember what's On the next slide It says 18 out of 23 So generalization So let me say the following So I define the gazolka measure Using one, two, three matrices Called A1, A2 and A3 And a third matrix called curly E And so if there was only one such measure Perhaps it wouldn't be so exciting But in fact it's part of a general class Of measures which are defined By just looking at matrices And so more generally you do the following We look at a full shift on K symbols Before K was equal to three Now it's equal to K And it has the property that these matrices Well before they were two by two Now they can be D by D And we also have a positive definite D by D matrix epsilon I'm not sure if the last one was positive definite But maybe it was And S matrix of course is some Typographical error which should just say matrix And given these objects You require this property Which is you take the matrices And you multiply them together And you end up with something which is identity And if you use this epsilon And you do it the other way round With the products of matrices You also get the identity And there's an extra strong irreducibility condition Which is something about taking the matrices And there shouldn't be a finite number of hyperplanes Which are preserved by them Just skip To define the measure in exactly the same way So using this finite collection of matrices You cook up this newer class of measures Funny class of measures And it's done in exactly the same way We define the measure of a cylinder Just by looking at the indices And for these indices we take the products Of the corresponding matrices from the list Take the transpose Multiply by the other matrix And multiply by the matrices we first thought of And when we end up with a well-defined And shift-invariant measure And these two conditions One way round or the other Give you the properties that are well-defined And invariant That's why we have to ask for these two conditions No, there's no relation Between the D and the K Any D that works If you can find matrices that satisfy these conditions You're in good shape I think you can probably find a continuous family Because there's no restriction on the entries And if you count the dimensions It probably works And if you take the extremely tedious case That D is equal to one, for example Then you can actually write Bernoulli measures like this But that's not so difficult Since there are only one by one matrices I don't think it's necessary Unless somebody points out You need it for these conditions to hold As far as I know I only require these conditions Okay And so you can just cook up these things For whatever reason And then exactly the same method of proof Gives you a similar result Which is that we have a shift invariant measure On, in this case, a full shift on K symbols And it has the property That it's a Godic and moreover It's a strong mixing Which is better than a Godic And moreover, it's exponentially mixing When you look at Lipschitz functions For example, it'll hold a continuous function It doesn't make any difference And in this case The alpha is more mysterious I don't know explicitly what it is In any other... In many other given cases The general argument doesn't give you Explicitly what alpha is In terms of the matrices But in special cases you can actually compute it And that was what it did And so I do know what's on the next four slides So What I would like to do is to skip through The next three slides For lunch This is the original definition Of the Kazulka measure Ignore all these slides You have to look at harmonic measures You have to look at harmonic functions Keep going and let me thank you For coming to the talk


My boobs are very pushed up. We're gonna basically see how many times he looks nervous, especially when they're on other... What? Ew, stop putting the camera right there. So we wanna get... Hold on, off camera. What? Your boobs look big today. What is good, Muggle Squad? And welcome back to our channel. So I am currently getting ready for the day as to why I'm in my vanity room. Bill is actually in all the way across the house. He's editing our thumbnail right now and posting our TikToks and doing a bunch of business stuff. So he can't hear me right now. So I thought why not do my intro right now? Get this Team Araya prank on a roll. If you guys have any, make sure you wanna hit the subscribe button and join my family and give this video a big thumbs up. We're halfway to two million subscribers, guys. Can we reach two million subscribers by the end of this year? Can we do it? I know we can, because our whole squad is so freaking strong. You guys are the strongest army around. We are so thankful for each and every one of you. But today is a good day for Team Araya prank. You know, Bill just pranked me last video where he gave me an edible and that wasn't funny. I don't really think this is a prank. This is more of a reaction. We're gonna basically see how many times he looks at this. So as you guys can see, my boobs are very pushed up. Like pushed up more than I've ever had before. I honestly, I feel suffocated. So basically I have three bras on right now and it feels like I am suffocating and that is why my boobs look super, super big. And they usually don't look this big, but that's what happens when you have three pushup bras on. So you don't even need a boot job but you can just freaking wear three bras and bing, bing, boom. But you're suffocating all day because I'm literally feel like I'm like, but I'm wearing this black shirt to kind of cover that I'm wearing three bras. I just want to see how many times he looks, guys.  He looked a lot. And I know he looked a lot when I wore those butt lifting leggings.  I'm not really like a cleavage girl. But, you know, flaunt what you've got. And if I had no big boobs, I would flaunt it more, which, you know, I have medium to small, but still. Not the fact. I don't ever really pop my boobs out like this. But today I am and we're gonna see how many times he looks at it. So make sure you give me a big thumbs up if you guys haven't already. We're gonna see how many times he looks at my boobs. We're just gonna be vlogging today. I know we wanna do like a TikTok video where we make each other our own October baskets. So it's like a Halloween baskets. And we're gonna actually post it on TikToks. We're gonna do like two parts. So if you guys wanna see those TikToks, make sure you guys go follow us on TikTok, Mariah and Bill, we are almost to 13 million followers. That is so crazy saying that out loud. Like 13 million followers. Wow, that is insane. You guys are so supportive and we're so blessed for all the opportunities that you guys gave us. But today, you know, Team Mariah is coming back. We're getting our reaction. We're gonna see how many times he looks at my boobs. I'm gonna go finish getting ready and then we're gonna get right on to today's video. Oh, I'm excited. I'm excited. I'm excited. Good morning, babe. What are you doing in here? Say hi to Marvel Squad. Hey, what's up Marvel Squad? Of course I was filming, give me a hug. Give me a hug. Appreciate me, give me a hug. Okay. Give me a hug. You look cute today. Thank you. I'm wearing a light makeup. Oh, I used my new working palette. Using my new perfume that I got from the buying video. I used my new makeup palette. Did you actually use the palette that we bought? Yeah. Color video. Guys, if you have not seen our recent video or not our most recent, but the color video, we bought a ton of stuff and I guess Mariah is wearing like half of it today. But what is good, Marvel Squad? Welcome back to our channel. So today we have a long day. I actually just told Bill we should go take Snap to get his nails trimmed because we are taking him to North Carolina in two days. So we have to get his nails trimmed so he doesn't scratch up Brandon's house because that would be the last thing we want to do. As guests come into a new house, if our cat scratches it up, that would probably be bad.  We got a lot on the list today. And we have to get going because this girl's taking forever to get ready two hours later. You know what? It takes a long time to get this ready. We got home from the gym like three hours ago. You know, like all of this ready? Okay. Yeah, it takes a long to get all this ready too. All right, take on, just get everything, like my face and like everything. Yeah, I know, that's what I'm saying. Also, I haven't looked at this in a while, but guys, we haven't even thanked you in a while, but shout out to all 1,427,792. Yeah, basically half way to two million subscribers which is insane. It's crazy. Literally you guys have been popping off, hold on, we gotta wait for it to go up, ready?  That was totally unplanned. It means the world that all of you guys like watch every single one of our videos that we post and are just showing love all the time. Like we cannot like tell you enough, it just makes our day. It's literally our lives. Every single one of you guys, literally, it just jumps up another like three subscribers again. You guys are amazing. But we're not gonna waste your time, guys. Make sure to go like this video right now, subscribe down below, and let's get on with the day. He has nails changed. Let's go. Look at how good this little guy does. I swear our cat is the best. The best cat when it comes to car rides ever, we're taking him on a plane to North Carolina in two days, and the first time he's been on a plane, he was amazing. I know, this one lady was like, you had a cat on the plane the whole time? Didn't even know. We were like, yeah. Dude, literally, snap is like a dog. And I feel like a lot of people look down on cats, but most people who don't like cats that come to our house and see snap, they're like, what are you doing? Oh my gosh. It's not working out well. They end up having a whole different view on cats. Yeah, no, but our cat literally is probably one of the best cats I've ever came across, and we're so thankful for that because he's not only the best cat ever, but like he's cute too, you know what I mean? Yeah. Yeah. Oh, look at this house, look at this house, guys. I love the Halloween spirit. Oh my gosh. It's that time of year. I cannot wait to start decorating for Christmas. Our house, I'm like, mark my words mom squad. I will show you guys. It's gonna be the littest house of the century. We're making it lit, and what's nice is you don't gotta put the lights up when it's cold. No. You get to just put it up whenever you want, and you don't gotta worry about it. I was saying, literally, I used to be on the roof when it was freezing cold in Chicago now. I could do yard work in the middle of winter. Literally. Think about that, that's insane. So they just took Snap back there. Yeah, they always say they're like, oh, this is a good kitty. But look at these cute little dogs right here. Look at this one. Look at how he is. It's a Marvel, it's a Marvel like mix dog. Look at the Orthys. Aww. I know, that's like from Wizard of Oz, you know, Dorothy's dog. So she handed him back, and once again, she said best cat ever, it means so much because a lot of the cats, they have to use welder's gloves because the cats bite them so much. It's because cats, like they like, they're a lot different than dogs. Like cats get just really nervous, especially when they're on other animals. So they get kind of like defensive, but Snap is like the chillest cat. That's why we love him so much is because he's so good. I also wanted to film really quick because this girl is destroying our brand new car. Stop. No, there were rappers in here from last week. Okay. But we haven't even used our car. We've been in Chicago for over a week. Like there's three Gino La Bar rappers and two- I ate this before the jail. Nice crispy. I'm sorry, can you forgive me? I will forgive you when this is gone. I don't even know what rapper this is. What is the clear one? What is that for? I don't know. Oh my God, oh my God. So guys, Bill ran inside to put Snap back inside before we go to Target. And guys, he hasn't really noticed. Like I've seen him look a few times, but he hasn't noticed. Guys, my boobs are popping, okay? I'm gonna have to unzip this a little bit more. Maybe then he'll notice, but guys, what the heck? I gotta do some flaunting now because he has not really noticed or even commented about it. Babe, my boobs are popping. How do you not notice this already? Like, come on, Bill.   And then they say that we're throwing stuff on them. We tried getting you noticed nonchalantly, but you didn't. So when I look like it, I go put my boobs in his face for him to see it.  All right, so we're in Target right now and me and Mariah have actually been away from each other because we're making each other like TikTok surprise little baskets. We can't see, actually I might be able to see a little bit what's inside hers. Here we go. Let's see. No. Oh. We can't see whatever is in the baskets at all because it's a surprise-all, you know, obviously. This is what I got from Mariah so far. Stop, get away from me. But I wanna tell him something. What? But we're trying to find Snap, like a little cat bandana. So we wanna get- Well, I'll camera this. What? The boobs look big today. I just noticed that. I was gonna tell you to zip them up.  I'm leaving them like this. Wait, what do you mean finally? Well, you know, I just, I don't know. They actually look good, like really good for once. I'm kidding, I'm kidding. I love you. For once. They always look good, but today they're extra. Like you just inflate to them or something. Like you just like attach to hose and you just like- Okay, okay. I'm not adding this on camera. Why I'm not adding this on camera? Why are you shushing me? But I wanna tell them, so we wanna get Snap a little cat bandana and we're gonna go to Carolina. So I want him to look snatched out. So we're gonna go to Paco after this because Target is a bust with them. I'm sorry, I'm not paying attention to it. You wanna get Snap a little cat bandana, right? Oh yeah, well, bandanas. We're gonna go to Paco. We're going to Paco to get bandanas and yeah, wow. All right. These look good. I'm looking at my basket. We look good. Look at my basket. So I just wanna expose Bill for a second.  Excuse me.  Why are you showing that on camera? That is nasty. Because you know what I'm showing?  Why would you leave it on the floor? Why would you not throw it away? I bet you people have blood fears right now. No, Bill is just a low key mad because I just exposed him and he didn't care about it. Mariah obviously doesn't care about the ones of you that are watching that have blood fears. So I on the other hand, do care about all of you guys. Whatever, but I was just showing I was exposing Bill because he's out here trying to expose me. But that's one thing because I didn't wanna hold it. That's one thing out of the 50 that Mariah has left in the past week that I've done. Okay, well I just wanted to expose him. So we just made it to Petsmart and of course the one place that we get stuck up in Marvel Squad is looking at the little kitties. Look at how, this is Andy. He's five months old and he's gray with a white like stripe on him. What if you brought him home for snack? Right? Get him a brother. Look at this one back here. Like that one's like a Halloween cat right there. Where? This one's about to come off. No way. He's moving this. Look at this one. Hi. I know, I'm not trying to wake him up, but. We're gonna have to adopt another one. Literally, this would be the exact cat that we would get from my mom if we didn't find Wave. Literally looks like reminds me of like, Babe. It does remind me of Wave. Look at the way he's laying. Oh, and the one back there is just so tiny. Literally five months old. Oh, it's shaking. I just wanna like pet him. I wish we could. Stop, we gotta walk away. Hey, at least they have cats out here because before they didn't even have him out here because of COVID. So, luckily they brought him all back out so we could see him. Oh, look at him. Hi, you're cute. What? Ew, stop putting the camera right there.  We're just out there. My new knockers? Yes. What else are we gonna film? Oh, we have to try to find bandanas and stuff. Like, bandanas and maybe some Halloween costumes. All right, so we just found the Halloween section and look at here. Wait, show them first the bandana, Babe. Okay, so this says happy Halloween. And even though it's for a dog, we'll make it work for a snap. This is a pet bandana, bandana. Okay, it just says pet. So we could basically wear it on any pet, and then. So these are for dogs, but Snaps basically like a dog. Dude, this is the same size collar as he already has. Do you think this will fit him good? I think that will fit him perfectly. I think it's too big. That's literally the same as that size. Or too small? Yeah, that's... Do you think he needs bigger ones? No, that's perfect. Because it can expand too. Snaps gonna be decked out for Halloween just to wake up. Guys, I am so excited. I love spoiling him. I wanna look at the Halloween, there's like little Halloween costume things over here. More? Babe, I think that's enough. I think we have enough. But I wanna get him... Look! Oh my gosh. This is what I have to deal with. We can be a magician. This is what I have to deal with on a daily basis, guys. We were gonna spend a thousand dollars at pet smart buying Halloween stuff. Just watch. Babe, you didn't want the BB, but let's have a BB. Wish me luck. It can be a bee. It can be a bee. So you like them? Yes, I like them like that. They look giant. What did you... Did you like take some... Well, I think it's the shirt. I think the shirt is like pumping up. You gotta push up, bro. No, I actually don't mind wearing a bra. Dude, they literally look like they're about to pop. I'm actually wearing one bra, but I'm not wearing... They look tight. Yeah, you like it a lot? No, but they look like tight. Like, they're just like smushed. Like, have you ever smushed a balloon? Yeah.  Like, that's what it looks like. That's what I was saying earlier. Should I start plumping up my boots like this more often for you then? So you did plump them up? No. But I'm just saying, like, should I try to make them more plump?  Stop. Hold on. Do you think you are? Why would I be pregnant? I don't know. Why else would I be in the... I'm literally about to get my period. They doubled in size since yesterday. That's all I'm saying.  As long as you like them, you know, like I just... I don't like them, I love them.  I could do it really good. Maybe I should get a boob job. You don't need a boob job. Maybe to plump them up since you're not like them. You do not need a boob job.  But obviously, look it. You don't need a boob job because they are looking pretty damn fine. I would only say that you need a boob job if you literally had washboard boobs. I'm glad that you like my boobs and I'm excited to give Snapp all his stuff. Turn on the camera because I want to show them the collar. But guys, we also... So, I know we showed you guys the one collar that we got, but... It was not a breakaway.  So literally it'll snap away and it won't leave him hanging to his death. Right before I was checking out, I was like, babe, I feel like I should go back and look for a cat collar because actually the dog... We actually were getting a dog collar but it would have fit Snapp. But this cat one... What? No. I don't want... Babe, I don't want them to see all that. That's all. Let it just be out. Let it just be out. Put the show on for me. Anyways, I was like, babe. We were one person away and I'm like, babe, I want to go look and see if there's another cat one. And I rushed over and I found it. I'm surprised because that one is a designated breakaway collar for Kat's and it looks way better than the other one. Well, there was one cat one that wasn't a breakaway and I'm like, oh no. And then I saw another one. If we leave Snapp for Malone, he's always jumping up on cat's. I know. The only thing... And there's been several... Sorry, not to interrupt you. There has been several times where his collar actually was off the middle of the floor and we're like, where's your collar? It's because... He jumped off of the collar and got snagged on like a handle. And if he didn't have a breakaway, he would have just been hanging there probably... Literally. Dad, but... So if you guys aren't, if you guys are cat people, make sure if you or you are want to get a cat, make sure you get the breakaway collars. I don't want to stress that enough because we do not want to call with your cat like choke to death. No. Look at how cute this booger is. Look at him. Oh my gosh. Here, pull it down a little bit so you can see it. Happy Halloween, Snapp. Happy Halloween. Happy Halloween.   He doesn't like going in his cage, but he loves it being in there. What do you think? The windows are open. Cameras on windows are open, man. Right. Cameras on windows are open. Water boat. I see. I know. You look cute. You look hot. So you want to... All camera. You want to know why my boobs look this good? Just put your face in them. No. Put your face, this is off camera. Just put your face in them, just feel it. What are you doing? Just feel it for a second. Ooh. Feel good? Yeah, can we film now? Do you want to touch them really quick? Just like grab them? Maybe I'll touch them on my nose. I know, but I want to tell you something. Can you just grab them really quick? Like really feel them. See if you feel anything different. I feel your bra and your boobs. No. Are you sure? Yes. Can we film? Okay, I'll tell you something. His whole point is weird. What? Oh my, what are you doing? Are you wearing a push-up bra or something? I wore three bras today. Three bras?  Hold on, show me. So how are you even... One, two... How are you wearing three bras? Because I am. Hold on, zip with this up. So be safe and film me.  That's why you were... That's why... Should we show them my cleavage with only one bra on? Maybe we should. Hold on, hold on, ready? Wait, hold on, I'm going to do a snap. It's going to go away. I'm telling you guys, that's why. That all makes so much sense. This is what it looks like. Look at it, ready? With three boobs. Hold on, hold on. This is actually going to be funny, ready? This is what it looks like with three bras on and this is what it looks like. That's what it looks like with one bra on. And okay, so you can still see if there's pull-up there. Oh my gosh, he's got a snap's hair. Babe, see, your boobs are not as small as I thought they were. I'm kidding, I'm kidding. But look, you should put the other bras back on. You should put like four on because they're... I'm just saying. How many times do you think you look? Oh my God, why do you have to do this for a video? Because I want to know. I knew you were being... I knew there was no... There was a reason why your things look so much bigger today. I don't even want to say it on camera, but guys, I probably looked 50 times. God, I knew you were going to give me back too. I just didn't know what. See, this wasn't a prank. It was a reaction.  I'm not a per... Babe, if anyone's pervert here, it's snap because snap watches us change and go to the bathroom. What the heck? And shower. Yeah, and shower. He literally, he loves watching a shower. He sits there and just stares at the water while we shower. Oh, I know. He literally is just like a muse. He is. I think he's just a muse of the water there and everything, but here, can you come down to my level? Come up to my level. Babe, why do I have to always do everything for you? I'm the provider here. I do everything. I bring in the dough. Gosh. You know that song on TikTok? A single mom that works two jobs. That's me, that's me. A single dad that works five jobs.   That was good. I'm gonna patent that. No one take that song.   I dominate. It's called domination. Whatever. It's called victory. I'm over it. It's called you were lurking at me. I knew you were doing something, but... Which is me, I love it, but... We had a productive day today. I actually thought we were just like five in today. We were gone for at least two hours. We had Snapped's nails trimming, Target run. We drove 20 minutes out of the way. To get him up, to get Snapped a new collar, bandana and a bee costume. So make sure to follow me, Mariah and Snapped on Instagram. So make sure to go follow all three of us on Instagram. So you don't miss out on any other aspect and part of our life. We do a lot of updates on our stories and a lot of just fun polls. We also incorporate you a lot in our YouTube videos that do answer stuff on our stories. But we hope you guys enjoyed today's video. So before we end the video, we're gonna give our end of video post notifications shout out to Amanda Weaver. Amanda, thank you so much for being a notification shout out. We love you. So shout out to you. If you guys wanna post a notification shout out, you guys have to comment down below letter by letter. Scare for Halloween. Scare because it is the Halloween season. As you can see, we got our Halloweeny, you know, pumpkins up there, up there. We got pumpkins everywhere, even up there. Look at that. Thanks actually for a TikTok. So make sure to follow us on TikTok because I am gifting that to Mariah right after we've done filming this. But guys, we will see you in the next video. But until then, peace. Love you guys.