Well, good afternoon, everyone. Are there any questions from last time? So if you remember, in the last lecture, we completed the proof of the theorem that a small perturbation of a linear map, of a contracting linear map, is topologically conjugate to the linear map. Okay? And what I mentioned at the end of the last lecture was that the definition of contracting linear map, so let me remind you, if we have a linear map that's contracting, if the norm of t, which is defined as the supremum over all vectors of norm one of t of v, is less than one. So we proved that theorem in a very general setting of Banach spaces, so it's a very general theorem. Then I mentioned that if you remember, in the finite dimensional case, in particular in the two-dimensional case, the contracting condition is fairly strong. So for two-dimensional maps, t from r2 to r2, we have... we've seen many contracting... many examples of contracting maps like this, but we've also seen many examples of maps which are not contracting. This is not contracting because vectors are not contracted, like a vector at a certain distance from the origin may map further away from the origin temporarily, right? Because if you wait long enough, it's going to map closer to the origin. In fact, every point converges to the origin after iteration. So really the dynamical properties are very similar, and in fact, as we proved in the first part of the course, these two are linearly conjugate. They exhibit a very strong form of conjugacy, right? There is a linear map that conjugates these two. However, the theorem, as we stated it here, does not apply to this situation because the theorem, as we stated, assumes that the linear map is contracting in this sense. Do you remember what was the condition that implied that a linear map was linearly conjugate to a contracting map like that? Excuse me? Exactly. If all the eigenvalues have known less than one. Okay? So we can replace in the two-dimensional case, but in fact, we proved it in the two-dimensional case, but it's true in any dimensions, that if the eigenvalues all have known less than one, then the map is linearly conjugate to a contracting linear map that satisfies this condition, that the operator norm is less than one. Okay? So what we're going to show is that in fact, this linear conjugacy allows us to define a norm on the system such that in this norm, this linear map is contracting so that we can apply our theorem also in this case. So let me be a little bit more precise. So let me state our proposition like this. Let invertible linear map, which linearly conjugate to a linear map B, which is contracting. So I'm going to use that as an assumption. So as I said, in the two-dimensional case, this is always true as long as the eigenvalues of all have known less than one. In fact, this is also true in the higher-dimensional case, but because we did not prove it, I will actually use directly as an assumption the fact that we have a linear conjugacy to a contracting linear map. Then there exists a norm star on Rn such that the norm of this operator of the linear map A in this norm is less than one. So recall that if P is the conjugacy, let P be the conjugacy, the linear conjugacy between N and B. Then we have An equals P BN P minus one. If you remember, we saw that it's an easy calculation to show that if P conjugates A and B, then it conjugates all the iterates of A and B. And so the norm of An which is equal to the norm of P BN P minus one, which is less than or equal to the norm of P times the norm of BN times the norm of P minus one is less than or equal to... Sorry, maybe I should have...  So let me call this lambda less than one. So we have a specific value here. Then what we have here that B to the N of course is less than or equal to lambda to the N. So this is less than or equal to P times norm of P minus one times lambda to the N which we can write so An is less than or equal to C lambda to the N for all N less than or equal to one for C equal to P minus one. So this does not show that A is contracting because of course A in this norm is not necessarily contracting but it shows that it's eventually contracting. As we know, because the fact that it's linearly conjugate to a contracting map, of course means that also for A all the... everything must converge to the origin and the forward iteration. And this gives a little bit more an explicit estimate of this convergence to the origin of every orbit. It converges like this. So there's a constancy which in principle might be big. So this constancy, this kind of bound is very important. I'm sure you will have seen it in other situations before. What does this mean? This means that this C, in principle this C might be very big, right, arbitrarily big but it's a C that is fixed once and for all. It does not depend on N. So if C is very big like one million, okay, then when N is small, this is not... might not be contracting at all. The norm of A, the second iterate of A or the third iterate of A is allowed to be very big because this is one million times whatever this is. A half or half squared or half cubed or whatever. But as N gets bigger, then at some point lambda to the N becomes smaller than one over C and this becomes less than one, right? And so eventually when N becomes larger and larger this still goes to 0 exponentially fast at the rate lambda. When N is very big, the value of the constancy does no longer plays a big role. But it plays a crucial role in dealing with N small because this is not really contractual, okay? So we have shown that A is, let's call it, eventually contracting. And now we're going to use this fact to construct the new norm in which A is contracting. Now we fix two constants. We fix some lambda tilde between lambda and one. And we fix some N greater than or equal to one, sufficiently large so that C lambda over lambda tilde to the N minus one. So this is the condition. So notice because we've chosen lambda tilde bigger than lambda, then this is less than one and then for N sufficiently large we have this for whatever constancy we have, okay? So this is just the key little trick and then we define the new norm by V star equals the sum i equals 0 N minus one of lambda tilde minus i ai. And then using this definition we will get what we want. So in this case we just want to estimate. We use directly the definition. We have that AV, the standard of AV is equal to by definition the sum i equals 0 N minus one of lambda tilde minus i ai of AV. And we write this as the sum. I take one lambda tilde out and I write this i equals 0 to N minus one of lambda tilde minus i plus one of ai plus one of V. And I can write this as, so I change the indexing a little bit and I write this as lambda tilde sum i equals 0 of N minus one of lambda tilde minus i ai of V. So I've changed the indexing here from i plus one to i and that means I have some extra terms and if you check what extra terms we've got we've got one minus lambda tilde V here and one plus lambda tilde minus N minus one of a N of V here. And this is exactly the norm. So this here is equal to lambda tilde, the norm of V star minus. Okay, so here I have written the new norm of the image of AV as the old norm plus these two terms which have one negative term and one positive term. So this is lambda tilde is less than one. So as long as the bound is lambda tilde V star I have a contraction by a factor lambda tilde which is less than one. So all I need to show is that the sum of these two terms is negative so I need to show that the positive term is less in absolute value than the negative term. Okay, so I'm just going to write that here. So here we use the assumptions on lambda tilde and on N here and we get exactly the estimate the lambda tilde minus N minus one of a N of V is, so now we use the estimate that this is eventually contracting and this gives lambda tilde minus N minus one times C times lambda to the N and this gives exactly equal to, sorry, here I want AV and here this gives C lambda to the N minus one times lambda tilde to the minus N minus one times lambda V which is exactly what we want, right? Because then this assumption here says, so this is exactly C lambda over lambda tilde to the N minus one times lambda V and this is less than one so this is strictly less than lambda, than lambda V, yes, and this is strictly less than lambda tilde V which is exactly what we want for you. I may put what? I may put an equality. Well, but V equal to zero, here I may say that V is non-zero, yes. Okay, so this shows that we have constructed by this simple formula we have used the dynamics because you see this formula uses the dynamics, right? So we're defining a norm on the vector V in terms of the dynamics for the first number of digits and the fact that A is eventually contracting allows us to then plug this into the formula and show that this is in fact in this norm the map is contracting. Can I erase? Yes? As long as your map is linearly conjugate to contracting linear map then it has a norm in which it's contracting so the norm in which A is contracting is called an adapted norm because it's adapted to A because it's more natural norm because A is morally contracting because everything is converging to zero so in some sense that is really the adapted norm and we can prove a lot of the results where in this norm we can look at the linear map in this norm and then we can prove that theorem about the Lipschitz perturbation in this norm and then we can apply that to these cases as well. Okay, so to finish today this section on contractions I want to talk about some very interesting applications and in particular about some local versions of the theorems that we have been talking about. So I want to talk about something called local linearization and of course about structural stability for contracting maps. So I'm going to state some results.  Exactly, we can conclude everything exactly the same as for the contraction mapping theorem.  So then there exists unique fixed point P in U and Fn of X converges to P for all X in U and as n tends to infinity. Now we have two assumptions here about U can we relax either of these two assumptions because you could have for example X complete metric space think of Rn for example or think of R U is a subset of X which is not complete what's an example of a subset of R that's not complete some open set for example the open interval 0, 1 open interval and then what happens to the theorem is the theorem still true? No of course not right because you could have a contraction like X goes to half of X and every point converges to 0 but 0 does not belong to your set. What about the second condition is this is essential that F is assumed to be defined on the whole space on all of X. We could still have it that it's a contraction but if you could still be complete F could still be a contraction on U can we have F of U not contained in U? Why not? What do you mean by L and X? Yes I know but I don't understand your example. You can take easily an example in which you have a map on the real line that takes for example the unit interval it shrinks it but it maps it somewhere else and then it shrinks it again all towards infinity the whole time so it can be a contraction on U and even on all details of U but of course U does not map into itself so it might not have a fixed point. Probably you were thinking of similar kind of example like that. This is just an observation so I will not prove this because this is just a kind of trivial proof let me leave it as an exercise. It's just a question of making this observation that it's a contraction. But there's an interesting application here which is also local version. I'm going to write a whole list of local versions of theorems we proved in the first two lectures. So, proposition. Let F from Rn to Rn be a C1 map, P be a fixed point and suppose that Df of P is less than 1 then P is an attracting fixed point is a locally attracting. Do you remember what a locally attracting fixed point means? It means there's a neighborhood of point where everything converges to the fixed point. So what is the proof of this theorem? That's right again the mean value theorem or I mean yes exactly you can apply directly as a mean value theorem to show that this is a contraction in neighborhood of P and to show that there's a neighborhood that is mapped inside itself. What is the key property? Do we use the fact that it's a C1 map here? How do we use the fact that it's a C1 map? Remember that there's a difference between the assumptions of this proposition. It seems very simple but here we're only assuming this at the fixed point. We're not assuming this everywhere. This is a very important difference. In the previous proposition we said suppose the derivative is contracting everywhere. Suppose you mean it's at the fixed point. So even though this is an elementary result it's an example of a very important class of results where we make assumptions about the dynamics at one point and we deduce something about the dynamics in the neighborhood of that point. It's quite a significant step. So how are we able to pass from this information to the information about the neighborhood? In other words how do we show the neighborhood that maps into itself and on which F is a contraction? Yes. Yes, we have C1 and the differential is continuous. Exactly. So by continuity of the derivative the norm will also be strictly less than one in some neighborhood of P and then we can apply this result. This is a simple but very important step. To prove since F is C1 there exists a neighborhood U of P such that dFx is strictly less than one for all x in U. You're right, you're right, yes. I guess we don't really need x to be a complete metric space. As long as U is complete, yes. Okay, that's correct. Of course, we do not use the completeness of x in any way if we assume that U is complete.  And so we apply the local contraction map in theorem and we get the result. So again this comment about the norm this is a norm that we're given on Rn. As we said before, it's a little bit restrictive to assume that the linear map is contracting because we know that there's many linear maps that are still morally contracting but are not strictly contracting but they have an adapted norm. So is it sufficient to assume that this is true for the adapted norm? Do we still get the same result? In other words, I'd like to say suppose this is a C1 where P is a fixed point and the F of P is less than 1 in some adapted norm. So we suppose this is an adapted norm. The F of P is a linear map, right? Rn is a linear space, okay? We have a norm on linear space. In fact, we have many norms on the linear space. Not only that, but all the norms are equivalent. You know what it means? That they're equivalent. Equivalent means that the ratio between the two norms is bounded above and below uniformly for every vector, okay? So the important property about equivalent norms is that if you have a converging sequence, it will be converging in any of these norms. In particular, if the orbits converge to the origin, they will converge to the origin in any of these norms. These norms will converge to zero in all of these norms, okay? Which is what we're trying to show is that everything is converging to the fixed point so in a neighborhood of P, you have some contraction and everything is converging to P. So it is sufficient. So this norm is a norm on Rn. The norm on the derivative is defined in terms of the norm that we define on Rn. So if we change, if we take any norm on Rn, it's sufficient to get that the derivative is contracting in some norm in Rn to conclude that P is locally attracting in that norm but obviously if it's locally attracting in that norm, it's locally attracting topologically. It doesn't depend on the norm whether it's locally attracting or not. So this is an interesting result because this is now no longer in the world of linear maps, okay? We are slowly moving out of linear maps, okay? That was part of the proposition about Lipschitz perturbation of linear maps. Here also this is a nonlinear map but we're using the information we know about linear maps to say that if the linear map is contracting at a fixed point, then because the nonlinear map is in some sense in a small neighborhood of P, the linear map is an approximation to the nonlinear map, the fact that the linear map is contracting means that in a neighborhood of P the nonlinear map is also contracting and we get the same result. This is the model of this result, okay? Technically it's fairly simple but it's important to understand the spirit of the result. Okay, and a question then is, for example, is what about topological conjugacy, right? So let me state a local version of the topological conjugacy between the linear maps and the perturbations. Then we will have a very nice application of that. So actually you know what, let's just take a couple of minutes break, stretch, relax your minds a little bit and then we'll come back in two minutes and I will do this last part of the lecture. Okay, so to motivate just these last few results which I will state, let me ask the question, a natural question here, which is we now have two contractions, right? We have this linear map which is itself a linear map from Rn to Rn and it's a contraction and we understand the dynamics. And then we have the nonlinear map F which in a neighborhood of P is also contraction and in some ways it is a small perturbation of the linear map because the linear map is an approximation of the nonlinear map. So you see immediately we might think back at the motivation for the result that we proved in the last lecture which says that if a nonlinear map is a small lip sheets perturbation of a linear map then they're topologically conjugate. And we really, in some sense the main motivation for that was exactly this kind of setting, can we say that the nonlinear map in a neighborhood of P is topologically conjugate to this linear map in a neighborhood of the origin? For example, this is called local linearization. In other words, is the dynamics in a neighborhood of P conjugate to a linear version of it. So that's what I'm going to do now. I'm going to state these results formally. First of all, I need to state what I mean by a local topological conjugacy because these two maps cannot be globally topologically conjugate necessarily because we know that this map has a unique fixed point at the origin. The only thing we know about the nonlinear map is that in a neighborhood of P it is contracting because we've taken the neighborhood U of P with this property. But because this map is nonlinear outside this neighborhood it could be doing anything else. It could have other fixed points. It could be doing lots of crazy things. So the most we can hope for is that this conjugacy is between a neighborhood of P and a neighborhood of the origin which is the corresponding fixed point for this linear map. So let's define what we mean by a local topological conjugacy. So we'll state it in general. In a general setting of metric spaces we have two metric spaces and two maps, continuous maps. Suppose that P in X and Q in Y are fixed points for F and G respectively. Then we say F and G are locally topologically conjugate if there exists neighborhoods in P and Q of P and Q and a homeomorphism H from N, P and Q such that H composed with F equals G composed with H. And I need to add one more comment and that's the fact that these two neighborhoods are not necessarily invariant so this needs to hold whenever both sides are defined. So I will explain exactly what we mean here. So we have our two maps F, X to X and here we have a map G, Y to Y. Here we have a fixed point P. Here we have a fixed point Q. The maps F and G might not be topologically conjugate to each other on the whole space but we want to formalize the notion that in some neighborhood of P and in some neighborhood of Q the dynamics is topologically conjugate so it's the same. So what's the natural way to define it? Well, let's suppose we have a conjugacy in some neighborhoods so let's suppose we have some neighborhood here N, P and some neighborhood here N, Q and some homomorphism H. So what is the only problem here? The problem is that we do not know that N, P is mapped inside itself. In the case of contracting contractions we could more or less assume that or arrange that but this definition I'm giving is not assuming that there's any contraction. I just want to make a completely general definition and in many cases we want to apply this definition to situations which are not contracting and the situation which we're not contracting for example a saddle point, remember in the linear maps we have saddle points where we have one direction contracting the other one expanding you might have for example that the image of N, P is like this, this is F of N, P so it might be that there's some point X here that maps outside here which is F of X and H is not defined outside this neighborhood H or F of X so if we try to define this left hand side here it's not really defined at this point because H composes with F it means you need to take X, apply F and then compose H but it's not defined so whenever both sides are defined which can be, you could write in fact let me leave it as an exercise to find out exactly when this is defined because it is precisely defined for all points X in which F of X continues to be inside N, P then it's okay, right? So at least the left hand side is defined at every point so notice, wait let me write down exercise, describe explicitly explicitly the region where the conjugacy is well defined so the homeomorphism is well defined but the conjugacy is not well defined everywhere for example the left hand side so let me write as an example e.g. left hand side is well defined on the set of all X in N, P such that F of X belongs also to N, P which we could also write as N, P intersection F of N, P so I mean this is not as crazy as it seems it's not as abstract as it seems I just write it like this in the definition but there is a well defined region this is still a neighborhood of P so I'm just doing this to emphasize the subtlety of defining local topological conjugacy because you don't know that these domains are invariant by the map and so you have to be a little bit careful about exactly what you mean any questions about that? Is that clear? This little issue about the invariance of the domain in particular this still makes sense it still shows that as long as there are still some neighborhoods in which this is defined a neighborhood in which this is defined and as long as points stay inside this neighborhood then they are topologically conjugate so they match up with orbits that stay inside this neighborhood so for example if these were two contractions and N, P was all mapped inside itself and N, Q was mapped inside itself then we would have no problem here and everything would be well defined so we really have a local topological conjugacy between all the points in those neighborhoods and it's easy in the case of contractions so theorem, I write it as a theorem because it's quite important it's just a local version of the theorem we proved the other time so suppose A, N to R, N and A is a contraction in some adapted norm I always write in some adapted norm but after a while you know that assuming this is the case is always equivalent to assuming that it's the case in some adapted norm and suppose F from R, N to R, N is a non-linear map F equals A plus delta F with delta F of 0 equals 0 which means that 0 is a fixed point for the non-linear map also because F is the sum of the linear part and the non-linear part the linear part of course the origin is a fixed point I'm assuming here just for simplicity that delta F also fixes 0 so 0 is a fixed point for 0 so we have that A has a fixed point at 0 F also has a fixed point at 0 and there exists neighborhood N of 0 such that the Lipschitz constant of delta F restricted to this neighborhood is less than or equal to the minimum between A minus 1 minus 1 and 1 minus what's the conclusion of this result? Exactly, very good then F and A are locally topologically conjugate at 0 and what is the proof of this result? So you see the assumption is local remember this is the same setting as the general one in Banach spaces in Banach spaces we assume that this was true everywhere this was the boundary delta F was bounded and the Lipschitz constant was equal to this was this here we're assuming that this is true delta F of 0 equals 0 I'm assuming that F is a C1 map F in R and Rn F is C1 so I'm assuming the same properties of the global theorem but only in the neighborhood of 0 but then if you look at the proof then it's very easy to just repeat the same proof or what we can do we can define a map define a map F hat from Rn to Rn such that F hat restricted to this neighborhood is equal to F restricted to this neighborhood and F hat equals A plus delta F hat with delta F hat bounded and satisfying the Lipschitz condition and Lipschitz of delta F hat is less than or equal to these conditions here and then we can apply the global theorem that gives a global topological conjugacy between A and F hat and then we just restrict this topological conjugacy to the neighborhood and we get the local topological conjugacy so then apply global result to get topological conjugacy between and restrict a global conjugacy H so H restricted to N is the required local so let's now come to really the interesting application of this so let me write one more definition definition let F Rn to Rn C1 map and P is a fixed point then we say that F is locally linearizable at P if F and dF of P are locally topologically conjugate 0 at P and 0 so what does this mean this means we assume we have a map here F nonlinear map Rn to Rn we have a point P we don't know what the dynamics is around the point P but we have a linear map that is associated to point P which is the derivative of F of P derivative of F of P is itself a linear map from Rn to Rn and it has a fixed point of the origin and it has a linear dynamics linear dynamics we understand quite well at least in the hyperbolic situation remember we have basically classified at least for two dimensional linear maps we have a fairly complete classification so the question is is the dynamics in a neighborhood of P is it topologically conjugate to the linear map that's what the local linearization is that's what I mentioned before this is the definition it is if they are locally topologically conjugate so can we find neighborhoods of 0 and neighborhoods of P in which these two are topologically conjugate so with everything we've done so far what can we prove about topological conjugacy can we prove something about topological conjugacy here about local linearization what results do we have available first of all we've been concentrating on contraction so let's assume that this fixed point is a contracting fixed point in an adapted norm for example then this linear map is a contracting linear map do we know that these two are locally topologically conjugate so while you think about it I will state this result and then we will discuss the proof it's just application of results that we have proved that I have been stating so theorem N C1 map P is a fixed point suppose Df of P is less than 1 in some adapted norm then Df is locally linearizable at P and why is this true you could say this is the main result of the section I did not state it at the beginning you could say this is almost the main result now it will follow just as an almost trivial corollary of everything we've done before but in some sense it's the main motivation because again it is an assumption about a single point the derivative at a single point right and the conclusion is about the neighborhood of that point right the previous proposition I wrote was a weaker form of this remember when I emphasized this fact that we are going from the point to the neighborhood all we proved there is that if this is contracting then P is a locally attracting fixed point now we are saying much more not only is P locally attracting but the attracting nature the map in the neighborhood of P is topological conjugate to the specific linear map that is there there is a much stronger version ok come on what's the proof yes we only need a weak form of that but that's the basic idea we want to apply the previous theorem that I just stated so you are right in the Taylor formula that we want to write F in the neighborhood of P as the linear part plus the nonlinear part that's what Taylor formula does and that is exactly how we have been writing all these other perturbations of linear maps as F equals A plus delta F which is the nonlinear part exactly exactly how do we prove this so so for simplicity let us assume that P is equal to 0 it just make things a little bit easier for the notation otherwise we can just write otherwise define F tilde equals F minus P which will have a fixed point at 0 it's exactly the same thing we just translate it's a local result in the neighborhood of P so we can just pretend that these change the coordinates and it just makes it a little bit easier to do ok so let's assume P equals 0 so in some sense these are two different copies of Rn we have a nonlinear map that has a fixed point at the origin of Rn and the linear map that has also a fixed point at the origin of Rn the relation between these two is that this linear map is exactly the derivative of F at 0 and this of course will be a key point um so we write F we write F equals A equals dF equals F equals A plus the derivative sorry A plus delta F equals the derivative P plus delta F so this is linear part of F and this here is the nonlinear part so all we need to show to apply the previous theorem is that delta F is a small Lipschitz constant and then we have exactly the situation that we had before where F is just a small perturbation of the linear part by a small Lipschitz constant right so this is delta F so then is equal to F minus dF at the point 0 sorry we decided that the fixed point was 0 so I'm just going to write 0 here just the derivative of 0 ok so delta F is just F minus the linear part it's just exactly how I wrote it here so what's the derivative of delta F is the derivative what's the derivative of delta F at 0 let me write it here it's equal to the derivative of F at 0 minus the derivative of the derivative which is the derivative itself the derivative of a linear map is just the linear map itself so it's just dF of 0 so this is equal to 0 dF of 0 minus dF of 0 is 0 so the derivative of delta F is 0 so we have additional information about what this non-linear part looks like but that is why it's called the non-linear part because all the linear part is in here the fact that the derivative of this is 0 is saying that this is strictly the higher order terms of delta F and if the derivative is 0 and the map is C1 then the derivative in the neighborhood of 0 is small ok so so for all epsilon greater than 0 there exists neighborhood neighborhood there exists neighborhood N of the origin such that the derivative at X the derivative of delta F at X is less than epsilon ok just by the C1 nature of the map delta F is C1 the derivative at 0 is 0 it's small and so if the derivative is small that means the Lipschitz constant is small because the Lipschitz constant is just basically the derivative and so Lipschitz constant of delta F restricted to neighborhood is less than epsilon that's all we need so for epsilon sufficiently small for epsilon sufficiently small ok and so and thus the neighborhood sufficiently small ok the conditions of the previous theorem hold and F restricted to N and let's see F and dF of 0 are locally topologically conjugate so let me emphasize that for for these nodes for this course to be self-contained I'm doing everything here for contracting linear maps but this theorem is true much more generally so this theorem holds basically for any hyperbolic linear map ok remember hyperbolic is any map with the eigenvalues or not on the unit circle any invertible hyperbolic linear map so it's a much more general theorem that's also why I wanted to write the definition of locally linearizable which depends on the definition of local topological conjugacy in a more general setting ok however as it is stated here we really have proved everything every step of the way so it's really we have complete one last corollary of this we're almost finished is again the question of structural stability we can now handle suppose we have the same assumptions suppose we take a small perturbation of f is it structurally stable at least locally is the new perturbation still topologically conjugate to f or at least locally topologically conjugate we can now also answer this question so let me again give the definition so we have f same assumptions of that f ln to ln c1 p is a fixed point so f we say f is c1 locally structurally stable if there exists epsilon and neighborhood of p such that what is the definition of locally structurally stable do you think what that is epsilon and what is this neighborhood n of p speak louder I can't hear you do you remember what structural stability what's the normal definition of structural stability all last semester I get telling about structural stability that's right so structurally stable interior of its conjugacy class okay how do we explicitly formulate this being in interior of its conjugacy class what conjugacy class and what interior with what topology are we talking about so we want to say that if it is c1 close so if you look at the c1 topology there is a c1 neighborhood an epsilon neighborhood such that all the maps that are topologically conjugate to the original map f that's structurally stable that's being in the interior of its conjugacy class interior means you can find an epsilon ball around it that it's all contained in that same conjugacy class right so here I say there's an epsilon such that if I take two maps that in the c1 topology are less than epsilon so I take g inside some epsilon around f then f and g are topologically conjugate except that in this case it's too much to hope that f and g are globally topologically conjugate because the only information I have is that I'm just looking at the fixed point I'm defining the notion of structural stability just in the neighborhood of the fixed point so that's where this neighborhood n comes in and I want f and g to be locally topologically conjugate right so f and g such that if this is true then f and g are locally are topologically conjugate f f and g are locally topologically so it means that inside and the dynamics does not change very much so what's the theorem here theorem we have the same assumptions theorem f rn to rn c1 p is a fixed point contracting fixed point in adapted norm then f is locally structurally stable at p proof? what's the proof you can give me the proof yes so you're right because the last semester I emphasized that structural stability depends on which topology you choose and which kind of conjugacy you choose so it turns out and I mentioned that but of course generally let's say the correct topology is the c1 topology because if you take the c0 topology nothing is structurally stable in general ok if you take the c2 topology then it's very difficult to be structurally stable everything is structurally stable and the right conjugacy class is generally the topological conjugacy class because if you try to take differentiable conjugacy class then again nothing is structurally stable ok so here that's why I write it c1 locally structurally stable if you take the neighborhood in the c1 topology and the conjugacy is a topological conjugacy ok so from now on when we talk about structural stability we will think of it like that the c1 topology and the topological conjugacy class because that seems to be the most natural so c1 locally structurally stable ok so what's the proof it's a couple of applications of this theorem yes we take another g satisfying this condition both are equivalent to the linear that's fine but the linear will no longer be the same linear map right so what's the philosophy of course since it's c1 first of all there exists a neighborhood in which the derivative is contracting in the whole neighborhood right so step one since f is c1 there exists a neighborhood n such that df of x is less than or equal to let me write it less than equal to lambda less than 1 for all x and n yeah because df of p is strictly less than 1 so we can find a neighborhood where it's also strictly less than 1 bounded away from 1 then we can take a c1 perturbation and because it's a c1 perturbation the derivative will be close so we can still have that so if epsilon d1 fg less than epsilon then we can have the df um dgx dgx is less than equal to lambda plus epsilon which we can write as lambda tilde which is still less than 1 if epsilon is small enough because what does being close in the c1 topology means the derivative are close in particular the norms of the derivatives are close so g is still a contraction so g is a contraction on n also because f is a contraction in a neighborhood then you can assume that n is mapped strictly inside itself and so for sufficiently small epsilon we can also assume that g maps the neighborhood and strictly inside itself so we have a contraction on n which maps strictly inside itself so g has a unique fix point so g has a unique fix point q also in n so to prove that is locally structurally stable we need to show that f and g are locally topological conjugate in n so we need to show that f is stick to n is topologically conjugate to q to g is stick to n both of these have fix points so they're going to map the fix points to each other so we need to show that the topologically conjugate to each other this conjugacy will map the fix point p to the fix point q and we need to show that it extends to neighborhood of p and q so what's the last step now to do this what do we know about the topological conjugacy class of these maps in neighborhoods of p and q why is it enough so it's enough you're saying enough to prove that df in p and dg in q are topologically conjugate why is that enough linearizable yes exactly so by this theorem we have that f is locally topologically conjugate to df of p at p and g is locally topologically conjugate to dg of q at q right so f at q is topologically conjugate is linearly is locally conjugate to the derivative of g at q and f in a neighborhood of p is topologically conjugate to the df of p ok so as you said correctly it is enough to show that p and dg of q are topologically conjugate and is that true that the topologically conjugate so sorry I wrote this in a slightly in a slightly different way so I added that as an assumption so is it true that these two are topologically conjugate these two linear maps these are two linear maps right they're both contractions so we proved this for two dimensional linear maps in the case n equals 2 we proved that the eigenvalues remember if they have the same number of eigenvalues that are contracting and expanding then the two linear maps are topologically conjugate ok so if they're both in the case of n equals 2 we proved this final step so sufficient so sufficient to prove that the f of p and dg of q are topologically conjugate but in this case they're both contractions so for n equals 2 we have proved if two linear maps the same number of contracting eigenvalues they are topologically conjugate ok in this case both df of p and dg of q are contracting so they're both eigenvalues contracting and so they are topologically conjugate ok for the two dimensional case now for the higher dimensional case it's also true but we didn't prove it so I actually I copied as I was looking at the notes I didn't realize so to solve this little problem I actually stated this result in a slightly different way the assumption has an extra assumption so extra suppose df of p is structurally stable in the space of linear maps and then we have this we have that extra assumption in the general higher dimensional case then that covers this last bit because we know that dg of q is very close to dfp in the space of linear maps because we've taken a close a c1 small c1 perturbation so by taking a small c1 perturbation we can make sure that this is arbitrarily close to this as a linear map then automatically it implies these two are topologically conjugate and we get the result okay for invertible linear maps so I'm always assuming also that these are invertible sorry, maybe I forgot I'm always assuming that the derivative is invertible yes sorry, I should have probably stated explicitly in each result let us assume always the derivative is invertible in this case non-invertible is kind of a quite degenerate situation so every one of these results I stated assume f is c1 p is a fixed point df of p is invertible and df of p is contracting so the right way to see this is to have this extra assumption and then this assumption takes care of the last step to show that these two are topologically conjugate equals two we don't need this assumption because in that case we have proved that any contracting linear map is structurally stable in the space of linear maps invertible contracting linear map with distinct eigenvalues okay, so yes but again it's true without that assumption it's because what we did last semester is to not get into very technical issues I always assumed that the linear maps had distinct eigenvalues but essentially all the results are true in the most general case so I try to make sure we have everything proved but sometimes it's good to have an assumption that shows that the results are a little bit more general okay, very good so this is really all I think we have done a quite thorough study of linear maps of contracting maps sorry, of contracting maps so this local linearization and the structural stability come at the end there's a corollary of everything we've done before because I wanted to present this this way but in some ways you can think of this as the main motivation of the whole section these are the main results of the section and if you look back at it when you revise the notes you'll see that everything that we did in the section from the contraction mapping to the Lipschitz perturbation basically everything was setting up the results that we needed to then be able to prove these quite easily with that machine and finally again essentially all of these results hold not just for contracting fixed points but for fixed points whenever the derivative is hyperbolic so they're much more general but the proofs are considerably more sophisticated okay, so starting from next time we will change gear completely and we will look at expanding maps maps which expand and we will see that the techniques and the kind of maps have very different behavior and we will address the usual problem of topological conjugacy and structural stability but whereas here we constructed these topological conjugacy using some fixed point results there we will use a completely different and new technique okay


Cut. Cut what? We didn't know. Action. You're the director now. I quit. I love you guys. You might be thinking to yourself, who is that chicken the glasses?  It is not often that I wear my glasses on this channel because I don't enjoy wearing glasses. You should wear more. I like it. I don't, I don't like wearing my glasses. They always just make me feel like a little crazy. I'm just not like a big glasses wearer. I just, I'm not. I prefer to wear my contacts every single day. I am a negative four in both eyes.  Okay, give them back. I am the same prescription in both eyes, which I consider myself very lucky, which means I don't have to put two different contacts in each eye. I get to put both in the same eye. It's great. I am nearsighted, which means if I took my glasses off, I can see close to my face. So what I wanted to do was like a bunch of fun stuff without my glasses, like throwing and playing catch and like throwing a dart. We tried all of that, but like I can do most of it. As most of you that wear glasses or contacts and are nearsighted would know as soon as something gets close enough to your face, it's not a challenge. You know, if like my eyesight isn't terrible, it's not great. But something that I do know that I can't do is like in the morning when Julian's like showing me a picture of something on his phone, I'm like, I have no idea what you're showing me. We've lived together for like, what, six, seven years and every morning you're like showing me something. I'm like, I can't see. I don't know what you're talking about. Why are you showing me that? But yeah, this is going to be a little bit of a different video in case you're going to tell me because I literally just want to sit here and like see how useless I would be in like an apocalypse scenario. The second I take my glasses off, I feel like I start making like, I have no glasses on face or you're just like, I just want to play a game. Like what can I see without my glasses? Fair enough. This video would probably be a lot funnier if I was far-sighted. You want my glasses? Oh, who is that? Welcome to my beauty channel. I'm making the, I don't have glasses on eyes and you're making the, I have someone else's glasses on. Number one. Okay, it looks like a picture of Ariana Grande, but she has a white wig. Wait, wait, no. Yeah. It looks like the hair that she has in the sweetener like album pictures, but I've never seen that picture of her face that looks like that. Just so you know, you're off to just a really, really horrible and unpromising start. Yeah. It looks like a, like a Bratz doll version of Ariana Grande. Bratz doll? Yeah. Oh my God. That's funny. Why is that funny? I can't see it. What is it? Oh my God. All I could see was like the exaggerated like eyes and like. Thank you, sisters. Next off to a really unpromising start. Here we go. See, I'm like, I don't want to squint all the way because then I feel like I would be cheating, so I'm just going to try and keep my eyes completely open. So what I can see is a face, the shape of a face, something dark here that looks like hair and then maybe something blue on the top that looks like a hat or maybe that's just your desktop. That's my desktop. Shit. Who has hair that goes out like this? Or maybe it's someone wearing like headphones. Or it's someone with like a really blunt bob. Like I can't tell any gender or any facial features whatsoever. I'm going to say Dita Vontis. Anthony Padilla. Okay, that was a pretty good guess. Thought that that might be Anthony Padilla.  I was right about the hair though. That's not my bob. Yeah, it is. That's like a bob. I couldn't tell that it was curly, but I could tell that it was hair. I like to hear. You said it was headphones. I can't tell from here. Ready? That just looks like the Godfather.  It just looks like a moon cookie. Can you show me another picture of this person? Yeah, yeah, sure. Is this a hand in the foreground? Yes. Okay. I think it might be a man. Okay. If I had to guess, are we doing YouTubers? Like you shouldn't do this because then I'll be able to guess. I'm not missing anything. It either looks like Ricky Dillon, Jacksepticeye, or Joey Garsepa. I think the third guess is the closest.  This looks like a chair or something angular in the background. Should I put my glasses up? Why are you laughing?  It was this one. Oh. I'm supposed to tell what that is with no glasses on. It just looks like a moon cookie. Ready? Is it a cat? Is it a cat? You nailed that one. That's a cat. There we go. All I can see is a face and something red and something white. Okay. To me, it looks like a baby wearing, like, a bullfighting, like, red thing, but around his neck. Like a small cape. Like the matador cape? Yeah. It looks like a baby matador. Okay, so baby matador is your guess. But wearing like a, I'm assuming what is like a festive bullfighting outfit. Yeah, okay. But I guess it'll get glasses. It's not a good guess. Yeah, but like, I could see his tuxedo. So it, like, What about a tuxedo? It looks like a matador cape. No, like a festive bullfighting outfit. But he, like, it just looked like a giant baby to me. It looked like a baby head. I don't know. He just looked like a baby without my glasses. All right. Like, I can't tell a person's age without my glasses on. All right. It looks like a blue, a sloth in, like, blue face. Like a sloth with its face painted blue. Do you like my guess? Oh my God. Seriously? Okay, but that's, that's not what it looked like to me. It just, it looked all blue, but like I could only see slanty, like, lines. Yeah. Well, nice job. Thank you. Ready? Yeah. It just looks like a floor. It's not a floor. It looks like a, like a gray dog wearing one of those curly colonial wigs peeking outside of like a door frame.  Yeah, from this side.  Really? Yeah. It's a floor. It's not Ben Franklin wig. Here we go. Okay.  It's not a Kardashian. Cher. Not Cher, but very, very close. Cher from Stuck On You. No. Sandra Bullock. I'm just guessing people with long dark hair. Wait, what's your last guess? Cher. No one for that. Sandra Bullock. Correct. Wait. Oh, Julia's hair. I've got a good one for you. Ready? It looks like maybe there's blonde hair and like a black shirt. Their mouth might be open. What? Nothing. I think their mouth is open. It looks like there's a cavity right here.  Alright, give her a look. Voldemort. Okay, also another good observation. These are all like very poignant. Are you just showing me pictures of you? All I can see is like something black coming up and then it's like almost like twisted. Like they have no neck.  It looks like a cartoon. Like Daria. Honestly, it looks like if you took a pencil and you drew this, but like a human face was in it. Like a teardrop shaped head. Okay, so his head's a teardrop. Yeah, but like not a real person. It's like a cartoon. Because it is no human's head is shaped like that. Trying to think of another cartoon that looks like that. Like Bob's Burgers. Bob from Bob's Burgers. Okay, Bob from Bob's Burgers. Can I look? I feel like the lighting in that picture makes it so difficult to read that that's a real person.  No, it just, it looks like this to me. I'm telling Shayna. That I thought he was Bob from Bob's Burgers. Yeah, or Daria. Or a cartoon or no human has that shape of head. No, you said that. I didn't say that. Maybe one of those like howler monkeys. Or like a bird. All I can see is like something really dense and dark right in the middle. It's like something light colored around it. But I can't, it doesn't, it's not a person. A lizard. It's something just like something pointy's face. Every time I feel like you're judging me. Oh, see I could tell it was something pointy. I didn't think it was going to be Kermit. That looks like somebody. Somebody has red right here. And then it looks like this. So like home alone.  But with dark hair. Somebody doing the home alone face. So, okay. Where are you red doing the home alone face? Yeah. Has dark hair. Yeah. Okay. Oh my God, that's my mom. Why does this look like this to me? Isn't that crazy? Yeah, that's bizarre.  It's so is.  That's my mom. That just looks like some, like a butt. Oh, like wait, don't laugh. I don't know what it is. Like the bottom, the bottom of the picture. I feel like I can see two round things. Like a butt. And there's like something darker going up the middle.  What is so funny? So where her, her shirt strap is and her shoulder. That looks like a butt, but then I couldn't see her face. I can only see her hair. It looks like someone's back. Oh my God, Julian. I'm going to get into so much trouble. Well, honestly, like I did a little better than I thought. Like some of those I got when I just like the sloth with the blue face. Like I did not think that I was going to be able to get stuff like that. So I'm not completely useless if I didn't have glasses or contacts. But I'm like, almost useless. Do you have anything you want to say to all the people that you guessed? I'm especially sorry to Liza Koshy. It was really hard, but it was really fun. And if you have not good vision like me, it's a fun thing to do because I swear. Like my eyes were making up colors that didn't exist and patterns that didn't exist and things that didn't exist, which is why I have to wear glasses. It's funny because you were born with perfect vision and you're just you wake up every day and can see clearly. Your words not mine. Musty, nice! I have glasses and contacts and can tell the difference between Anne Hathaway, Sandra Bullock, and Julia Roberts. I want to see Chris Mulberger playlists because he's like legally blind. Good job. You tried your best and you didn't succeed. This was really fun though. I don't know if it's fun to watch, but it was fun for me to make and I had a good time and I think it's funny. Make sure you subscribe to my channel. I put out a video every Wednesday slash Thursday. It must be so nice just to wake up every day and be able to see everything without your glasses and contacts. It is. I have perfect pitch. Give me a note. C minor. I don't even know if that's right. Perfect pitch. Okay. I'll see you guys next week. Bye.


Now, let's go over to our man, Mr. Dave Mazza. Dave is the managing director head of product at Direction, and of course, he can basically lead us into these different volatile markets that we've been looking at quite a bit. If you haven't test driven yet any of the Direction products, really easy to do. You're right on our website at TFNN. Just hit that banner. Dave Mazza, what's going on? Hey, thanks for having me back. How have you been? I've been great, man. Great to have you back. And last couple of times, we've been talking a little bit about, we've got some rotation in the marketplace, right?  We've got some action here, Dave. Yeah, I know you're right. We've been talking about this for a few weeks, and there's been this kind of push-pull relationship. On one hand, it's the stay-open, reopening trade, the stay-at-home trade, but really, it's also the harsh versus small and growth versus value. And today, you're absolutely right. Like we've seen in some pockets over the last few weeks, today's been an all-small cap day where some of the riskiest, higher-beta stocks are really leading to charge higher. Just take a look at some of the financial names, of course, the energy names are really ripping, and those names have been out of favor. So in days like this, this is when we see traders really look to TNA or even TZA if they want to play the other side of it to take advantage of the small capitalists. You know what's amazing is that I remember the first time, I think it was, let's see, probably maybe six weeks ago that you had brought it up that we felt like there was some rotation that was taking place. And it's really wild, because at the beginning of a rotation, it's like, okay, is it not taking place? But this morning, it was pretty clear. I mean, the NASDAQ, you can see that the tech stocks are getting hit a little. Inside of the small caps, well, the banks right away, the banks right away just start going higher. So it's pretty wild, man. Yeah. Well, you're absolutely right. I think traders have been distracted not in a bad way necessarily, but because of earnings season. And we know, if you look at the really that fang plus group, an aggregate 50% earnings beats, which is pretty wild in any market environment, but especially one with as much uncertainty that we find ourselves in this global pandemic. But again, now that 90% of S&P companies have reported, investors are turning their attention again to other opportunities. And right now, it's saying, can we look at some of these beaten down names? And what's interesting is we're starting to see this coming to play again. So what I'm going to continue to look at is some of these leading indicators that I use. So I like to look at the semiconductor relative to the NASDAQ or the S&P 500.


We were talking, we talked at the break, about when you start to see how things that seem to be so necessary for the body and all these activities and all this, you start to see that as really a distraction, and you start to hone in on the homing beacon, so to speak, then it just is a glorious ride, it's a glorious life, because that's part of why we're here is yeah, it's really quite a spectacular life, because when you don't put your energy in thinking about a lot of things that seem to be about the body and time and protecting and maintaining, a lot of energy goes into that. When you free your mind up from that and you just turn your mind in another direction from all that, then you have an amazing amount of energy, and amazing, you might just say you're so open and receptive to miracles coming in you and through you, that your life takes on miraculous qualities, and really, that's what we loved about the fairy tales, you know, we enjoy watching characters and fairy tales that kind of have a magical, mystical, carefree life in which there's joy and glee and excitement, and there's a sense of the adventure in it, it's not like the world, the linear world of human beings is very much about problem-solving. And so much energy is put into problem-solving, and we might say that from a Course in Miracles perspective, it's really trying to solve problems in a horizontal way, and really it's not going to actually solve the problem, that you know, it's almost like the world was a setup to have so many problems and so much energy spent in trying to solve the problems inappropriately, like in form, that eventually it gets very depressing and very tiring. Jesus says the world is very tired, and that is a good description for the linear world, because you put all this energy and nothing seems to come of it, just more problems arise, you get another set every day, and then another set, and then another set, and then there, at some point there has to be something that rises up in your soul, just as there has to be more than this. I can't really get accustomed to this, I can't really think that this is how it's meant to be, it just, that's how Bill Thedford and Helen Schuckman joined together, you know, Bill was the head of the department, and Helen worked in that department, and they just came together, and he just was expressing his frustrations one day, and he just said, there has to be another way, and he was quite surprised when she said, you're right Bill, and I'll help you find it, you know, that was, there's the collaboration in mind, there's the joining in that. I find the people, like with Francis and I, how we were brought together, and there's so many people that I've been brought together in their lives, a few days ago, we went to see a new movie that just came out, it's called Begin Again. I really enjoyed it. It was Mark Ruffalo and Kira Knightley were in it, but it was a beautiful movie of going past horizontal love to much more of a divine love. There was huge collaborations, there was such transparency, such an authenticity, and I just had people that I've known and lived with for many years, and how many, there's about 20 of us in this movie theater watching this movie together, and then the movie ends, the credits roll, and there we all are, you know, as the whole theater empties out, we're just there. We were just laughing before the movie started, and that just was the kind of a buzz of energy, but just, it's just coming together in that same purpose, when that is the purpose of your life, and then and then watching this new way unfold. It's not really a spatial thing, it's not like living in community in a spatial way, like Jane Marie and Peter are out, and every day is a new day of discovery. You just gave me their business card. What was the top of the business card? On the road? On the road. Unplugged. Yeah, come what may. Come what may, and then the picture of the road, on the business card. On the road, unplugged. Come what may. And it's this sense of loosening from patterns of thinking your life, you know, thinking you know what your future is going to be. I know when I went through in the parable of David through high school, you know, I just looked even back at those yearbook pictures in high school and in college, and I kind of had the kind of a glazed look on the face, kind of a bewildered, glazed look, like I don't know what's coming, but it was like a look at my face like nothing really that I was seeing on planet Earth was really appealing. It's why I think a lot of times with teenagers, there could be this sense of like a look, great depression, or an anxiety around the future of thinking, oh my gosh, what a daunting task going into adulthood. Some of us had had some nice moments in childhood, you know, I think of playing by the creek in unstructured time and sparkly summer days where we just go out and lose track of time and space and just be so absorbed in the play that it was really idyllic. It was just wonderful. And then looking ahead, I think, hmm, so now it's just to become a productive adult. What, is there anything out there that looks attractive? No. And then having of my mother was at the same school where I was, so she knew the guidance counselors and sent him out on all these field trips and adventures, find something, find something, but no, there was nothing that could be found. And then when I got into university, the search just continued.  I think you're better over in this direction, that direction. No, not appealing, not appealing. And yeah, it took me ten years of university full-time before I think I must have had mine. There must be another way moment. I mean, I had lots and lots of skills accumulated after ten years of university and doing quite well in university, but like it was like an emptiness, like, no, no, this I'm still not seeing anything flipping on the radar chart that is a sign of life. I mean, I'm looking at the whole gamut and then I started to open up a little bit to like helping professions, feeling like there was something around altruism and something around helping. Wanting to be truly helpful, but not really seeing anything really clear that was clearly lighting up as a way to be truly helpful. And then that's when in 1986, in pretty much a state of disillusionment with the world, that's when I started to get into humanistic psychology and transpersonal psychology and then the course dropped in with those as forerunners and Carl Rogers, Virginia Satir, lots of amazing thinkers and reflections of my mind that there was a pathway beyond this world, but it was totally transcendent. It really wasn't based in any of the thinking of the world or the all the different disciplines that I had studied. I had studied so many. So everybody comes to that point in their own way. And did you study also other spiritual things before the course? Yeah, that was part of, I hear a lot of stories about how people were kind of channeled in. We were just talking at breakfast this morning. Karen was saying it was Eckhart Tolle's The Power of Now. So I'm mentioning of course the miracles, of course the miracles. Read a quote here and there. Sounds pretty interesting. Curious, or curiosity can arise. And then it was Disappearance of the Universe, you were saying, Peter. People, I had my inroads. I think I came across Jerry Jim Pulski's Love is Letting Go of Fear. Very small book with little cute little caricatures in it. I was like, hmm, this is interesting. It kind of came to me in a number of different ways where I started to hear it and then I was out in California when I actually came across the book itself. And La Jolla, very close to where Bill Thetford was and the Lechitz. But and Francis shared last night too about about kind of getting into the course and then teaching it more conceptually and then one retreat when we had a seven-day retreat together. It was such a powerful seven days that it just like opened up a whole new world. Almost like that song, A Whole New World. A new fantastic point of view, you know, just starts to, you get a glimmer. Wait a minute. This just isn't some kind of a new theology. This is like a gateway. This is almost like a portal into a way of living, an actual living experience. And to me, that's what it's been. It was actually my heart lighting up and the swirl of joy and glee and energy that I felt in my heart that I just said, wow, this is it. I've hit the mother loom. I don't know what I did or how I got here, but that gets you into this. This like, it's almost like being me up Scotty. It's like a vortex of energy that just starts to pull you away from everything that you thought your life would be and sends you off into a whole new trajectory. Absolutely unimaginable trajectory. And that's for me, that's that's absolutely essential in learning the world. I had to have something. It's like, show me, Lord, show the way you've got to, you know, like, what's the one in the movie? Show me the money. I'm going to show me the joy, show me the love, show me the spirit. I need to be drawn in. I need to be taken in by something that's so expansive and so transcendent. Then I can get my heart over to it. That was the thing I was always like in a holding pattern. When I wasn't seeing anything that really totally sparkled for me, I couldn't throw myself into it, you know, whatever it was.  And then I guess for you, it took the same thing. It took an experience to start to send you off in another direction and start to say, well, maybe this isn't so so successful or so really what I truly want. Yeah, it seems like the material things have accumulated and built up, but it's just emptiness in it. And for me, you know, the spirit apparently guided me to just first let go of some of the things I hold on to tight as my security or something that's just I couldn't let go of. And those things, because I put my security in those things, I completely lose sight of what is truly the source of, you know, that sustain me. Like I couldn't see it. It's not that I'm not living in divine providence or I just couldn't see it because I trust everything that I build up myself and, you know, everything in the world and possessions and relationship and career. So it was a very, you know, seeming it seems radical, but actually my experience was very gentle. Like the spirit is saying, put those aside first, because your mind needs to be convinced of something completely different that you actually are blinded of right now. So you have to put those aside and even like with everything, my relationship with money, with career, with all the people around me have been completely translated, retranslated by the spirit. I was just remembering last year when I was in Australia, traveling with a friend, we were traveling and just the experience was so apparent and obvious to me that everywhere I went, I met my mother, my father, my sister. It was just so unbelievable, like the patterns that, you know, the way they took care of me and the way they asked me questions were just my mother there. But in a way that is not in the same body or they weren't playing the roles with me and yet I could recognize that the spirit was saying, you know, don't, you know, have a fixed relationship with a body, relax all those roles and boxes in your mind and take what is given and really see, you know, there is experience to really be convinced that what is given is my will. What is given is the best for me.  And I have a separate will than this. And I know what they are, so I need to strive for that. But just to be able to relax the mind and accept what is right in front eventually really leads to this experience to recognize, oh my God, this is my will and everything that I have is what I want and is what I need. And there is nothing else. So just really open up even to the idea of relationship, like feels really strange to think that I have a relationship with a body because that is really about behavior. Do you have a relationship with this person? It's saying, do these two bodies behave in a certain way in this box? And that is like, I don't really quite know how to relate to that question because experience wise, I am constantly in relationship, but the body shifts, like with my mother, the kind of relationship I used to have with her now is with anyone. Like I can have a momentary feeling of recognition. And yet with that particular body of my mother, I don't necessarily have the same way that we interact with each other. It could be more, we can talk about metaphysical ideas and we could talk about something else, but not a mother-daughter kind of conversation and dialogue. But when I was on travel, someone could just look after me and feed me the way that she used to do. And I just see like the relationship is completely not with a single body, it's just a momentary experience. Momentary experience. And then when people really ask questions about a relationship, sometimes I will think, maybe that's the question is about whether the body behaves in this way, whether that fits into the box. And then I will direct my answer, but in my own experience, it's just a continuous experience. Like with David, for example, it's just this deep collaboration and mind together and we just feel the inspiration to speak together and to talk and that's just a very deep experience. And I can't really fit that into a box. Depending, it just feels like everything, all the boxes and the definitions just been washed and there's one vast experience that's ongoing. Yeah, and you start to feel more and more as you go into it. It's a collaboration, it's really with the Holy Spirit that when you give yourself over to the Spirit to be used, you are going to have a loosening from all attachments to form. I think of the beyond all idols section of A Course in Miracles, it's back towards the back of the text, but there's a line in it, God knows not form. That's a one sentence, great one liner, but metaphysically, if God knows not form and I want to know God, you can see where it's going. God knows not form. If I'm going to give myself over to God and go for complete alignment with God, I'm going to move back towards abstraction. And I talked about that last night, that love, light, the King of Heaven, God, they're all purely abstract. There's nothing on earth that even is a hint of abstraction. Everything is very specific and concrete as part of linear perception. And then when you get into this unified perception, it feels much more expansive. You don't feel, you're not looking at form the same way. Because I was just gonna, you just explained it, I was just gonna say, well, what about the script is written that I keep talking about? If the script is written and you need to be in relationship with these one or two people, or in your case, maybe many, does that fit on the different level? Are we talking about different levels now? Well, when people have asked me about the script is written, which is a line from the word book, the main emphasis is on the last word written, which is past tense. It's really, the script is written as just another version of lesson number seven, I see only the past. So, in other words, nothing to figure out about that because it's a linear construct and it's just saying it's the past. And imagine if you could free your mind from saying, see, I don't really have to work on the script anymore. Imagine you- I don't have to write the script. I don't have to write the script.  Yeah, well, that's why I'm gonna take it with, here we are, we're right into unlearning the world. And I mentioned in beyond our idols area was God knows not form. There's another great line in there. He says, your will is universal and cannot be content with form of any kind. Wow, your will is universal. He's talking about the will of God. He's talking about Christ will and the will of God. Your will is universal and cannot be content with form of any kind. Then there's another line that I like from that section. When you decide upon the form of what you want, you lose the understanding of its purpose. When you decide upon the form of what you want, you lose the understanding of its purpose, like an eclipse, like if this is content or we'll say this hand is love and you get all mesmerized by the form and it's like an eclipse where the form comes before the content and this book has a lot to do with form and content. You'll see the topic come up again and again and again. Love is content and form is the distraction away from that content.  So you're blinded then. You're blinded. And to the extent that you open yourself to forgiveness and say, Lord, teach me how to forgive illusions, you will, the form will fade and fade and fade away until all questions about form will just evaporate in the mind. You will see that why would I give my mind over to such tiny, petty, little, nothingness of all these questions? All the questions around form in this world seem to be the most important when people say, well, come on, now be practical with me. What's the bottom line? How much money, how much return do I get on my investment? And where's it going? Is she gonna marry me or not? You know, it's a, we just. I'm just thinking about that work. Yeah, exactly. It just goes, everything is so focused, riveted on the form and the content is obscured. For example, we have a great sharing. We're here with you now sharing, sharing, sharing. I get back on the plane, I go back and then I have, I fly to Mexico to collaborate with another friend of mine, Maria Felipe. So I'm going down there to collaborate. Then I fly back. Then I fly to Brazil to collaborate with another friend of mine, Armel. Then I fly back, I'm Brazil. And then I have a festival and there's a woman flying in from Europe who's just like, happy as a little child, just very believable, Lisa Cairns. And she's flying in. So it's just collaborating. And then, after that, I go back to Mexico, collaborate with a whole group of people. And then I take off for about almost two months, two and a half months, and Francis is part of it, but there's also Jason, Kirsten, Michael. Configurations, the spirits using all the forms. So it's like playmates, little playmates that are coming in just as part of these configurations. It could be one, it could be two, three, four. You know, it's just, it's really the form is irrelevant. It's the glee factor, it's the joy, it's the love, it's the happiness that's really the content. And the form is just given. It's almost like you could think of the form as almost like the props that will work. Like I've traveled in 31 countries and I speak to all these cultures and people that I don't speak their language. And the spirit sends me the most magnificent translators where we are so connected, so telepathic. It doesn't matter their age, either. One time I was in Switzerland and they were having the most difficult time finding me a translator that didn't know what they were gonna do and a French translator and they finally found this 20 year old student who was in a translation school, I think, in Paris. And they pulled her out. They said, could you go translate for this guy over in Geneva and everything? And she comes and she's sitting next to me and I happen to have a French translation of A Course in Miracles in front of me. So she's got maybe half an hour to prepare before. She knows nothing of The Course in Miracles, nothing about metaphysics and everything.  And she's just so wide eyed and willing. I give her the book, she opens it up and she starts glancing through it and she comes to this word and she said, it was, she came upon the word atonement which probably is the most important idea in A Course in Miracles, the sole function of the miracle worker and the teacher of God is to accept the atonement for himself. It's that important. It's the only function really that we have. And she finds the word and she goes, hmm, because in the French translation they still have put a word that is not really a very good word for atonement and because it still has some aspects of like paying a price, it still has, she's 20 years old, she's not even a core student, she's picking up that there's still some sacrificial tones in whatever word that they've chosen that's made it all the way through in the French translation. So she's looking at me and she's explained to me that this does not seem to be the right word. And she says, I said, well what is the word that you're feeling? And she said, correction. And I said, yes, you're my dress later. This is sent from the Holy Spirit. And that's what I've noticed that I have all these great collaborations with translators, it wouldn't be the same without the translation. It's a collaborative effort. Miracles are a collaborative venture. Jesus says that in the Course. And imagine just being open in your life to feel like, wow, that's the case. So I just want to be open to the adventure of it and the collaboration of it. And I want to welcome everyone who's sent because no one is sent by mistake. It's a very huge function that we have for giving the world and no one is sent by mistake. Even us, we're down here and this book is 20 years in the making. That's right? It only took 20 years to finish this book. It took about a year and a half or two years to start to bring it together. But what is different? What makes this book any different from most books? Well, with major publishers of books and everything, when you get a major publisher, one of the thing is major publishers want to make money on things. You'll notice that about things in the world. That everything through the ego is a commodity to make a profit on. And what makes this strikingly different is this book is actually just a website that has been on the internet for free for years and years and years. This is just a different form of the same content. Jesus said something that really struck me in the Bible. He said, freely you have received, now freely give, freely give. So with books like this, everything that comes out in terms of books that has anything to do with me is already available for free. People just have to go search for it, but it's already been made available free. It's not kept apart to be sold for a profit, to be sold to make money or anything, that that motive is not in it. It's totally just, this website is called Teacher of Teachers because it's a tribute to the Holy Spirit. And on the web it's called Teacher of Teachers. Here it's called, oh my gosh, it's a book. If you can put it in your hands, it's unwind your mind back to God. So it's the motive, it's the purpose underneath that's very, very, very important. And that's something that's been important to me. It's like everything that I receive that is always for me for freedom, I want to give it away, I want to extend it. And I just have given different ways to do it. And this is one, it's kind of fun. It's like a novel, it's just a different form of the very same thing. The title is very, brings you right into it. I mean, just the word unwind, it's very alluring to me. She's just like, she's just like, she's just like that. Okay, the Holy Spirit's doing its job. I've wanted that out. I've heard that comment a number of times actually. People like a number of people have said that, yeah. You know, we keep saying, I thought it was funny given what we're learning about time, that we keep saying it took 20 years, you know, and this book is the culmination. So I'm thinking of it as that book on that table created the last 20 years, here, right? Instead of it took 20 years to write it, that book created the last 20 years of comments on the website. Yeah, you just take all the causation out of everything and then you can really have fun. And it's also like of all places, we're kind of premiering it right here because Karen was like, oh, come back, come back. We were back, we were here a while ago. I don't know how long it was when Francis and I were here. May 10th, you were here. No, I was just talking about when Francis was here. When Francis was here. Oh, Francis, April of 2013. Yeah, April 2013. April 2013. And now, then we were back here again with the Quantum Love Tour, kind of doing it through the Quantum perspective and now this, so, yeah. It's an invitation, you know, really. This is like an invitation. It's like I always liked that about what the belief do we know, you know, it's Fred Allen Wolfs with the hair, you know, at the very end, saying, don't just take my word for it. Find out for yourself that we're in an adventure of discovery, of discovery of our Christ self, of pulling back all the parameters, all the limitations, every limiting belief. There's one little blurb on here that is interesting in the middle of the book here. It says, message from the Holy Spirit, right there in the kind of lavender words, right in the middle in the back.  Only the release from these false beliefs will bring you true happiness and lasting peace. And that's got that unwind word in it that came to me from the Holy Spirit. It was like a very clear directive to me that I would be undergoing seemingly an unwinding process. And the length of that process really is not determined by anything in the world. It's really our willingness, our readiness, and our willingness to be unwound. It has nothing to do with anything else. We don't have any benefits from wealth or things, appearances in the world. There's nothing that gives us an advantage. But you can't get away from this book if you're a searcher because you've got the title in the front and then you've got your picture that draws you in in the back because you could see the Holy Spirit over you on there. Just the joy. And I see the joy of the Holy Spirit in your face. It's just really beautiful. Really, I'm not trying to... She moves to you a lot, right? It's working for you. The front cover and the back cover are... I don't know what's in here. What's in it? It's very good. I will tell our publishing team that it will be built-in. I'm going to be thrilled. The Rochester Metro Chiefs want to go to the fireworks. She wants to go to the fireworks. Yeah, we're going to talk about... I've read a head-on, like looking up topics in the book and read where you laid out each of the sections with topics. And somewhere in there, I believe you talk about how in the course it's quoted that all we need is a little willingness. But then you mention that somewhere in the annual for teachers something about we're stepping up our willingness. Can you talk about that a bit? Yeah, that's where he mentions great willingness. He talks about little willingness all the way through the course, and it takes great willingness to see that all events, outcomes, circumstances are helpful. I mean, that's coming back more to the Bible. All things work together for them who love the Lord. They work together for good. There are no exceptions except in the ego's judgment. An amazing unified perspective which has to be simultaneous to be true or even to resonate because from the linear perspective it just seems like there's victims and victimizers, there's good fortune, there's bad fortune, there's good outcomes and there's bad outcomes. All those seeming outcomes along the timeline are just interpretations and judgments. Like, for example, a stock market crash. Whether it's way back and the big crash way back after the 20s, the end of the 20s or even more recently with the recent recession was that around 2008 or whatever it's from the ego's perspective these are crisis times. You know, this is in a world of bigger, better, faster, more in a world where everything is supposed to grow. Even churches are supposed to grow if you have a church that's flat lines or like this, it's like you are not working for the Lord. The Lord does not make a shrinking church. Oh, really?  He was a... Hugh and Gale, he passed away I think but he and his wife formed out in I think this maybe it was Arizona, the disposable church. They formed the disposable church. I'm like, isn't that wonderful?  It has nothing to do with buildings. It has nothing to do with congregations. It has nothing to do with attendance. Tithing, 10% of your income, my gosh is that the best we can do? My gosh, we are the givers of life as the Christ. We are created in its likeness and image of God. We are life itself eternal life and so the best that we could be as a church would be as a reflection of that eternal life. So over the years, the first kind of symbol I got when I had a friend of mine said you're going to be traveling all over the world and you're going to be going more places than you ever know and you need a non-profit foundation to help funnel some of the money. It's going to hold you back money unless you find a way to just let it be used in a non-profit way. So I said okay, then go ahead and do it. So he went ahead and he came to me for a name and the name that I received was Foundation for the Awakening Mind because that's what's happening. This mind has fallen asleep in the Bible it says Adam woke up I would say Jesus Christ appearing as the resurrected Adam basically the way the truth in the life that's one with God that to me is the wake up right there it just doesn't say Jesus was Adam awake but I can kind of read between the lines saying here's a witness of mind awake. So I titled the Foundation for the Awakening Mind back in 1999 and then that foundation is still going today and it's still extending beautifully with websites and all kinds of different things and then when we got the guidance to do a church and it was like oh my gosh I can't tell you the different people in the world that wrote in and said please David don't go down that road oh my God basically like when Jesus came they said nothing good can come from Nazareth when we were guided to do a church people said nothing good has ever come from a church everything rotten has come from churches but we were guided and then soon a name came in and eventually it evolved known as living miracles we were just talking recently we like that combination of words and symbols we've been giving living miracles and then it became pretty much a global ministry and it seemed to grow and grow and grow and at times shrink and shrink and do whatever things do but we've never really been concerned about the form of it because it's the content it's the love and the forgiveness that we need to experience to reach that love and to experience that love that's the most important thing so it's kind of fun every day we really don't have an idea we don't really have any ambition for the world we don't have goals we don't ever have meetings where we sit down and go okay what are our goals for this year the goal would be to let go to believe there is a year okay we need to forgive the concept of a year you see how different it is from how the world seems to operate which is always bigger, better, faster, more how can we grow how can we expand we're just not interested in that a state of mind is really what is our full focus and attention to be in the kingdom of heaven is merely to put your full focus, your full attention on it I thought you were going to say to be or not to be start quoting Shakespeare huh yeah to be divested in the form to be in the glee the joy at the moment yeah what I'm doing David is also that I cannot sell it to in front of my family I cannot sell what I'm doing if you know what I mean yes right they aren't buying it there's one thing that I can show you know what I can show is oh yeah they have their own invitation of it so that's that's difficult sometimes a lot of times it is just my struggle really my struggle to find peace when I believe in I've seen it I cannot refuse that I've seen it I know that I want to go this way and then the other side is so terrible, difficult yeah you've clearly been called by God and when Jesus says I'm calling you out of the world he's basically saying I'm calling you out of the belief that you are in the past the past is over and gone and the way that we experience the past is over is we need new witnesses you know and just you coming over to America just all the witnesses that you've had since you've been here these are like witnesses coming your mighty companions coming and marrying Jane Marie you know it's like it's a new life you've told me that too it's like I don't just want to marry another woman I wouldn't have given up everything that I gave up just to go marry another woman that would not be enough it has to be part of a much bigger plan a much bigger extension where basically let's just bring it back to the mind that the mind is just now onto a new purpose it's going away from the ego's purpose of hatred and it's coming to a new purpose and you are simply calling forth witnesses and once I started to see that for myself I was like okay then I don't ever have to be like Lot's wife in the Bible and turn back and turn into a pillar of salt which is pretty graphic I remember when I first watched that on the big screen you know and you're a little kid and you see all the people around you don't look back don't look back and then she does it she turns back and then there's this block of salt this pillar of salt and you're a woman and that was just really striking about how whenever we look back to the past we try to make any sense of anything from the past we can't you know I've had to just let go myself and just let the witnesses keep coming in I was doing a movie together at the monastery a while back and I got this piece of mail and I opened it up and it was an email it was actually an email and I looked around and I said this is amazing and I said I started to read it to everyone there so I'm gathered with all these deep spiritual friends and I've lived and worked with all these years and the email was going you need to find a new set of mighty companions and I read it to all my existing mighty companions because this is an email I just take everything to come in it's like you need to find a new set of mighty companions and what it was it was saying it was talking about breatharianism you know it was talking about why do we even need to eat at all and it was like literally a call into the mystic and she continued to write oh my gosh and write and write and send and send boxes and so on and so forth but the initial symbol was I saw it as a beautiful symbol of opening to transcendence to complete transcendence we can't stay fixed we can't stay locked into anything we can't take anything and make it comfortable and familiar and say oh this is this is that we can't look back and we can't sing that old song those were the days my friends we thought they'd never still I have not to look back anymore we can't have any sentimentality and even with the way we see the landscape now we can't just assume and project a future based on the way we're seeing the landscape now that would be just trying to use the past to make project another future and the course is saying don't do that I was thinking about that too like Israel what's going from Egypt and when you're in desert they look back to Egypt oh it was better than Egypt you know old habitual comparison so that's and I'm for sure that's not what I want I'm longing so I'm longing so for to live the life with God just to discover more and more what I have seen I've clearly seen things and I'm longing so for it so yeah I don't want to look back yeah I join you in that that's you have no idea maybe you have an idea you have no idea how it would be that when you wrote me I made you all the way man that was really I feel that about everyone I meet I feel like we are lifelong companions it's like we the earth and all of its heaviness and gravity is is the ego and we are strapped in with our booster rockets and we are packing a bunch of extra rockets because we are taken off from this gravity we are breaking the gravitational field I always thought of that one time I used to remember growing up watching those big Saturn V Apollo missions with the big three stages and all that fuel and this tiny little capsule I was like that's me I'm strapped in and they've got Apollo rockets have got like three big compartments I've got like five ten I don't know I've got extra people that you have way more fuel than you need just in case I'm going beyond gravity I'm going beyond the weight of the heaviness the judgments of this world that tell us we're just human beings that tell us nobody's perfect nobody is perfect but Christ is perfect and Christ is our identity so why should we hold on to limiting thoughts you know I love that line in the Course in Miracles I think it's in the teacher's manual where Jesus says God's teachers are not perfect or they would not be here and so they come to teach perfection over and over and over three times Jesus uses the word over over and over and over until he learned what perfection now that's a goal I can go for no wonder I could never be in high school and college you know what color is your parachute I don't know how high how high can you go the sky is the limit well I don't even like that the sky is the limit I can do better than that Jesus is telling me to teach over and over and over until I've learned it and I said alright now that's something I can go for the rest I was really just sitting on the sidelines and saying I'm not yet I'm not going to put my heart into it unless it's truly fulfilling unless it's truly reachable unless it's inevitable you know unless it's the will of God why would I put my energy in it so that's what was beautiful and that's why this comes out because all this is the same as the website it's just an expression of Latin I be single really really go for it and then nowadays we have what are called like the crystal children and they're called by many different names but they're showing up more and more in my awareness they they don't seem to have that baggage like most of us have all seem to come with we seem to come with an awful lot of baggage like a big load of it and these crystal children they're showing up and you can feel it in them they just have a look about them they have a way about them they're just symbols of transcendence some of you the last time I was here I think we watched the movie Tron over at Andy's house we watched Tron and I did Jason and I and Nikita were there we were doing all these metaphysical interpretations of Tron and you remember in that movie that the ISO the ISO's were discovered inside and they were you know destroyed they were wiped out but then the one ISO remained and she was very important and she was part of the escape from that captive electronic world and I feel like that's we're just in a time right now where it's very exciting because these beings are showing up they're just ourselves they're just reflections of ourselves but without all the baggage and they're symbols of inspiration that way so for our community I think we're just going to see more and more of these ISO's showing up just because they're showing up right on time right on cue you know just heralding the awakening what's the difference in the personal children you've heard about indigo children indigo? yeah same thing I mean I think of them in the same way yeah yeah okay