What's up navigation traders happy Friday today's Friday April 5th. Welcome to this week's video update exclusively for Pro members Let's let's jump into the community real quick and talk about who got caught being hot this week This week's winner goes to Federico Lombardo. Congrats Federico. You got caught being hot Federico jumped in answered some traders questions posted some good things asked some great questions as well We like to recognize one member of the tradehacker community every week for helping other traders That's what it's all about So congrats and we sent you a private link Federico to pick up some tradehacker swag. So enjoy that And let's jump into the alerts for the week So starting with the first which was Monday Our first alert was rolling adjusting trade in smh So we rolled our short strangle in smh from april which was down to 17 days to expiration out to may And then we adjusted our puts up from 92 to 104 And so let's take a look at the platform at smh So we've got two pieces on here In smh we've got the uh, let's see we've got the This one here where you can see price is well out of range, but keep in mind If we look at just the puts we've still got a decent amount of premium left in those So we're not looking to adjust again quite yet We definitely need some down movement in smh Like the rest of the market this thing's been on a tear to the upside And along those lines, I just I want to make some comments about the overall market I mean look at what has happened since the beginning of the year. I mean the market has just ripped higher You know, this is smh, but if we look at For example, the the s and p 500 it's a very similar story And anytime you have a one directional extended move like this That is the risk To the methodology that we teach right we love two-sided action We love high implied volatility But when you get an extended up move In stocks not only does that Breach your call side, but also you are now we're looking at a period of low implied volatility But just keep in mind that that's this doesn't go on forever Okay, I know in a period like this where we've seen very little pullbacks in in over three months We've seen just two little pullbacks basically In in three months This is gonna hurt. I mean we carry short delta. And so this is going to be painful I mean we've given up a lot of our profits that we saw From just a you know just a couple months ago because we've carried short delta because we've seen this extended upside And now we're at you know, we with our range of short delta to theta We like to be in that one to one range up to five to one as far as our short delta versus our theta We are currently right at about that five to one ratio So what does that mean a one it has hurt our p&l right? It's it's it's grinded away our p&l some of our p&l But we also have we're we're we're pretty short And so a couple things if we get a small down move a that's going to benefit us greatly And of course on the flip side if this thing continues to rip higher that's going to be painful But this isn't going to go on forever. Nothing moves in one direction forever So you've got to stay mechanical and just understand that periods like this are going to be a little bit painful But that's what you have to understand when you carry short delta the way we do so This isn't anything we haven't seen before And while it's still painful every time it happens You've got to you've got to keep a mindset of understanding what's going on and don't panic and and and don't do anything stupid Just stay mechanical now if this thing does continue to stay high or continue higher I mean we're going to be fairly aggressive You know in the near term as far as either cutting loose some short delta or adding in some long delta Those are the two ways to kind of keep your portfolio balanced So look for that, uh, you know depending on what happens now obviously if this thing rolls over You know that's going to be the best for us And and that'll help our portfolio, uh tremendously, but you've got to stay mechanical Especially when you get to extremes Like we're seeing at this point All right, so well back to smh. Sorry got off on a little tangent there But smh so we've got this one piece that is is out of our range And then we've got our other piece here Which is still within our range after after we rolled but on both of these looking for a little bit of downside to benefit Next trade was a Opening trade in netflix. So this is a pre earnings long strangle Just like we teach in our earnings course The the focus of that module is really pre earnings long straddles, which would be just the same strike With a long strangle because netflix is a higher price stock We just widen those strikes out a little bit. So it's a technically a pre earnings long strangle And we're targeting about 15 percent, uh profit on this trade So we're looking for an expansion in implied volatility Leading up to earnings as well as you know, obviously a decent price move in either direction Will benefit that trade. So let's take a look at where we're at on netflix It's just been kind of bouncing around here. And uh, if we take a look at the implied volatility We have not gotten that expansion at this point earnings is coming up on 4 16 So, you know, we're we're anticipating that we're going to get a little bit of a pop and implied volatility And then obviously if we get a decent move in one direction or another that's going to benefit that trade So just holding steady for now Next trade closing adjusting trade in wheat. So we had uh, uh an iron condor on in wheat We went ahead and booked that one booked over 40 of max profit on that piece of the trade And then we're still holding our short call vertical From a previous iron condor and prices moved down in wheat. So it's come back into range And now we're just holding on to this To see if that can come back down and if it gets down to about around 450 455 as far as the price We'll go ahead and book that and and take a profit on that iron condor overall We've also got a another iron condor in wheat which is an alert coming up here So I'll just go ahead and address this one here. We put on another iron condor price It's fairly centered here. Just waiting for some time to pass in that one Next trade closing adjusting trade in invidia. So we had this iron condor on in invidia Price of invidia has been on a quite a wild ride We had a couple opportunities where we actually had orders in to get filled when it was pretty centered Unfortunately price ran away from us and now and then it breached the call vertical side Uh, excuse me. It breached our upside break even so we closed out the put vertical side here And we're still holding the call vertical side, which is Out of our range now at this point out here. So looking for some downside to get back into range With invidia and like I said, you know, I mean this thing has just been on a tear for the last few weeks So hoping for a little bit of a pullback next week that would benefit us there Next trade rolling adjusting trade in XLK So we have a long put vertical in XLK that we've been holding for that short delta exposure We went ahead and rolled that from April out to May And uh, and we adjusted our strikes accordingly. So let's take a look at XLK That's the healthcare. Excuse me the technology ETF And uh price is still uh in range here just looking for a little bit of downside to benefit that piece Next trade closing trade in Disney So we had a short strangle in Disney booked 30 of max profit in just 12 days on that one So that was a good trade if we take a quick look at a chart of Disney Got a nice implied volatility contraction We put this on back here and then got that nice contraction and and exited there for a for a nice Profit in just 12 days in Disney Next trade was an opening trade in WBA Walgreens boots alliance So we put on an iron butterfly here Because it's a smaller a lower price stock We could have done an iron condor or a short strangle But we went with an iron butterfly just for a more narrow range and a higher credit And we put this on they announced earnings and the stock was down over 12 percent And it kept that instead of getting that implied volatility crush that you typically get after earnings It was just a little bit of a contraction So we put that on in anticipation of further contraction in implied volatility Which we sure which we certainly have gotten here And so as long as price continues to stabilize we'll take this off. I mean You know, we're at a point. We're at about 12 percent of max profit If we get you know on monday If this opens up and implied volatility is still contracting and price is still pretty centered We'll book a you know a quick 15 percent of max profit. We'll go ahead and take that one off You know looking for about 150 160 bucks Uh before we take that off we want to make it worth the transaction costs and everything else So, uh, hopefully we get a quick winner in that early next week, but we will see Next trade a closing trade in ewz So similar to disney we had this one on for just 12 days Booked over 30 of max profit got a quick implied volatility contraction in ewz Which gave us the ability To book that one And that's the key with these with this type of trading. I mean you've got to just keep Putting on occurrences putting them on taking them off putting them on booking winners putting them on making adjustments And and and you'll and that's how we're profitable over time If you look at um, uh ewz, we put it on back here got that quick implied volatility contraction And booked that 30 of max profit winner So we're not we're out of ewz. We're gonna wait for implied volatility to pop back up Before we would Put that put another trade back on Next trade was an opening adjusting trade in zw, which is wheat I already mentioned that we just added that new iron condor on in june with 51 days to expiration at that point Next trade closing adjusting trade in spy. So we closed out the put vertical side of our iron condor and spy Price had breached our upside break even There's very little value left in the put vertical. So we closed it out Still holding that call vertical side. So if we take a look at sp y Uh, you can see here price has moved out of range now. Keep in mind this this gray area is kind of the one standard deviation move So there's still a very decent chance price could move back into range here This is in april and so we've still got 13 days to expiration Remember if this was a naked position, we would roll this when we get under 21 days to expiration But because it's defined risk, we're willing to hold this until we get Closer to expiration. We'll hold it even into expiration week if needed And then in this case we would probably roll that out or close it depending on where we're at with our overall Short delta to theta ratio. Uh, so we'll see what happens. We'll we'll give that more time Next trade it was an opening trade in lulu So we did a long call vertical in lulu just adding some long delta to our portfolio and with the anticipation Uh, that we we'd like the upside in lulu And here here's kind of the thought process behind this Uh, if you were in the community after they announced earnings This price after earnings gapped up well above the expected move and when that happens, we like to put on Short puts or short put verticals in anticipation that price is going to stabilize or continue higher Now we never got filled And I still just continue to watch this for the next few days and and with implied volatility continuing to contract We didn't sell a put we didn't sell a put vertical Instead we bought a long call vertical which is which is virtually the same When when it comes to the the risk profile graph The difference is uh instead of Selling a put vertical we bought a long call vertical In anticipation that implied volatility may may pop up a little bit which would help us but either way they're they're they're very They're very very similar. So either one would have been fine We did get a nice pop yesterday and then it's uh price is coming down a little bit But looking for just a little bit more of a continuation to the upside And if we if we get that we'll we'll book this early next week Uh trying to book, you know, 30 to 50 percent of max profit on this trade So hopefully we get a little bit of a continuation to the upside on that early next week Next trade was a rolling adjusting trade in es So we've got this long put vertical that we've been holding for that short delta exposure and just kind of keeping this on as as a You know a short delta hedge And so price had moved up higher Well out of range with with a low probability of getting back and sort of that when that happens We like to roll to extend duration this keeps that short delta in our portfolio And gets us back to a positive theta position And so let me just explain what I mean there, you know when price gets out here Then then you get to a point where that theta actually turns negative. So we want to get this back into range And you'll see here That you know now we're in a positive theta position So we're collecting that theta as long as it stays within our range Obviously the best would be a quick down move and that's going to benefit that trade the most But that's just uh, that's just a short delta piece that we have in for our overall portfolio Next trade was an opening trade in jp morgan So to add some additional long delta to keep our portfolio balance We added in a long call in jp morgan and we did this The only time that we'll ever buy a long call Is pre earnings and the reason is is because we can expect or anticipate a possible Expansion and implied volatility so it slows down that theta decay and sometime actually, you know If we get a big spike in implied volatility, it really helps our position So that's what we did in jpm So let's take a look Let's take a look at the chart first So basically what we're looking at is we've had this quick up move in jpm And then we had this little bit of a pullback on this day here and that's where we got in And then prices popped up a little bit Not enough to take off yet, but if we just get a little quick move higher in jp morgan leading up to earnings We'll book a profit there now. You can see we've got a we implied volatility really contracted today Which hurt our position, but again, hopefully we get you know that typical Pattern with implied volatility expanding into earnings next week And hopefully we can book a quick winner in jp morgan They announce on 412 before market. So we want to be out on 411 at the latest So we'll be keeping an eye on that and send alerts as needed You can see we're we're basically where we put this on no profit or loss at this point But if we can get a quick move, hopefully book a quick profit on that one Next trade closing adjusting trade in iyr So we closed out the put vertical side of our iron condor price had breached our upside break even and it actually had breached it It was out there past our break even for several days But we were just giving it a little bit more time Before we closed that out, but it continued to grind higher. So we went ahead and closed that out Still holding the call vertical side on that one And so that's what this looks like here Looking for a little bit of a downside movement to benefit that So we'll address that here in the next week or so By either closing or rolling that one out And then next trade was a rolling adjusting trade in qqq So we have two sets of short call verticals in qqq This one in april we went ahead and rolled that out to may And adjusted our strikes to get it back into that positive theta position and keep that short delta exposure in our portfolio So here's that one here You can see we just did that today So pretty much the same spot where we did it and then we've got this other one that is still within range here So both of these just looking for some downside to benefit those Next trade was a closing trade in six b the british pound So we had this one on we made one adjustment one roll And then because implied volatility just continued to stay high with the kind of uncertainty around brexit But we we stayed in price stayed pretty stable And then over the last few days got a really nice contraction in implied volatility If we look at fxb, which is the corresponding etf you can see just this week We've got a great contraction and implied volatility a lot of that uncertainty Is coming out of the market as we get closer to that kind of brexit date that they are anticipating and so That gave us a chance to book over $500 in profit on six b now implied volatility is still nice and high Especially compared to a lot of other things so if if if implied unless implied volatility just contracts massively on monday We'll look to put on another position in six b, but we are completely out of it at this point and last trade Natty gas so we we rolled one of our strangles from may which was at 20 days to expiration Remember when we get under that 21 days? We want to look to roll out And that's what we did here. And then we we adjusted our strikes. I want to explain this a little bit We just adjusted our puts we left the calls the same And we just adjusted our puts down from 3.25 to 3 The main reason is because these puts are so deep in the money that the open interest Starts to get a little bit sparse that deep in the money. So we just wanted to move that down And and keep our exposure in that gas here. So Let's take a look at natty gas. So we've got these two different pieces on This is the one in This is the one in may And you can see we just need we need some upside movement in that gas and we'll roll this one Next week as well. Remember, we're under that 21 days to expiration. So we want to roll that out But just want to spread out our roll. So we're not doing it all in one day So just looking for a little bit upside there And then this one here Same thing. We need some upside movement, but this is the one that we just rolled. So we're at the 2.8 call 3 put So just looking for some more theta to decay And some more time to pass and a little bit of up movement in that one All right, so those are all the alerts. Let's take a look at some of the other positions that we have on Oil we've got 41 days to expiration on these options. So a lot of time here Let's uh, let's separate these So we've got the 53 call 57 and a half put because the price is out of range But if we look at the put if we look at the put Still got a decent amount of premium left in there. So not looking to Make any adjustments there and then this one slightly out of range same kind of story Still have a decent amount of premium in those put So we don't need to do any rolls or adjustments at this point Just looking for some downside movement in oil to benefit that I'll see z n the notes. We've got two pieces on here We've got this unadjusted strangle here, which is got a little bit of profit Just waiting for a little bit more time to pass before we do anything on that one And then our other piece is this straddle And this was a originally a strangle that we ended up adjusting into a straddle And on this one, we've got 20 days to expiration. Excuse me 21 And so I didn't do the roll today I just wanted to give it over the weekend, but early next week look for us to roll that one out to june as well Collect a credit and continue to manage that one Assuming price is kind of where it is right now. We'll just keep the same strikes keep that 123 and a half Straddle and we'll just roll that out to the next expiration to to extend duration on that one I mentioned wheat Excuse me DIA we've got this short call vertical. It's still in April If if price doesn't come back down We'll we'll look to roll this out to get this back into a positive theta position next week And then we've got a full iron condor That's already out in may and you can see price is just kind of hanging out here Near the upper end of the range on di a Eem we've got this short call vertical. We need some price to go back down to get back into range here So, you know again looking at this This gray box is kind of one standard deviation move. So if price doesn't Come down pretty quick. We'll look to roll this out to mate again to get back into a positive theta position on that one IWM we've got an iron condor on here. Just waiting for some more time to pass. We're still within range here I mentioned IYR J and J man. This one's a little frustrating If we would have just you know, we we had a decent profit It was just we were just waiting for a little bit more upside when price was up here to book a profit unfortunately J and J did not follow the rest of the market Which is a little frustrating came back down into center and so now we're just we're whole we're Currently in a little bit of a losing position here but just waiting for potential expansion Inimplied volatility Which we've gotten but uh, but this price movement has gotten this back down into range So we need a decent price move either up or down before we get to earnings And earnings is 416. So we've got a little bit of time there But hopefully we get a little bit of decent price movement to get a get a profit on that one I mentioned JP Morgan. I mentioned lulu I mentioned netflix. I mentioned nvidia. I mentioned cues. I mentioned smh. I mentioned spy. I mentioned wba xlk And then lastly xrt So xrt we've got this adjusted strangle price is just kind of hanging out here near the upper end of the range Like a lot of these just need a little bit of downside movement and some more ivy contraction and uh Before we get out of that one. We're still in may we're we're out in may in this so Plenty of time 42 days to expiration in that one So nothing to do there So that's it guys. That's all the alerts. That's all the positions. Everybody have a great week of trading stay mechanical Just you know, keep your keep your uh positions going do the necessary adjustments roles Uh, you know keep keep your short delta to theta ratio In check between that one to one and five to one at least that's what we like to do Uh and and stay mechanical this one directional move to the upside and stocks cannot last forever And uh, hopefully we get a little bit downside Next week that'll both help our current positions and give us the opportunity to enter some new positions So we'll see you next week. Have a great weekend


Mark your own would you like a quiz with which to follow along? Oh you did find it. I knew you would I Like number one. I like number one. I like number one number nine one is a nice question Suppose this mass begins from rest, okay Mass one is 2.8 kilograms mu is the covalent friction is point one two the angle of inclination is 52 degrees Find the time it would take for this mass to slide a distance of six meters down the slope So what this question is asking me to find is? T I Know vi is zero. I know D is six Do I know v final? Nope, I think What I'm really being asked to find is The acceleration because if I find the acceleration then I think I can use d equals vi t plus a half a t squared and I can find t So even though this question is saying find the time I've been I'm gonna be spending most of my time. I think Finding the acceleration. How am I gonna do that? I think I'm gonna label my forces M1g normal force friction I Think I would break M1g into two components parallel and perpendicular M1g perpendicular M1g parallel and we said that this angle and this angle and that being the same which they do So to find the acceleration I'm gonna ask my side. Who's winning? Well, which way is this thing accelerating? Down the hill Who's winning then? Okay, so my equation is gonna be M1g parallel So I'm gonna list all the forces that are parallel to the other one is friction winner or loser What if I was sliding up the hill? Which way would friction be acting? In fact, I would have two winners.   That's a deafening silence makes me a little worried friction is what times what I Don't know the normal force. Oh, but look look look look look I know another force the same size as the normal force in an 8mg. It is M1g perpendicular Let's do a bit of trig parallel is opposite High pot news. I think parallel is sine M1g Sign of was it 52 degrees? Yeah Minus mu and I think perpendicular ends up being M1g Cosine of 52 degrees and that equals M a and this one's kind of nice Cara. Is there a mass in every single term? then in this case Masses cancel I'll take that. I think the acceleration is gonna end up being 9.8 sine 52 Minus what I say the coefficient of friction was point one two nine point eight Coasts 52 and I get an acceleration of something or other Let's see. Oh, I went through this yesterday. You made it Glad you made it. Let's try this again 9.8 sine 52 Minus point one two times 9.8 times the cosine of 52 and I get a lovely acceleration of 6.998. I was tempted to round that to 7 But since this isn't my final answer, I'll actually carry Mitchell 6.998. I'll carry a few extra sig figs Probably won't make a difference, but I'll be paranoid Now I'd like to find t. I know that D equals vi t plus a half a t squared, but vi is zero Which means that? D equals A t squared over 2. There's the one half. I think what this really means is that T equals 2 times D divided by a Square root which is going to be 2 times 6 divided by 6.998 square root 2 times 6 divided by answer Square root of that you guys get an answer of 1.31 seconds people nodding So that's for full marks. If you're wondering how I gave out part marks on this one. I did something like this Did I give you quiz to version 1 or version 2? What's it say on the top? Version 1 that's how I gave out the part marks But if you got the final answer One mark, but a half mark if I saw that half mark for that one mark for the acceleration Half mark for writing that half mark for that half mark for the answer question to Atwood machine Did they give me both masses? Then I'll treat this as one mass. I'll go winner minus loser equals mass of both times a they only give me one mass They'll give me enough information that I can target that one mass and find all the other stuff and use the other stuff To find the missing mass Anyhow when I did that I got an acceleration of 1.75 meters per second squared And I got a tension of 61 newtons Yes, I think atwood machines are kind of from last year, so I didn't do a big song and dance about that one if you would 61 newtons. Yeah No, yes, and I used m1 because tension was winning and it was easier to get tensioned by itself a student of mass m Places a bathroom scale on the skateboard. What does the scale measure does a scale measure mass? Now what is measures normal force so what this question is talk as they talk about a scale I'm always thinking to myself self normal force normal force normal force He then rolls forward on level ground while standing on the scale, which is still on top of the skateboard What will the scale read more than m less than m or m? Yep, I think so Now the student rolls down the skateboard Sorry rolls the skateboard down a ramp which is inclined at an angle I was typing this up on a Mac years ago, and I couldn't find the theta But I could find that funky symbol. It's a fi I think but anyways at some angle while standing on the scale which is still on top of the skateboard What will the scale read more than m less than m or m? Brett I agree Convince me Now if you just write that on a quiz, I'll give you marks on a test So here's what I said scale measures normal force and I said the normal force is equal to mg parallel Probably if you drew a little ramp with a good free body diagram that would really help and mg parallel I said in this case is mg coasts and It's always going to be less than mg because cosine is a decimal, which means if you're multiplying by decimal your answer is always going to be smaller If you answered something along those lines if you use the black a be approach if you said well If it was vertical normal forces zero and level if you use the black a be theorem I'll give you full marks there if you're not quite sure how to mark this one lawyer with me afterwards and The last one Assume no friction. I would have no problem including friction by the way. It means you got an extra force one extra force If mu 1 equals 10 and mute. Sorry mass 1 equals 10 and mass 2 equals 4 and the angle of inclination is 56 degrees What did I ask you to find tension? Oh? acceleration and then tension Okay What's the first thing I'm going to do here? Well not dealt the pictures there, but I labeled my free body diagram So I said on mass 1 it's going to be gravity down tension up on mass to gravity straight down tension along the rope and Which is the bigger mass? I? Know this one is when enough this one is bigger It would be a bit trickier because depending on the incline this might not be enough to make it down the hill But if the bigger mass is hanging in mid-air, that's going to be the winner on a hill So I said okay friction is in that direction although did I include friction? Why do I have friction here? Oh? Did I do a part C? Oh for part C gotcha So I added friction later on for part C. That makes sense for now. I said M1g is winning Tension losing tension winning and M2g parallel is losing Winner minus loser plus winner minus loser equals the mass of both of them times a Tensions cancel M1g M2g parallel was M2g sign I Divided by the mass of both of them to get the a by itself and I got 4.7 meters per second square Then to find tension I could have used this mass, but this mass is much cleaner Tension is what no M1g is winning tension is losing M1g is winning tension is losing equals M1a because Joel looking at a single mass But it's pretty easy to get tension by itself. I plus the t over here. I'm minus the M1a over there and Tension Zay was 51 Newtons Part C What would the minimum magnitude of the force of friction have to be in order to keep this entire system stationary if it's Stationary, what's the acceleration? Zero I said okay really what we're saying is This Oh, I just looked at one mass tension, which was 51 minus M2g That has to cancel out friction You could done a bigger equation. I just looked at the one man.  I don't think I did a great job of explaining that but I'm seeing a bunch of you nodding your heads like you got it anyways, so Is that right? Yes So if you can be so kind as to give yourself a score out of whatever this is out of 17 So any questions from last day's homework? Gave you a couple of nasties. We looked at the lawnmower question. There is going to be a lawnmower type question on your test I'll tell you that right now where The picture looks something like this although I May also instead of pushing down. I may have you pulling up with a rope instead In which case your normal force will be a little bit different back. It'll be less not bigger Anyhow any questions from the homework that you wanted me to go over? I assigned one two three and I think that was it. Yeah Yeah Sorry to love to so Here's my mass We have good old Mg down now. It says it's being pulled upwards at an angle like that Brett With 300 newtons of force Definitely there would be friction in this direction. Is that okay, Brett? and because I'm pulling up I'm cancelling out some of gravity. I'm pretty sure the normal force is smaller than gravity. In fact, I know it is Because you're lifting up as you pull up and The normal force is always cancelling out mg. Well, no you're since you're cancelling out some bit It's only cancelling out what's left. Is that okay? I Would also then call this fx and Fy a says find the normal force Well, the normal force this time it looks to me Kayla that Fn that plus fy equals mg the two ups cancel out the down Normal force plus fy equals mg. Is that okay, Brett? Which means normal force is Is going to be mg minus Fy and fy I think it ends up being signed But I are you okay on the the trig part if I let you take that part on your own? Yes, right. Here's your angle opposite. I think it's ended ends up being signed in this case I think this is backward y is signed again because we're not at a wonky angle B Friction is going to be mu times whatever your answer was in part a I shouldn't put an a there You'll think that's acceleration. It's going to be mu times the normal force, which is from there Who's winning? not the 300 The 300 only be winning if you're accelerating upwards and diagonally, but which way are you accelerating to the right? Which forces the to the right force? Yeah, okay, it's going to be winner minus loser equals ma FX I think it's going to be the 300 times cosine, but I have to do the trig to double-check that Friction you just figured out in part D Mu times a normal force and you figure the normal force out in part Is that enough of a kickstart? I think Yeah, I'll let you crunch the numbers by the way, um What if it was not accelerating what if it refused to move what would your acceleration be? Zero what could you tell me about these two forces them? They're the same size that might give you a different way to find friction Certainly if you didn't know the mass that would give you a different way because you'd know the force that you were tugging at it Wasn't moving you can figure it out that way any others There's only three questions, so any of you have any questions from the great big ultimate forces review now? Remember I would like you to hand in This When you write your test if all possible I'll take it afterwards But it's really dumb to do this after you've written the test if you haven't clued in if you go through this You'll have seen about 95% of the questions on the test and The answer keys are online so you can double-check The answer keys are attached, but the actual answer solution key showing all my steps is online Any of these that you glance that you're going and by the way some of these are nasty which is great The five not too bad a little bit nasty. Oh I like number 10 17 is pretty tough Anyhow any of these you want me to go over Number three went too far. Mr. Duke should have stopped there Okay, I Think I sort of like number three in that I think there's a question on your test I think it's multiple choice. I think where I say to you. Hey, I didn't give you one of the masses. What you're gonna do Okay, here it wants you to find the mystery mass How am I gonna start out here Caitlyn? Same old same old absolutely Mitch you want to sit up and follow along maybe just be thinking Maybe you want to get out the review if you've lost the review I might have a couple extra copies floating around or else. It's online. You print it up Uh Caitlyn I would go like this M1g tension yes, and M2g Normal force tension friction be one extra force if they did Do I know both masses? Then I can't use my winter minus loser equals m1 plus in fact I have to start this one out by looking at an individual mass which mass will look at the one that I know I'm gonna write a force equation just for this guy just for this guy Who's winning? tension Who's losing? No one although if I had friction it wouldn't be that hard care for me to add in friction you times a normal force I don't know the normal force. Oh, but look look look look it's mg in this case and Because I'm just looking at one mass Cara. It's gonna be M1a Do I know mass one? Sorry m2a I'm looking at this mass my bet. Do I know mass two? Yes. Hey. Oh, I can get the tension. I'm pretty easy Tension is going to be four times one point five pack. I can do that in my head tension is gonna be six Newton's Can you see where I'm gonna go now? What Now I'm gonna go look at mass one Who's winning? Who's losing? M1g minus tension equals m1a Do I know m1? No, do I know g? So I'm gonna write this as 9.8 m1 because we're used to having the number and then the variable from math 9. I'm gonna treat m1 like an x Do I know tension yet minus six? Equals do I know m1? No, do I know it yet? 1.5? M1 Caitlin how would I solve that wouldn't I get the m1s to the same side? How? I think I'd minus the 9.8 over because I don't want to have nothing over here So I'm gonna minus the 9.8 m1 and I'll get Negative six equals whatever 1.5 take away 9.8 because those are like terms, right? It's like if like m1 is an x It's really 9.8 x minus 6 equals 1.5 x 1.5 take away 9.8 negative 8.3 m1 Caitlyn final step. How could I get the m1 by itself and Wonderfully the negatives cancel I get a positive mass, which is good because we're not dealing with antimatter here or anything like that, right? Negative 6 divided by negative 8.3 and you end up with a mass of 0.7 222 B So if they don't give me a mass no big whoop Look at the one mass that you do know find everything you can from there Bring whatever you found over to the mass that you don't know and solve for that single solitary mass Is that okay? Yep Any others I Think there's one more like that somewhere else in this by the way We also find it was one or two more where you're finding a mass then we're gonna begin the next unit Which I'm gonna teach as though it's brand new. I covered a bit of it I don't think that's your kamosy covered any of it. So I'm gonna teach it from scratch So we're gonna start this next lesson. I don't think I'll get through this whole lesson. I might we'll see Lots of terminology here lesson one unit three work and energy work first of all work is a physics term and we got a bit of a problem here Joel because It's also an English word when you guys think I'm doing work You think I'm getting tired getting sweaty that that's not what we mean in physics It has a specific physics meaning now. It is a scalar Would I be a good teacher if I confiscated that game for a couple of days as A wonderful lesson about the important of education versus going and standing in line in a store and missing Things I might right now. I'm on the fence in other words. Don't push me, right? It is defined as the product of the force component in the direction of motion and the distance moved Say what work is defined as this? Force times distance. Oh Except both of these have to be in the same Direction in the same Direction the equation then symbol for work is a capital W and the equation is force times distance Fd what are the units for work? Well, what do I measure force in? Newton's what do I measure distance in? It's a Newton meter, but this one is so important. We've given it its own name We call this a jewel symbol capital J named after a scientist whose last name was Example one How much work do you do when you lift a 35 kilogram mass? To a height of 1.5 meters well Work equals force times distance Which force was he resisting? The force is mg and you know what we actually frequently call a vertical distance a Height we often use back. We'll talk about Work done against gravity. We're gonna call it mg h The mass is 35 G is 9.8 Isn't it negative it's a scalar and H is 1.5 meters. How many jewels of work did Joel do? I know it feels strange using the word work and Joel in the same sentence, but well, you know use your imagination What do you get? 500 540 514 point what so not 514 Round off properly 515 Five hundred and fifteen jewels of work. Okay Is that a lot you know what? 515 you wouldn't want to do that 20 times in a row. You'd be tired So that lets you know kind of 500 jewels of energy is a reasonable chunk How much work does a hiker do on a 50 kilogram backpack if he carries it to the top of the 750 meter mountain? Well, here's the nice thing don't write this down But no matter what the mountain side looks like I really don't care because the only force that this hiker is Resisting is gravity, which means all I'm interested in is the vertical displacement I don't care about any of the horizontal junk that went on none of this matters. All I know is work Is force times distance and if you're carrying something up a mountain side the force that you're doing work against is gravity and The distance up a mountain side is H. It's going to be 50 Times 9.8 Times 750 What do you get? 367 thousand five hundred three six seven five zero zero now we better do our sig figs So how about three point six eight times ten to the one two three four? jewels 50 kilograms How many pounds is 50 kilograms? You're not sure multiply it by 2.2. What's 50 times 2.2? What's 50 times 2.2? anyone Sorry 110 so that'd be like piggybacking someone who weighed 110 pounds up a mountain side. Would you be tired? Probably in other words 367,500 jewels that's a fair chunk of work the ball at the end of the example three the ball at the end of the rope is Following a circular path Because of a constant force exerted on the ball by the rope in a direction towards the center of the circle How much work is done on the ball by this force? Look I'm doing this Okay How much work is this string doing on the mass on the end of this? Why is the answer zero? Why is the answer? It is zero by the way. Why is the answer zero? First of all, what did I say work is what times what? But what was the condition about the force and the distance? Same direction now read the question. Which way does it say the force is acting? Towards the center let's pause this suppose I could freeze it right here. The force is acting towards the center But which way is the ball moving? That way are the force and the distance in the same direction then we would say mathematically I'm doing no work I'm doing no work Push it now if I have Trevor do that for a long time will you get tired? Will you get tired? Yes. He's not doing any work though. Wait a minute. He's getting tired has nothing to do Thank you Trevor has nothing to do with work work is what times what? Force times distance Kara was he exerting a force? Yes, was the wall moving a distance? No No work being done Well, why is it getting tired then okay now we're gonna get a little bit specific actually Trevor you were doing work But not on the wall when you were giving Trevor a piggyback you were doing work and not on the wall What was happening was your muscles vertically were compressing and shrinking compressing and shrinking and there's the force and in the Same distance as gravity. Okay, so I should I should be clarifying that in general a Force in the same direction of motion like f1 on the next page Does positive work a? Force in the opposite direction of motion like f2 does negative work You can't have negative work and positive work positive work means you're gaining energy Negative work means you're losing energy. Oh I'm a physics nerd and I threw that in on the side need to figure out how to do this mr. Pen in 1977 turn your paper sideways. Let's see what it says here in Northamshire, England David Perley's race car crashed and his speed dropped from 174 kilometers per hour to zero Kayla in point six six meters He had 29 fractures three dislocations his heart stopped six times But he survived as far as we know this is the human being who has experienced the most G forces and lived to tell about it How many G's did he experience? All right, let's see if we can calculate this guy's acceleration What's the initial 174 kilometers per hour what 174 what I Can't do math with kilometer. How do I go from kilometers per hour to meters per second times by 3.6 or divide by 3.6 which one I heard both which one I mean again, I'm hearing both which one I'll give you an easy way to figure it out What's the speed limit on the freeway? You have your driver's licenses you're getting what's the speed limit on the freeway the easy one No freeway not hot freeway Hundred you've all been on the freeway speed limit there is a hundred. Yes Can you go our vendor at 100 times 3.6, please? Okay, it's okay with me here. Are you traveling so you're saying on the freeway you're traveling 360 meters every second You're traveling three football fields every second on the freeway. I Don't think you times by 3.6. You know what you do that by the way, that's how I figure it out every time I say well a hundred times 3.6. No on the freeway. I don't go for football fields a second not a chance So it's divided by 3.6. What's vi? No, it's a hundred and seventy four divided by three point six. Come on. What's vi? Sorry forty eight point three What's the final? What's D? What equation has those and a in it? I think it's going to be vf squared minus vi squared all over 2d What acceleration did this race car driver experience and survive? Sorry, I can't do this up here because I got my screen flipped sideways Which means my well if I do that it'll just terrify you guys What do you get? Where's your calculator? Sorry Okay, what'd you get? Would you get a little more participation would help boys and girls you get six hundred ninety nine? I? Don't know I can't do mine up here because my calculator sideways. What do you get? It's why I want more if you try this by the way Brett how many numbers are on top to you to put the top in brackets. Yes. What do you get? I? Need a consensus answer here folks. What'd you get Brett? No, no, no, no, what'd you get? Okay? It's way 1,767 anybody else yes anybody else Caitlyn would you get Sorry, how many G's is that divided by 9.8, please? 180 G's in that car accident that's the equivalent of 180 people doing a dog pile on him and he survived Okay So he's the person at least four years ago when I last checked He is the person who has survived the biggest deceleration and lived to tell about it Or at least lived long enough to tell people there may have been people that survived for a brief moment after an accident In something more than this, but they didn't survive long term here in this the nerd part turn the page Or next page over So we said this work can be positive or negative Work in the same direction of motion is positive work work in the opposite direction of motion is negative work Although work can be positive or negative It doesn't have direction. This doesn't mean Direction it means losing or gaining Child is pulling a wagon along the driveway and they're exerting a force F along the handle of the wagon How would you calculate the work done by the child on the wagon? Here's the wagon There's my force F Here angle theta What direction is the wagon moving? Dagonally upwards No, which way is the wagon going to roll? So I need the force that's in the same direction because work Equals force times distance. Yes, but the force in the distance have to be in the same direction in fact What you would want to use is fx and I think you're gonna find it the cosine of theta Equals fx over f in fact Emily if I wanted to get fx by itself What would I do and as a matter of fact? The actual formula for work is f times d cosine theta What if theta is zero degrees? What is the cosine of zero degrees? Can you try that on your calculator right now really quickly? Rihanna, what is it one because that means you're in the same direction anything? Oh, what if theta is a hundred eighty degrees? What's the what if you're pulling backwards? What's the cosine of a hundred eighty degrees? Negative one. That's why we said in this direction is positive work in this direction is negative work. Oh What if you're pulling straight up? What's the cosine of 90 degrees? zero no work and Matt some of that should be familiar to you from our circle from last year But that's one of the many many reasons why the trig values have the values that they do work equals what times what's force times distance ah? Only if the force is constant what if the force is not constant What if the force is changing? Well as it turns out work is also the area Under a force versus distance graph Here's a graph We're graphing force Versus distance and it looks like the force is increasing when we started out It was one Newton, but eventually the force finished up at three newtons And the question asks how much work is done moving this object from two meters to six meters? I guess we're pushing on something Well, there's two meters There's six meters You know how much work was done and moving the object? That much work the area Now I always ask this because maybe there is a math nerd here. What shape is that that we've just colored in first of all Is it a square? No, is it a triangle? No, is it a rectangle? No This is a side that has this is a shape that has two parallel sides and two non parallel sides What do we call that I? Heard someone say it Trapezoid called a trapezoid now Believe it or not guys with me back there Okay, please stop because every time I can't that's back there you chatting Believe it or not if it's about physics good, but still there is the formula for the area of a trapezoid there is one Does anybody know it? I'll tell you why you don't know it. You don't need it Every single trapezoid is also automatically a triangle and a rectangle draw that line in and Call the top area area one and the bottom area area two the reason we don't make you memorize the formula for the area But trapezoid is nice. It's one half a plus b times h for what it's worth is well Do you know the area of a rectangle? What's the equation for the area of a rectangle? Really? Length times width we usually say but base times height works Do you know the equation for the area of a triangle? What's the area of a triangle? It has base and height in it Brianna Because a triangle is actually also half of a rectangle. So that's why it's divided by two So since you know both of those in theory We don't memorize a trapezoid Area one is going to be base times height divided by two. It's a triangle. How long is the base of that triangle? Look at your graph sorry For what's the height of the triangle Jacob?  Can you go four times one divided by two? Yeah, what? Okay, so the top area has two joules Area two is a rectangle which is length times width or base times height I really don't care how you think about it. What's the length of the rectangle Emily? What's the width of the rectangle Emily? What's the area of the rectangle Emily? So the total work 10.0 joules if that was one person and they were pushing something they lost 10 joules of energy and That object gained 10 joules of energy You got 10 joules tired of Doing this Okay, so there's actually now two expressions for work force times distance if the force is not changing Or it's the area under a force times distance graph Which makes sense to me by the way because didn't you tell me that area was length times width Force times distance if I look at the units it should be the area energy Energy is the ability to do work if you have the ability to exert a force over a distance then you have energy It's amazing. I said the word energy and I saw four yawns You were the final don't worry Kara was the initial There's two in between it's amazing Energy is the ability to do work If you're asking does an object have energy what you're really asking is can it exert a force over a distance? Does this gun have energy my Nerf dart gun? Can it exert a force over a distance on the projectile? Yes It has stored energy we're gonna call that potential energy. There's all sorts of different types of energy the energy of motion we call this ke stands for Kinetic energy, so I'm gonna fire this dart gun at Matt Does this dart have kinetic energy? Yes, when it hit Matt hits Matt it can exert a very small force over a distance But it's moving anything that's moving and has mass has kinetic energy Has kinetic energy Okay, you tried to catch it. I figured you'd be athletic enough to dodge or catch it That's why I didn't pick Well, no Brett would have just flopped and taken the charge, right? Yeah, it's true okay If I had a baseball And I said I've had enough of jewel and I'm gonna throw this right at his jaw As hard as I can Does the baseball have energy when it hits him will it be able to exert a force over a distance? Anything moving has kinetic energy We're gonna be looking an awful lot at kinetic energy this year heat Heat is another form of energy Heat is usually created by friction Can heat do work? How do you think your car engines work? I think steam engines work. Yeah, absolutely heat can do work It can exert a force over a distance Gravitational potential energy this is stored energy due to an objects position It's stored the word for stored in physics is potential. It has potential energy But it hasn't been used yet I Wanted in fact you can give any object at all some gravitational potential energy just by lifting it up Which would you rather have land on your head this or that? So certainly when we come up with an equation for gravitational potential energy, I bet you mass is in it bigger mass more energy more I'll bet you a height is in it because if I raised it higher it would hurt even more Chemical potential energy gunpowder is a great example of chemical stored energy Your food the food that you eat is all chemical stored energy Your body breaks that down stores it in your cells and your cells can transform that into Kinetic energy to move your arms to move your heart It can transform it into heat your blood temperatures all are regulated at a certain level Other forms of energy sound energy Elastic energy, that's the elastic the energy stored in a rubber band Nuclear energy, that's the energy stored inside of a nucleus all sorts of different types of energy If there is a way for an object to exert a force over a distance If it has the ability to do work then that object has some energy example for Sometimes when we do work we make things move it says fill in the proof below to find an expression for motion energy So if an object is moving it has energy it has kinetic energy. What's the equation? Well first of all work equals force times distance and Force equals what times what from this past unit f equals f equals ma.  If I multiply by a half and I multiply by two don't they cancel each other out So I haven't changed the equation, but I got a good reason for doing this. You see two ad Vf squared equals vi squared Plus two a d. What's two ad? What's two ad the same as if I get the two ad by itself? I can replace the two i ad with the vf squared minus vi squared and If your initial velocity is zero you get a half mv squared See it That's the equation for kinetic energy kinetic energy is equal to a half mv Squared you don't realize it yet, but that equation there has a huge impact on athletics because of the squared Because of the squared so hockey Who's my hockey players here? Okay Hockey stick if I want to give more kinetic energy to the puck I Can make the stick twice as heavy that will let me give it twice as much energy Or I can swing ties as fast now if I swing twice as fast that to actually get squared I'll have four times as much energy Have sticks gotten heavier or lighter over the past few years Why lighter? Because it's all about velocity the more velocity you can the faster you can swing your stick The V is squared that gives you way more bang for your buck Otherwise, why wouldn't you just make a stick out of aluminum like a baseball bat in college? Well, you want it lighter to get more speed Tennis rackets tennis rackets have gotten lighter over the years, but the serves have gotten harder. Why? Again, if you double the mass you'll have twice as much energy But if you double the velocity because the two gets squared you look four times as much kinetic energy They'll go four times faster and go four times faster car accidents if You're in a car accident at 20 kilometers per hour Versus a car accident at 40 kilometers per hour twice as fast. You'll do four times more damage If you're in a car accident at 20 kilometers per hour Versus 80 kilometers per hour What's 80 divided by 24 times faster 16 times more damage? You're probably not gonna live half mv square No vector symbols because kinetic energy is a scalar By the way, just to give you some frame of reference about energy and jewels One jelly doughnut contains about 10 to the 6th jewels of stored chemical energy and One mosquito push-up Is about 10 to the negative 13 jelly doughnuts. It's about one herd just for the nerd. Yes. That's a mosquito push-up Yes, I looked it up turn the page Let's assume. It's a mosquito doing a push-up. Would that not be what a mosquito push-up is? No Next page so a golfer wishes to improve his driving distance. What would have more effect doubling the mass or doubling the speed? Doubling the speed. Oh have golf clubs gotten lighter over the past 20 years. Yes Are people hitting it harder?  How do you hit hard? It's all about club head speed Nothing to do with strength nothing to do with how heavy the driver is Club head speed the more club head speed you can generate the harder the further the ball is going to go. Oh Since kinetic energy equals a half MV squared Doubling the mass gives you double the kinetic energy Doubling the velocity gives you four times the kinetic energy four times the kinetic energy Baseball players so I have baseball players in here Okay, they always talk about bat speed That's what sends the ball out of the park how strong you are how heavy the bat is It's all about how fast you can get the bat moving and if you can get the bat moving faster Any increase you gain by the square of the increase which means way more bang for your buck Sometimes when we do work we lift an object and we give it height energy Also called gravitational potential energy Fill in the proof below to find an expression for height energy Also called gravitational potential energy or PE for short work equals what times what? What did we say at the beginning of this lesson work equals what time what Katie? Okay, so let's fill that in force times distance If you're lifting something which force are you having to resist? MG and we call a vertical distance a height gravitational potential energy is mgh Both of those are on your formula sheet in fact I'm pretty sure those are the only three formulas in this hole. No never mind There's one more formula. We'll look at power in this unit, but on your formula sheet The four formulas in this unit Work equals force times distance kinetic energy equals a half MV squared Potential energy equals mgh and the one for power which we'll get to next day Usually in this unit we're going to focus on mechanical energy Mechanical energy has two main types Kinetic and potential We almost always say an object is moving or it isn't and it has some height or it doesn't we're going to pause here What's your homework? You're writing a test next class Sometime during your one of your blocks. I'll do a tutorial today after school and Study


one Caleb here with the late in sports cards next tonight it's gonna be for Tom B one of the sports card Almanac good luck man thank you very much volume number two personal box edition let's do it Tom box number 86 yes it is good luck Tom Tom be here let's do it fair for us fair fair man okay Tom that's a nice shield one-up one Michael Carter NFL rookie patch autograph gorgeous shield from a limited 20-20 football there 20-20-21 that is nice Michael Carter one-up one from a limited gorgeous shield Jets card there you go Tom sick garden coming your way thanks again Tom appreciate you hopping in we'll have him protected and right I'll see you


The simplest way to break the hydrogen bond is by imposing some thermal energy, but it is expensive. We know that when we heat an object, we tend to increase its temperature, but that's not always the case. Sometimes you can apply heat to an object. That heat, that thermal energy goes into rearranging the molecular structure, which is otherwise known as a change of phase. This occurs for water when you change its phase from solid water, which is ice, to liquid water, and then also from liquid water to water vapour into its gas phase. Here we have a simple experiment that demonstrates the change of phase of water. We start out by placing a dish containing ice onto a burner and watching it with a thermal camera where the colour indicates the temperature. This purple-blackish colour is below zero Celsius and this white colour is above 100 Celsius. The ice is initially below zero Celsius, which we can see by its dark colour. As the dish of ice begins to warm, some of the water changes phase from solid to liquid and we can see the convective circulation of the liquid water and the temperature of this liquid water is increasing. Solid water, however, remains at zero Celsius, even though it is completely surrounded by much warmer liquid water. The size of the solid water decreases as the ice melts, but the temperature of the ice does not increase. That is, the heat that we are pumping into the ice is not changing its temperature, but it is leading to an internal rearrangement of the molecular structure and it's changing the phase of water from solid to liquid.


Thanks, Stefano. That's a great pleasure to me to speak here again, and especially because in the audience we have people that finish PhD in Maceo after a period. So today I will speak about a problem that I've been working for a while. And I would like to talk, I divide the talk in three parts. The first part is a main question that I want to discuss. The second is an example where we can get uniqueness of equilibrium measure for some partially hyperbolic host shoes. And finally I would like to discuss about maximum entropy measures for random local details. So let's begin with the main question that I would like to discuss. So let's assume that we have a map, a C1 map from a manifold. And given a point, let's denote by lambda 1, lambda 2, lambda d. The Lyapunov exponents at this point, if they exist, they don't need to exist. So if they exist, I will write this. So let's denote by ek, the set of points, such that the Lyapunov exponents exist at this point. And we have exactly k negative exponents and d minus k positive exponents. So ek is an invariant set, maybe it's an empty set, but I don't care. And maybe probably it's a non-compact set. So the main question that I would like to discuss is, can we prove that under some assumptions on the map f, the map f restrict to the set ek has at most one equilibrium state for any hold the potential fight. So I highlight the word restricted because I would like just to consider measures that see the set ek. So that's the question that I would like to start. So ek is the set of points with k negative and d minus k positive exponents. Actually, we can split the manifold. We can decompose the manifold as the set is zero. So the set of points with positive exponents only, e1 up to ed. So this is the number of negative exponents. Yes, this index is the number of negative exponents. So here we have the set of points with only negative exponents. And here, let's see, we have the rest. The set of points with some zero exponents or the set of points that do not have exponents. So each of these sets are invariant. I would like to discuss the map restricted to each one. So the conjecture is, if you assume something reasonable, I will explain what I mean by reasonable. Then any hold the potential has a unique equilibrium state on the set, restrict to the set among the measures that see the set. Again, the set can be empty or the measures that are invariant maybe don't have an invariant measure on the set. So by mild condition, I mean that condition about mixing properties of f on the set ek and conditions about the critical points if they exist, or the splitings, the Oseladette splitting. More precisely, for instance, a problem, a more precisely problem. Let's assume that you have a topologically mixing map such that the Oseladette splitings dominate on the set ek. Can you get uniqueness of equilibrium state for hold the potentials on the set ek? Or in a similar way, let's assume topologically mixing for local difference only. And let's consider, for instance, the situation of E0. Can we get that any hold the potential has a unique measure, unique maximizing measure, unique equilibrium state restrict to E0. So by restrict, again, I mean the measure that maximizes this, the entropy plus the integral, over the measures that give full measure to the set ek. So we get consequences if you can prove this. So maybe you can get real unique equilibrium states, not only restrict the equilibrium state, but a real unique equilibrium state in some situations. When we can get that the set ek has some, plays some role in the pressure. For instance, maybe if you can get uniqueness on the set ek, and we can, if you assume that the potential satisfied something like this, the pressure of the complement of ek is smaller than the pressure of ek, we can get uniqueness of equilibrium state. And another thing that we may get, somehow related with the talk of Felix and the talk of Juan, is that we have phase transitions when the measure decide to jump from one set ek to another set ek, from the set em to the set en. So maybe we have finite number of phase transitions. So that would be possible consequence for this. So we have now a lot of work that I would like to mention quickly, because the list is long, and most of them are here in the conference. So at symbolic level, we have breakthrough results by Sariq and Bouzi, and connecting maps with symbolic space. We have several works of Bruin, Pinheiro, Pezin, Saint-Zang, and we also have a different approach by Klemenhagen-Thompson via specification. So it's a different kind of specification, a weak specification property that one can use to get uniqueness of equilibrium states. And several examples from many people. Maybe an interesting example that I just realized, just read the paper, it's the example of Tazebi Rodrigues hats, where they get some examples of measures of maps, of partial hyperbolic maps on dimension three with two equilibrium states, but they have different signs. The entropy, the aponov exponent in the central direction, has one measure has positive sign, and the other measure has negative sign, and the map is topological mixing. For one-dimensional maps, we have several works. At dimension two, we have a recent theorem by Sariq that proves something like I stated before, and at dimension three, for flows, we have even more recent work by Lima and Ledrapie and Sariq. All of these examples and work are pretty much in the setting that I described before. I don't know a situation that are not in that kind of setting so far. I would like to know. I've seen people but so far I have not found any indication that that thing would not be true. I think that's true. So today I'm going to talk about two situations. The first situation is a partial hyperbolic host shoe that we studied with some co-authors, and the second situation is random maps. So let me start to describe what is the partial hyperbolic host shoe that I would like to discuss today. So maybe I'm going to jump this. I go to the picture. It's more... then I go back. So how we define the map? We have a cube, okay? And in the cube I will consider the direction x as a uniform contracting direction. The direction z is a uniform expanding direction, and the direction y is a central direction, and the central map is defined by this map, okay? So near zero you have expansion, near one you have contraction. So essentially the map does something like this. It contracts in the direction x, expands in the direction y, and then it folds in such a way that this side here, this pink side, enters in the cube again with face in the side of the cube, okay? So the formal definition is like this. So we define it on the... we divide the cube in three parts. So let's say r0 and r1. So the important part is r0, the part below the cube and r1 is the part above. So we fix constant, the expansion contraction constant. And we define, as I told you before, in the x-directional contraction, the z-directional expansion, and in the central direction, the map f. And the map f is the time one of our vector fields. So in the upper part of the cube, we have a hyperbolic map, okay? So we have contraction, expansion, and expansion again. But here, actually here's contraction. But here I do a flip in such a way that this map is pictured here. So what are the properties of this map? This map was introduced by Diaz and Orita, Sambarin and Hughes, and the map has some properties that you can check. The first property is that the point q, 0, 0, 0, is a fixed saddle with index two, and the point 0, 1, 0 is a fixed saddle with index one. So I'm going to show another picture very soon. So they prove that the map has a third-dimensional cycles, so the hyperbolic saddles q and p have different index, and they intersect. The stable manifold of p intersect the unstable manifold of q, and the stable manifold of q intersect the unstable manifold of p. The homo-clinic class of q is contained, it's trivial, it's q, and contained in the homo-clinic class of p, and the homo-clinic class of p is the maximum invariant set, and it's no wonder set of the map f. Okay, what else? It's possible to semi-conjugate this map with a horseshoe, and this horseshoe here is the horseshoe that do not allow the transition 1, 1. So we have a map that send a point in the cube in a sequence of 0's 1's, and in this sequence we do not have 1, 1. So this is a semi-conjugacy. And here is a very interesting feature of the horseshoe. So the map pi, the semi-conjugacy is not 1, 1. So we have some points that have a segment as a central manifold on the no wonder set. And this set corresponds, for instance, the pre-image of a cylinder like this is a collection of points that belongs to the same central manifold. So the homo-clinic class of p has infinitely many central curves that are not points. So this is something that they call now porcupine horseshoes. They have several spikes on the horseshoe and the spikes project by this map, by this projection in a single point in the shift, in the shift sigma 1, 1. So this is an interesting dynamic. Here is a picture of the heteroclinic test section that appears. The unstable manifold of p intersects the stable manifold of q. So in particular, if you take a point nearby here, the point eventually will follow this trajectory and will approach q again and will eventually return nearby here. So one problem that you may ask is what you can say about the invariant measures for these maps? We studied the invariant measures of this map, of this kind of horseshoe. And with Hennel Leplac and Isabelle Hughes, we obtained some results. The first result is the following. Given any continuous function, you may prove that that exists equilibrium states for this horseshoe. And you can prove also that there is a residual set of continuous function that the equilibrium state is unique. Pretty much because we are able to prove that the entropy is up as a continuous in this situation. One more interesting result that we have obtained is that in this kind of horseshoe we have phase transitions. So there exists a number such that if you consider the family, phi t, this family here, t times the logarithm of the derivative in the central direction and you study the equilibrium states of this family here, we were able to prove that there exists a number t0 and we can describe this number. So the number is the supremum of the entropy over the one minus the central, centrally upon of exponents where we take the supremum over all measures that are not equal to delta q. And this family has a phase transition meaning that for t bigger than t0, delta q is the unique equilibrium measure. For t smaller than t0, any equilibrium measure has negative central exponent. And for t equal t0, there are at least two equilibrium measures. One with positive exponent and another with negative central exponent. Actually the equilibrium measure with positive exponent is the delta q. So if you make a picture, maybe you will recognize this picture. We saw it before. The picture is something like this, a picture of the pressure. 2 phi t0. So here we have the point t0 where we get the phase transition. And the question that we pose to ourselves, so we can prove that we have uniqueness here, unique equilibrium measure. We can prove that all equilibrium measures that are for t in the left of t0 have the same sign of the exponent that have only negative exponents. So in particular they live on the set e2. Here the equilibrium measure would live in the set e1 and here they live in the set e2. But we were not able to show that they are unique in the left of t0. So that's the goal of the talk today to discuss a result in the direction of the uniqueness of the equilibrium state for potentials on this whole show. So the question is what conditions you can ask about the potential to guarantee uniqueness of the equilibrium measure? So there are some results that were obtained after we published the previous theorem. And the first result is that if you assume that the function is constant on central leaves then you have uniqueness. And also more recently, if you assume that the potential is constant on stable manifolds, in the case in the direction x, then you can show uniqueness as well. But so the approach that the second result of real cicada, the approach that they use is to consider a restriction of the map to center unstable manifolds and to show that there is a kind of Perron-Frabenius theorem for the map defined on levels of this whole show. So today I would like to talk about uniqueness of these, for all the potentials in general for this whole show. So we obtain uniqueness and we obtain uniqueness assuming the following conditions. So let's assume that the potential is holder and the pressure, the supreme of the entropy plus the integral over the set of measures that have only negative exponent is bigger than the supreme of phi. Then phi has a unique equilibrium measure. So that's the condition that we need about the potential phi to obtain uniqueness. Just a remark, the set of measures, in this example, the set of measures that have negative central exponent is every measure, every gothic measure minus the measure delta q. So we described this set in the before. So in particular I'm taking the supreme over all measures and accept delta q. And one situation where you can check this is when you have a potential such that the maximum minus minimum is more than topological entropy of this measure, 1 plus square root of 5 over 2. So let's now talk about different example, the same setting of same kind of question that I posed in the beginning of the talk. But the example is about random maps. The result is about random maps. So I'd like to start with the classical result about expanding maps. So for expanding map, we have that the topological entropy is equal to the logarithm of the number of pre-images. If you have a expanding map on a compact, connected Riemann manifold, you can show that the topological entropy is equal to the logarithm of p. p is the number of pre-images of any point. And you can show that there exists a unique maximum entropy measure. So that's the old result by Bohm. So a natural question would be to consider local defile morphism. Given a C1 local defile morphism, you may define the topological degree as the same, the number of pre-images of any point. And so we prove it under some non-uniform spanning conditions about the derivative. That exists a unique maximum entropy measure. So that is also a related result by Geron Bouzier. And let me explain what is the assumption about the map. The map, the local defile morphism is not a spanning map anymore, but we need that kind of expansion to get uniqueness of a maximum entropy measure and to get a formula for the topological entropy. So the underlying assumption here is that essentially that measures with big entropy have only positive level of exponents. So in this situation, the condition that I would describe in a second gives to me that any measure with big entropy will see only the set of points with positive exponents. So the set of points is zero. So what is the condition? The condition that I assume about the local defile morphism is written in terms of lambda q of the derivative, the exterior product of the derivative at any point where I assume that this inequality holds and k is smaller than the dimension. So pretty much I take the biggest rate of expansion for the derivative, for the k exterior product of the derivative among the manifold. And I consider k smaller than the dimension, so I take the maximum over all of these rates and I assume that this is smaller than the logarithm of p. So if this condition holds, so the map is not expanding anymore, but it has some expansion, and some expansion means this, and this implied this first assumption here that any measure with big entropy will have only positive level of exponents. So if I assume this, we can get that the topological entropy is equal to the logarithm of p, and we can get the uniqueness of the maximum entropy measure under mixing assumptions. Okay, so what's the problem that I would like to talk today? It was a result with Rafael Alvarez. So I would like to talk about the random compositions of local defiles. So what we can say about random compositions of local defiles. So what I mean by random compositions? I mean that we have an invertible map, theta, preserving a gothic invariant measure p, and I will take a family of c1 local defile morphism for each element of x, I will take a local defile morphism f of x, and with these two ingredients I construct a skew product like this. f at a point x, y is theta of x applied to f of x at y. So if you look at this and you start to iterate f, you can write for simplicity fn of x, y as theta n of x, and this component here gets more complicated. So fn of x at y is this expression. So it's what we call a random composition of the maps. So for each of these maps f of x we have a topological degree p of x, and the following diagram is commutative. So we have a projection, a natural projection from the skew product, from the product x times m on x. So I consider the set of measures, set of all invariant measures, such that the projection of the measure on the direction x, so this measure here is equal to p. I will consider only the set of measures, the set of measures that have marginal p, and we consider the set of measures among these that are invariant. So in the talk I just will talk about measures that when you project on x, you get p. p is fixed from the beginning. So everything that I'm going to talk will be relative to p. So let's discuss an example just to illustrate what I'm talking about. Take two local diffuses, f0 and f1, and let's consider the p0, the number of pre-image of x0, and p1, the number of pre-image of f1. Let's consider a particular measure p as the Bernoulli measure on the shift, such that the measure of the cylinder 1 is equal to alpha. So then we have an iterated function system with a Bernoulli measure on x, and we can ask about the maximum entropy measure of this system. But again, when I mean maximum entropy measure, I mean maximum entropy measure that has marginal p. So that we call a relative maximum entropy measure. So what we can prove with the result that I'm going to show in a moment. We can prove that if you take a map f1 as before, as in the result with Marcelo Viana that I mentioned before. So if you take a non-uniform like spending map in the particular sense that I discussed before, and if you take alpha close to 1 and you take any local d field p0, if you take alpha close to 1, then the topological entropy of f is equal to this average. 1 minus alpha log of p0 plus alpha log of p1. So you can translate this as if you take a non-uniformly expanding map in this sense, then you take non-uniformly expanding local d field in this sense and you take any other local d field if you start to iterate them randomly with respect to the Bernoulli measure. If the Bernoulli measure concentrates too much on the non-uniform map, the result will be non-uniform. The random map will be non-uniform. And you can get the entropy formula and the maximum entropy measure relative to the measure p. So that's an example just to illustrate what you're going to follow. So if we assume that the map f is exact, topologically exact, then we have that that exists a unique maximum entropy measure relative to the measure p. Okay. So here is the definition of what I mean by what is the relative metric entropy. So it's pretty much the same thing of the entropy. But now we're going to consider partitions that refine the partition by points in X. So the partition p is finer than this partition. So in particular, it's a partition, it's a partition on the fibers as well. So we can define the entropy as this, limit of this. And by this number here, this number here is defined as the integral over X of the entropy of the partition pn of X, the usual refining partition. And maybe I would explain what is it. This is the disintegration of the measure mu. Given a measure, an invariant measure, we can disintegrate it on the fibers using Hawke's integration theorem. And we get this system of measures. And to compute the entropy of the measure mu with respect to the partition p relative to the map theta, I need to average the entropy of this system of measures with respect to the measure p. Okay. So I will skip this. Maybe quickly mention what is the kind of functions that we consider. We consider functions, phi, that are continuous on the direction y, on the fibers. If you restrict the fibers, they are continuous. And if you look at the direction x, they are integrable with respect to p. So that's what we call L1 of X with respect to X, the collection of functions that are c1 with respect to X and continuous with respect to y. So under these conditions, you define the topological pressure as the usual. But again, you need to average with respect to p. And you may define the transfer operator as usual, but now you have a family of transfer operators. So for each x, you have a different transfer operator and the transfer operator depends on x. And you may define tools for these operators by the usual relation. So the eigenmeasure, we call eigenmeasure system, the maximum eigenmeasure system is defined as a system of measures such that this equality holds. A system of measures such that when you apply the dual to the measure mu of that ix, you get p of x, mu of x. This is exactly what you expect when you compare with the expanding case, the expanding situation. Okay. We assume that first assumption about the set of local differs. We assume that the logarithm of the derivative of the supreme of the derivative is integrable with respect to any measure. In particular, this is true when this is bounded. The example that I gave before. So that's the first assumption. And these assumptions are necessary for the, for instance, for the azela, that's theorem for the Brinkatox theorem and all other results that we have in erotic theory. So these assumptions are pretty much common to get the usual results in the deterministic setting. And here we have the similar that we mentioned before. For each number k, for each number k, I will define the lim soup of one over n logarithm of the stereo product of the derivative at y. I'll define this as ck. So this rate of expansion of k volume at the fx. And then we define ck of x as the maximum among y. And the assumption that we require is that the integral of the logarithm of px, so this is the average degree, logarithm of the degree, is bigger than this number. Okay? And also we need a kind of uniform continuity of the derivative at p almost every point. So this is a kind of uniform continuity of the derivative. Under these conditions, we can get the following. Under the condition 2, 3 and 4, we can get that the topological entropy of f relative to theta, so relative to the measure p, is equal to the integral of the logarithm of px with respect to p. And if mu is a measure that's a eigenmeasure for the transfer operator, so it's a maximal eigenmeasure system, then mu is a maximal entropy measure. Moreover, if you assume that the map f is topologically exact, then, and the degree of the map is bounded, the supreme of the degree, so this is p of x that appear above, then there exists a unique maximal entropy measure, and this measure is positive on open sets. Okay? So that's what we got. Here I check how we can get the condition 2. So remember that the main condition that we need to assume is this condition here, 3. So we need to show this condition if you want to apply the result. There are other conditions, but this is the most difficult to check. And if you want to check this in the example, if you want to check this in the example, you just notice that this is bounded, this number is bounded. You have only two maps, so this number here is L, this is a local diff, and then you may take the maximum among k of these two numbers, and then when you want to compute this, you can split in two parts. The collection of points that have a lot of ones, so remember the example. The example is we have two maps, and I have a Bernoulli measure that give a lot of measure to the cylinder 1. So when we iterate randomly F0 and F1, I see a lot of ones. I see a lot of functions F1 appearing at a rate alpha. So I may define the set of points that see, that the sequence of the sequences that have a lot of ones appearing up to the moment n. So by a lot, I mean at least a rate alpha tilde, and if I take alpha tilde smaller than alpha, I know that this set here, when n goes to infinity, the measure of this set will converge to one. And then if I want to compute this, I split in the set an, and in the complement of the set an. So then I can get that here, you get this estimate, and below we get this estimate. This part here goes to zero, because this measure here is going to zero when I divide by n, so I divide by n, everything. So this part here goes to zero, and then we get this kind of a bound for this number. If I make alpha tilde go into alpha, then I get this estimate here, that this number is smaller than this. And finally, if I take alpha going to one, I may get this going to zero, and I know that this is smaller than the logarithm of P1. Then we get inequality that we want, okay? I run out of time. Maybe I'm going to show, I jump to the strategy of the proof of the theorem one about the host show. So what do we do in theorem to prove theorem one? Essentially, we record the host show in an infinite shift. So remember the result about the host show, that the host show is naturally coded in a finite shift, but the code is not one-to-one. So using this structure, we record in an infinite shift, and the infinite shift is, the record is one-to-one. But we need to throw something away, and we show that what we throw is, has not equilibrium measure, if we assume the condition about the potential. So in other words, we check the condition under the condition about the potential, the scheme that we get is hyperbolic in this scheme, and we get uniqueness for this scheme. I have also this scheme for random local defaults, but I'm going to stop here. Thank you.