 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... Let me give a value to this norm. 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. Let me first write a proposition which is a local contraction mapping so suppose X is a complete metric space suppose there exists a subset U in X which is also complete U is also complete in the metric of X and suppose we have F of U is contained in U and F is a contraction on U then what can we conclude? Exactly, we can conclude everything exactly the same as for the contraction mapping theorem. This is a contraction mapping theorem on U so it's trivial here because basically because F of U is mapped into U then we can just think of the map F as defined on U it's a contraction on U, U is a complete metric space so we have exactly all the results of 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. So then by the mean value theorem F is a contraction on U and F of U is contained in U. 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