We have to come to terms with the fact that technology is playing a huge role in the future in the current setup and women contribute to 50% of our population. So it's really important that women are part of this whole technological future and it has been challenging I should say that because field is still very male dominated. So right now we have a digital women program in three different cities. It started in Berlin but then we opened in Munich and now we have a very successful pilot as well in Copenhagen. Lots of people coming from abroad but they were lacking a place to learn, to get trained. So this is why we need platforms like Ready School to connect people together. There's something so rewarding and satisfying with looking at how students progress, how enthusiastic they are to learn about the subject. I'm glad that I can pass on what the knowledge I've learned over the years to some women and hopefully empower them to go and explore and do their own thing. I'm from this favela in Rio de Janeiro and I have all these systems say for me, how you can do this, you're not for technology, and now I'm here. Many women here at Ready School get a chance, I worked with a computer program for six years but after the war in Syria there was no work, there were many problems and I've been looking for a practical area but in German. It's not just for me, it's for who comes after me. I need to be inside of this field because I really believe the technology is the key for change. There are 124,000 open positions in IT so bringing girls in coding is definitely a goal for the future.


Hello everyone, I am Dr. Rancida. Today I will be talking about Obturation techniques. The success of anaerotic treatment depends on three factors, that is the debridement of the canal, disinfection of the canal and Obturation of the canal. Debridement falls into your biomechanical preparation, disinfection comes under your irrigation and Obturation is where you fill your root canals so that you are obtaining 3D hermetic seed. According to the word Obturation, the word Obturation means to obstruct or to occlude. It is defined as a method used to fill and seal a clean and shaped root canal using a root canal spreader and a core filling material. Basically you are filling the canal and sealing it using a core filling material and also with the help of the spreader. Now you might wonder why is Obturation required we have already debrided the infection and disinfected the canal. It is to reduce the coronal leakage and the bacterial contamination and also to seal the apex from the periapical tissue fluids. And Obturation also entombs the remaining irritants in the canal. What does this word entombs mean? Imagine the canal after biomechanical preparation and irrigation there will be some amount of bacteria that might be present within the dentinal tubules or in the lateral tubules or accessory canals. But once the main canal is sealed the bacteria gets entombed inside the canals so that their virility will be reduced. Coming to the objectives of the Obturation technique it should be fluid impervious, fluid tight and bacteria tight sealed. These are the objectives of the Obturation technique. Now coming to the Grossman's requirements for an ideal root canal filling material. What all should be the ideal qualities that a root canal filling material should have? It should be easy to introduce into the canal. It should seal laterally as well as apically. The root canal filling material should fill laterally and apically canal. It should not shrink after being inserted. It should set slowly. It should be impervious to moisture. It should be bactericidal, radiopic and should not stain the tooth. It should not irritate the periratical tissues. It should also be sterile and easy to sterile. And it should be easy to remove also in case of necessary. Among all the root canal filling materials, Ghatapacha is the one which meets most of the requirements and is still considered a better choice for the Obturation. Coming to the classification of Obturation techniques. According to Grossman, the Obturation techniques can be cold lateral condensation, warm compaction. Under warm compaction it can be either through vertical or lateral. Then comes continuous wave compaction technique, thermoplasticized Ghatapacha injection technique, carrier-based obturation technique. Under carrier-based you have thermo-fill, simply-fill. You also then you have mixed pattern thermo-mechanical compaction, chemically plasticized Ghatapacha and custom-cool. These are the different types of obturation technique. According to Cohen, the Obturation techniques are classified as lateral compaction, warm vertical compaction, continuous wave compaction, warm lateral compaction, thermoplastic injection techniques. So basically there are two basic procedures for the Obturation. That is either you obtrude the canals using lateral compaction of cold Ghatapacha or vertical compaction of your warmed Ghatapacha. So either you're laterally condensing your cold Ghatapacha or vertically condensing your warm Ghatapacha. All the other methods are variation of the warmed Ghatapacha technique. The first one is the lateral compaction of the cold Ghatapacha technique. This is the most commonly used technique especially in case of your anteriors also. What exactly do we do in this lateral compaction technique is that see before obturation the first the important things that you should check is you should isolate the tooth and dry the canals with the paper points before obturating. After the biomechanical preparation you should select a master cone and you should check for the apical tug back.  If in case the Ghatapacha is beyond the apex you can either cut if it is very little beyond then cut the tip and reinsert the cone and take a working length again x-ray again if not insert in larger size Ghatapacha and then verify it. In case it is short of the apex then either place a primary GP and then smaller size GP and verify or increase the size of the canal. If it is at the working length then you can after you confirm it with the radiograph you can go ahead to obturate with it. So now you've done the BMP you've selected the master cone now what you should do you should coat the canal with the sealer. Sealer using lentilow spiral or master cone you can take the sealer either in lentilow spiral or in master cone and coat it all over the canal then you can insert the master cone until the working length. After you insert the master cone a spreader can be inserted alongside the master cone this spreader has to be inserted up to 1 mm short of the working length for 10 to 60 seconds it's like almost a minute then after that you can disengage the spreader.  See this is how you can do it. See master cone is placed up to the working length after that a spreader is inserted up to 1 mm short of the working length and laterally compacted so that the space you know that accessory canals are placed into the space that is formed between the canal wall and the master cone. The same method is repeated again and again till there is no space for the spreader to be placed the entire canal is filled with the master cone and the accessory can or Gattaparcha. After that the excess Gattaparcha should be removed from the crown. See the level till where the Gattaparcha is removed it is removed below the free gym level almost it should be below your CEJ so that you can do the access filling.  See this is the picture in case of your molars. See lateral compaction is not only done in case of your anteris it is done it can be done in all the tooth especially in cases of the oval route oval shaped canals it is very useful. The Gattaparcha should be cut at the level of CEJ in case of anterior tooth in case of posterior tooth it should be at the orifice are approximately 1 mm below the orifice. The other method of obturation is worm vertical compaction technique. This method was introduced by Schilder with the objective of filling the main root canal as well as the lateral and accessory canals here not only the main root canal but lateral and accessory canals are also filled with this technique. How exactly this technique is done is here heated pluggers are used and pressure is applied in a vertical direction on heat softened Gattaparcha. Basically softened Gattaparcha is condensed vertically by applying pressure using heated pluggers therefore what happens they'll be flow of your Gattaparcha which will fill the entire numen of the canal. See the master cone is placed in the canal a condenser is placed wood and the pressure is applied so the Gattaparcha is compacted at the apical third first then the middle third then the cornel third. So coming to the steps involved in this worm vertical compaction first and foremost the fit of your master cone should be verified your master cone should be 0.5 mm short of the apex the reason being short of the apex space when you apply pressure using the heated pluggers what happens your Gattaparcha will flow if your master cone is exactly at the tip of the apex there are chances that it might overflow out of the canal so you have to always be careful that your pressure that you're applying is controlled and your master cone is short of the apex there are different type of pluggers there are pluggers which prefit the coronal third pluggers with fit the middle third and the coronal third according to the thirds of the canal first and foremost a sealer is deposited with a handy lentilow spiral basically you have to coat the canal with the sealers then a largest pluggush be used to compact the worm Gattaparcha into the bolus so what happens when you're using the largest plugger the lateral canals in the mid route will get obturated then after this you can remove the excess GP that is there from the canal then again you can take a mid-sized plugger and try compacting it from packing the Gattaparcha this will causes the Gattaparcha to flow into the lateral and accessory canals in the apical third after that you can then change the plugger into the apical size plugger when you condense it the Gattaparcha will flow into the lateral and accessory canals at the apical third so basically you're sealing the apical third first and the lateral and accessory canals at the mid route and the apical third after your apical third is filled then the canal is filled back in your middle and coronal third will be obturated up to the level of your CG that is all about the worm vertical compaction technique coming to the worm lateral compaction technique this provides the advantages of both the thermoplastic technique as well as the lateral compaction technique what are the advantages is see this involves placement of master cone and laterally compacting as I told in warm vertical compaction technique there are chances that your Gattaparcha might overflow out of the canal so that risk is avoided in case of your worm lateral compaction where here we are not compacting the Gattaparcha vertically instead we are compacting the Gattaparcha laterally is using heat carrier such as Endotech 2 tips or Endotwin 2 tips basically instead of vertical compaction your lateral compaction is done this is the different types of heat carriers touch now heat is a heat carrier that has been designed by Schilder it's an electronic device specially developed for the worm Gattaparcha technique next method is the system be continuous wave of obturation the system be technique is a variation of warm vertical compaction technique introduced by Buckman this technique employs the use of Gattaparcha cones and pluggers that mimics the tapered preparation thereby permitting the application of greater hydraulic force during the worm compaction basically they're telling that the force that we applied during the condensation can be much more because of the tapered cones which fit exactly into the prepared canals this is the system be technique so coming to the steps involved in non- standardized Gattaparcha cone is chosen for its adaptation into the shape of the root canal space then you also place the plugger of your system be and select the pre-fit to its binding point you can check whether the Gattaparcha is fitting perfectly in Gattaparcha and your plugger is fitting into the canal perfectly a pickle 4 mm of the master cone is coated with the sealer and using gentle pressure the cone is seated into the place the system be heat source is turned to use and touch mode basically after you place your Gattaparcha you place your system be plugger and turn it on so that it gets heated then you turn it off you condense the Gattaparcha as the heated plugger approaches the binding point the rings which is released which allows the plugger to continue its apical movement because the Gattaparcha is softened so what happens the you can move your plugger in the apical direction the apical pressure is maintained on the apical plug of Gattaparcha for a 10 second sustained push to prevent cooling shrinkage while maintaining the apical pressure the rings which is activated again for one full second and the switch is released and the plugger quickly withdrawn so that you don't retract any of the Gattaparcha coronal flash of the Gattaparcha is then removed and a clean interface homogeneous mass of Gattaparcha sealing is obtained in the apical third after this sealer can be applied to the remaining canal length of the canal and the canal can be back filled the same steps used to fill the apical canal that checking the prefit of your pluggers then condensing the Gattaparcha is carried out for both the middle third and the coronal third of the canal once the canal operation is completed you can access do the access film thermoplasticized Gattaparcha injection techniques thermoplasticized Gattaparcha injection techniques this technique comprises of a pressure operator consisting of an insulated electrically heated syringe barrel and a selection of needles ranging from 18 to 25 God size you can select the needles depending on the size of the canal basically the plugger is designed so as to prevent the backflow of the Gattaparcha and the degree of the heat that is applied will be regulated so that there is no extrusion of the Gattaparcha in outside the canal example for such types of technique is optura 3 basically thermoplasticized Gattaparcha is a technique wherein you deliver the heated Gattaparcha in syringe and you can compact it using the plugger next one is the solid core carriers under solid core carriers there are various varieties of solid core carriers one among them is thermo fill the other one that i'll be talking about will be simply fill thermo fill is a carrier based Gattaparcha operation technique in which the core carrier is a plastic material to a plastic core the alpha phase Gattaparcha is coated onto it which will be placed inside the canal they are available in various taper sizes also they are available in ISO standard sizes also these operators are used in conjugation with the heating device basically you need a heating device so that the alpha phase Gattaparcha can be heated and placed inside the canal such heating device is called as thermo prep plus oven so how exactly is this done is after your biomechanical preparation you can verify the thermo fill various instrument you can place the instrument then check what size material do you require after selecting the appropriate sized obturator you can dry the canal and coat the canal with a layer of sealants then you can place the carrier into thermo plus oven first for approximately 10 seconds then you can place the carrier into the canal up to the working length with a firm uniform apical pressure you shouldn't rotate it the position of the carrier is then verified radiographically and the Gattaparcha is allowed to cool for 2 to 4 minutes before selecting the carrier at the level of the canal orifice the carrier is introduced into the canal placed up to the working length then it is cut off at the level of your orifice the other type of solid core carrier is simply filled sectional obturation technique this is also a carrier based technique in which the carrier it is it is also a carrier based technique which is used along with the light speed rotary instrument in thermo fill the entire canal was filled with one single obturator but in case of simply fill only the apical 5 mm of the canal will be filled with the simply fill technique here the apical 5 mm of the simply fill carrier is coated with the Gattaparcha and then it will be placed inside the canal after choosing the appropriate carrier size you have to follow the basic steps like drying the canal applying the sealant and then you can introduce the carrier up to the working length after introducing it into the canal you can rotate the handle of the carrier in a counter clockwise direction once you rotate this in a counter clockwise direction what happens it disengages the apical plug of the Gattaparcha from the carrier so what happens your apical 5 mm will be filled the remaining coronal space can be filled using the lateral compaction or thermoplastic size Gattaparcha methods the next method is the max python thermo mechanical compaction method this method uses the heat to decrease Gattaparcha viscosity and increase its plasticity how exactly is this heat generated in this technique is heat is created by rotating a compacting instrument here they are using a compacting instrument and rotating it at a slow speed contra angle and piece of 8000 to 10000 rpm alongside the Gattaparcha cones after placing the Gattaparcha inside the canal they're placing this compacting instrument and rotating at a speed of 8000 to 10000 rpm what happens this compactor forces the softened Gattaparcha apically and laterally that's how it fills both apically and the lateral canals what exactly happens in this case is after the step back method the canal can be verified the Gattaparcha cones can be inserted and you can verify the whether it is going up to the apex or whether it is short then you can also select the compacter blade according to the width and the length of your prepared canal by inserting it between the Gattaparcha and the canal wall next one is the chemically plasticized Gattaparcha technique this is nothing but you're using chemical solvents to soften your Gattaparcha so that you can condense and adapt it and fill your canals but the greatest disadvantages of this technique is the inability to control the overfilling what happens when the Gattaparcha gets dissolved there are chances that it might overflow beyond the canal with so there are chances of periapical tissue reaction and also there are chances that the material might shrink after setting resulting in a poor apical and lateral shape this is the reason this technique is not so commonly used these days the last technique will be the custom cone technique it is a chair side procedure employed for customizing the Gattaparcha in case of wide canals where the traditional cones cannot be used to adapt the canals in this technique a few Gattaparcha cones or softens along with the help of the eat and roll together between the glass laps or a spatula might be used to roll the Gattaparcha so that we can create larger size master cone is created which can fit into the size of the canal you can see in this picture smaller size Gattaparcha cones are taken and rolled together after eating and between glass caps and a larger size Gattaparcha cones is made to fit into the wide canals so these are the types of obturation techniques based on the size of the canal length of the canal you can decide the obturation techniques that is required and obtain a 3d hermetic seal of the canal thank you


It's a lot to the organizers for invitation. So what I would like to do in these lectures is to give you an overview of a number of classical results in holomorphic dynamics, in a number of classical results in one-dimensional holomorphic dynamics, and try to understand what they become in higher dimensions, okay? So I think it's one of the ideas of this conference of how to go from one-dimensions to higher dimension for groups, and I will try to keep this idea in mind of stating classical things in one dimension and see how they evolved in higher dimension. So the one-dimensional part will be very classical, so perhaps many of you have already heard about this with great details, so I apologize for that. I will give a number of proofs, at least for the one-dimensional case. So the overall plan would be something like that. So in the first lecture, so if everything is fine in the first lecture, I want to talk about the Fatou-Juliet dichotomy of dynamical space, so again in one and several dimensions. In lecture two, I want to talk about equidistribution aspects. So I want to talk about equidistribution, so there will be a little bit of ergodic theory, and lecture three should be about parameter spaces and bifurcations. So hopefully I'll be able to maintain the schedule, it's not clear whether this will not contaminate the whole three lectures, we'll see. So again I will try to give you some proofs in one dimension and some statements in higher dimension. Okay, so let's start with lecture one. So lecture one is part R and part A and part B and part A is dimension one. So I want to iterate an algebraic map of one-dimensional algebraic variety. So I take a certain f from x to x, where everything is algebraic and x is one-dimensional. And what I want to do with f is iterate it, so consider forward iterates of this. So I view this as a dynamical system and I want to predict the future of points. I have a point I would like to know what it will become in the future. So the first thing is that if you want to iterate an algebraic map on an algebraic rim and surface, you don't have many, many choices. So first case, so let's start with the easiest case, first case that x... So when you have an algebraic rim and surface, you can always compactify it by adding a few points. Okay? So you can always assume that x is compact. You can always assume that it's a compact rim and surface, okay? And f will extend homomorphically to the compactification. This will be different in higher dimension. So first case is when the genus of x is larger than 2, and in that case, just by looking at the Euler characteristic, you see that f must be an automorphism, and in that case, the automorphism group is finite.  So in that case, of course, for dynamical systems, it's not very interesting. We'll throw that away. Second case, genus 0, genus 1, when x is an elliptic curve, in that case, your map must descend from an affine map on C, and f descends from an affine map. In that case, the dynamics can be complicated, but still it's completely explicit. You can compute the iterates. So you will consider that it's not interesting from the dynamical point of view. So let's say it's easy. Even if, again, the dynamics can be chaotic. And the last case, which will be the one that we are interested in, is the case of rational mappings on the Riemann sphere. So that's the genus 0 case. So genus 0, okay? So in that case, x is the Riemann sphere, or equivalently the one-dimensional complex productive space. So we'll switch between these two ideas of a sphere versus productive space. I mean, I think most people here know about the relationship. And f is a rational map. So the degree of f is the maximum of the degree of the components. Again, the situation is not very interesting when the degree is 1. So I always assume that this is larger than 1. Okay? So this is my dynamical setting. I want to iterate a rational map on the Riemann sphere. So that's the setting for all this one-dimensional part of the lecture. Okay, so that was section 1, setting. So section 2, a number of basic concepts. So one important basic concept here is that of a normal family. So a normal family, polymorphic mappings from a certain domain. So omega is a domain in an open set in the Riemann sphere. So the Riemann sphere is just a relatively compact family of polymorphic mappings. So now this normal stuff is just all terminology. It just means that it is relatively compact in the space of polymorphic mappings. Okay? And it's also very classical. That's, it's the same as being a quick continuous, a quick continuous for the spherical metric. Okay? So there is one important and classical theorem here, which is Montel theorem. So that's one basic tool, Montel's theorem. Any family of polymorphic mappings avoiding three points is normal. So any polymorphic family, no, let me stay like that. Any family of polymorphic mappings from some set omega to the sphere minus three points is normal. Okay? So of course this must be a distinct point, right? And the proof in two lines, the proof is that the universal cover of this is the unit disk. The proof is that the universal cover. So consider the universal cover. So this is covered by the unit disk. And if you take any homomorphic map, if you take any homomorphic map from omega to C minus three points, C hat minus three points, you can lift it to the universal cover. And now a bounded family of homomorphic mapping is, okay, so perhaps you should assume that omega here is not the, is contained in C, right? It's a proper domain. So if it's proper, it avoids one point. And if it avoids one point, you can always assume that this point is infinity. Okay? So now it's a family of homomorphic mappings in a classic, a classical sense from omega to the unit disk. And then it's just, it's equicontinuous. And if you take the homomorphic mapping from omega to the unit disk, it is equicontinuous. Look, equicontinuous I mean on compact subsets, okay? So it's locally equicontinuous. Just because of the Cauchy estimates, okay? And that's the proof. So that's one basic tool. The other important tool in one dimension is the Poincare metric. I think it has been seen a number of times in this conference. So I don't want to say too much about it. So what do we want to say? So a Riemann surface is hyperbolic if its universal cover is the disk. And essentially all Riemann surfaces are hyperbolic except, so X Riemann surface is hyperbolic unless it belongs to a small number of examples. It is, so there's a sphere, a plane, puncture plane, C mod Z. So if X is not one of these examples, then it's a hyperbolic Riemann surface, okay? And then you get a Poincare metric. You get a Poincare metric on X just by pushing the metric from the universal cover. And the important property here is the Schwarz lemma. Schwarz lemma says that a homomorphic mapping between hyperbolic Riemann surface is always weakly contracting. So let's see a homomorphic mapping between a hyperbolic Riemann surface, between hyperbolic Riemann surfaces. Then the distance, it is a weak contraction. So the distance with respect to the matrix of X prime between images is less than the distance at the source. And also you know, even more than that, you know the equality case. And actually it is a strict contraction. So it is a strict contraction unless, unless F is a cover. So of course, you imagine that in dynamical situations it is very useful information to have contractions. Okay. So now let's start to do dynamics. So what's the Fatou-Juliet dichotomy? I think everybody in this room knows what it is. Okay, so let's define the Fatou set. So again, I take F, a rational mapping of degree at least two from the sphere to itself. And I define the Fatou set. So the Fatou set is the local equicontinuity set. It is the set of point Z in the sphere such that there exists a neighborhood of Z such that the family of iterates restricted to this neighborhood is a normal family. So you see that by definition it's an open set and it is invariant under F and also invariant under F inverse. So it is open and totally invariant, meaning that F of the Fatou set is also F inverse of the Fatou set is the Fatou set. And the Julia set, it's just, it's just a complement of the Fatou set. So it is a set of points where the family is not locally equicontinuous. Now I realize that I should have prepared a number of pictures, but I didn't. So you just have to imagine pictures of Julia sets. I think there may be some in the back of the room now. In every mathematical room there should be a picture of a Julia set, but for some reason here there's no. Okay, so let me give you a few examples. No, let me not give you a few examples. So you know that you get these fractal pictures, and the, well, most famous factor is the circle corresponding to F being Z to the power of D. So, example, F of Z is Z square. So what happens to points under iteration of Z square? Well, if the modulus is smaller than one, you go to zero. So it's equicontinuous. If your modulus is bigger than one, you go to infinity, which is also equicontinuous for the spherical metric. And if the modulus is equal to one, you don't know where you want to go. You are separate, well, you are in between the two. So certainly in the neighborhood, the family cannot be normal, right? So the Julia set is the unit circle, okay? And that will be our only example of Julia set. Okay, if yesterday, this morning, we heard the name quasi-circle, okay? So if you perturb a little bit this, and actually, epsilon can be as large as one-four. Then the Julia set is homeomorphic to a circle, is homeomorphic to a circle. So you see a deformation, homeomorphic to the circle. So not only it's homeomorphic, but it deforms, it deforms continuously. So you start with a circle, you start with a circle, and when you start to move epsilon, so I should have used this one, all right? You see this circle moving to a picture which is not smooth, which is a quasi-circle. So this could be a quasi-circle. So a quasi-circle has no inward pointing cusps. So when you draw one, you should make cusps point on the outside. Anyway, so when you deform, when you take small epsilon, you see a picture like that, okay? And when you go, when epsilon is larger than one, real and larger than one-four, you have a counter set. And you get plenty of pictures. Let's proceed without the pictures. And so the Julia set is closed, and also it is totally invariant. So the first thing I will do is to characterize, well, to talk about this closed total invariant set. So let's say that E is an exceptional set, is exceptional if it is finite. So it is a subset of the sphere. If it is finite and total invariant. There's a proposition here, easy proposition. There is a maximal, a maximal exceptional set. Let's call it E naught. And the cardinality of this set is at most two. So what it says that you can have, it is, let's say also that a point is exceptional. The point Z in the sphere is exceptional if it belongs to an exceptional set. It means that equivalently, it has finitely many prime edges.  And what the proposition says is that there are at most two exceptional points, OK? That's the worst you can do. And the proof is? Yes, OK, I don't want to, I don't want, but you don't need to assume that. I mean it's just, yes, yes. Perhaps you need to add it in the definition. I don't think so. Well, the important, perhaps I should have just talked about exceptional point. An exceptional point is a point which has finitely many prime edges. That's the important concept, OK? And the proposition is that there are at most two exceptional points, OK? OK, the proof is easy. The proof is just the remand of its formula. So if you have an exceptional set, you get a mapping from C from the remand sphere minus the exceptional set from itself to itself. So F induces a mapping like that. You apply the remand of its formula. So the Euler characteristic of sphere minus E d times is equal to d times itself minus a certain ramification term which is non-negative, right? So this cannot be a negative number, otherwise it would go to infinity. So it means that this number has to be non-negative. It has to be non-negative. But this number is just 2 minus the cardinality of E. Just 2 minus cardinality of E, OK? So you get the result. So this has consequences. Which consequence? So another proposition. So let me just, OK, just a classification, perhaps a small classification of the possibilities here. It happens in generic cases, the exceptional set is empty. OK, you have no exceptional point. Now if the exceptional set is just one point, you can always declare that this point is infinity. And what is a rational map with infinity total invariant? It's a rational map with no pole. It's a polynomial. So E, one exceptional point corresponds to polynomials. And two exceptional points, you only have the examples of z to the d and z to the minus to conjugacy, OK? So a generic case, no exceptional point. One exceptional point means polynomial. And one last, and there's one last case. OK, proposition. If z is any point which is not exceptional, then if you take the union of prime edges of z and take the closure, then it must contain the Julia set. And the proof is just Montel theorem. The proof is that if I take z not p in the Julia set and a neighborhood of p, then the union of forward images of this neighborhood, it's a certain open subset which misses at most two points. And it is invariant. So it is open. It is forward invariant. And it misses at most two points. So it means that its complement must be an exceptional set. So the complement of this is exceptional. So z is not here. So it means that you take prime edges of z, you must enter n. So when you take prime edges of z, you must get close to p. OK, so that's the end of the proof. And the corollary of that is that if now... So you know, I should have made a remark that the Julia set is never empty. It's just because the degree is at least two. So you know that some complexity should arise under iteration. So the Julia set is not empty. And the corollary of that is that the Julia set is actually infinite and also perfect. And the reason is that if... The proof of that is that if z is a point in the Julia set and we know that it's not empty, then the Julia set is just the union, the closure of union of prime edges of z. And it's... An exceptional point is always in the Fatou set. As I said, one possibility is infinity for a polynomial and the other possibility is zero and infinity for zd. So you are in the Fatou set. So if you start with a point in the Julia set, you know that it's not exceptional. So when you take prime edges, you have this property and then you get equality here. So it says that the Julia set is perfect and infinite, right? So we get one first property of the Julia set. So there are plenty of other... So we saw this property for limit sets this morning, right? Another property which I won't prove is a theorem by Fatou. And Julia, I would say Fatou and Julia. No, excuse me, I will say it later. Sorry, sorry, sorry. First I want to talk about periodic points. So that's section three, four, periodic points. So finding the periodic point of period n for f is just solving an equation of degree d to the n, right? So you know that f has d to the n plus one, periodic points of period n, counting multiplicities of period dividing n. This is just because you solve the equation fn of z equals z. It's just an algebraic equation. So you notice you have plenty of periodic points, right? So now if z naught is periodic, if z naught is periodic of exact... So now when a point is periodic of period n, it's also periodic of period 2n, 4n and so on. So there's a minimal one and I will say that it's the exact period. So if z naught is periodic of exact period n, then you define, you can look at the derivative of fn at this point. This is a number. So you have to consider the spherical metric here, but since you consider it at the beginning and at the end, you just can consider it as a derivative, right? So this number is called the multiplier of z naught, okay? And just by applying the Taylor formula, you know that the multiplier determines the action of f in the small neighborhood, fn in the small neighborhood of z naught. So it determines the... At the first order, it's the main... Excuse me, I missed the word. It's the main character for the local dynamics of fn at this point. So there are a few cases. So let's assume... So you can always assume that n equals 1 just by replacing f by fn. So first case, the multiplier is less than 1 in modulus. Then the point is attracting. It attracts a neighborhood. Z naught is... Let me keep the... Let me keep the... Z naught is attracting. There is not much to say. You have a point z naught here. It has a certain basin of attraction, and all points nearby go to this one. And then it belongs to the fatal set. Then if the multiplier is bigger than 1, and the point is a repelling, so in that case, you cannot describe the whole dynamics of a point in the neighborhood because it will go far away, and then it does something. In this case, you know that the sequence of the derivatives at z naught goes to infinity, so you must be in the Julia set, and you cannot say much more than that. Third case, more interesting one. Multiplier is 1. Then you say it's neutral. Then you can have several subcases. So let me describe a few of them. First there's a lemma. So I would like to know whether a neutral point belongs to the Julia set or the fatal set. So there's a lemma here. So a neutral fixed point, neutral periodic point, fixed point, so again you can always assume it's fixed just by replacing f by fn, belongs to the fatal set if and only if it is linearizable. So linearizable means that you have z naught here, which is excuse me, and you want to know if it is conjugate to the action of the linear part neighborhood of the origin where lambda is just the multiplier of... So I'm going too fast here. You know that the multiplier determines the local dynamics of z naught. So you can always imagine that z naught is zero, of course. And you want to know if it is conjugate or not to the multiplication by lambda. And what I say is that this is true if and only if the periodic point is in the fatal set. So one implication is clear. So the linearizable implies fatal is clear because if the map is conjugate to a multiplication by a complex number from this one you have a quick continuity. For the forward implication, there's a little trick and the trick is this one. You consider hn it's a sequence of mappings and this is this one. 1 over n sum of... So I assume that z naught is equal to zero. So that's a local statement. It has nothing to do with rational mappings. It's a statement for local mappings defined neighborhood of the origin. So I assume that z naught is zero and f is just a germ from c zero to itself with a multiplier of lambda. And I will consider this sequence of mappings. Not good here. So we'll define a sequence of homomorphic functions like that. So it's a cesarean sum of this sequence. So if I name this gk, you remark that gk plus 1 is just lambda gk plus 1 is just gk composed with f. And also you can remark that all these mappings have the relative 1 at the origin. So hn prime at the origin is 1. Now if z naught belongs to the factor set it means that fn is equicontinuous. So it has bounded the relative in the neighborhood. So lambda to the minus k, lambda is a modulus 1, right? So this is gk is equicontinuous. So hn is also a normal family. So you can extract hn is a normal sequence. And now by construction hn plus 1 is the component composed with f minus lambda hn goes to 0 by construction. This is just because when you compose with f it just corresponds to a shift in this sum and you have this 1 over n. It's the same trick as you know constructing a fixed point by taking a cesarean. When you have this when you want to prove this Markov's calculating theorem that when you have a compact convex set it has a fixed point. You just take cesarean and if you take any cluster value of the cesarean it must be a fixed point because the terms just cancel. And here it's exactly the same. When you compute this you get that this is going to 0. So if you take any cluster value h of this sequence satisfies h composed with f is lambda h. And also it has the relative 1 So it is a linearizing map. It's a local diffeomorphism which conjugates f to its derivative. So that's the proof of my lemma. Proving that a neutral point belongs to the Julia set is exactly the same as proving that it's linearizable. So a corollary of that is that so in the neutral case if you have a neutral point you have two cases so you write the multiplier as the exponential of 2 pi theta. So either theta is rational or it's irrational. So if theta is rational you see that it's a parabolic point and if it's not I don't know you say that it's just an irrational neutral point. In that case you see that the linear map has finite order. So it cannot happen for f because for my rational mapping my rational mapping has a degree bigger than 1 so it's not a finite order so a corollary of that is that a parabolic point it is not it cannot be linearizable in my case and actually it belongs to the Julia set. Actually you can say a little bit more about the dynamics near a parabolic point so the typical example is like that so the typical example is f of z is z plus z square so the derivative at the origin is 1 we just have one perturbative term and the dynamics near the origin so on the real axis you know the dynamics on the real axis it is z plus z square something like that it is attracting on one side and repelling on the other side if you start with a point here you do something like that it will go to the origin and on the other side you go to infinity in the complex number it's the same so when you have this so that's 0 0 is attracting on a certain cone on the left on the side so you have an attracting base in here and you have a repelling base in here and actually you can even say a little bit more excuse me the picture is like that you have a certain cardioid shaped picture where the dynamics is attracting like that and it is repelling on the other side so my row here is going to the fixed point so you have a certain base in an attraction and 0 is on the boundary of the base and it's repelling on the other side so certainly this is a point in the Juvia set because you have no neighbourhood with the dynamic system in the irrational case the situation is more complicated so you certainly know that the problem of linearizing holomorphic deformorphism is very complicated so let me just say a few names here so Zigo so in the forties proved that for certain values of theta I don't want to be too precise here certain values of theta every holomorphic mapping with multipliers like that is linearizable F is linearizable and Kramer in the forties I think proved that for certain values of theta F is not linearizable so there are certain values of theta which ensure that F is never linearizable unless it is already linear right? of course you can start with a linear map so actually excuse me so what Kramer proved is that for theta this mapping is not linearizable okay? and actually for quadratic polynomials so we know exactly that's a theorem of Yoko's famous theorem of Yoko's that we know exactly the set of parameters such that the mapping is linearizable there is an open question so for Yoko's we have a complete answer to this problem for quadratic polynomials and also and this is based on the work of Bruno and so there's a sufficient so the situation right now is that there are sufficient conditions for linearizability which is called the Bruno condition it is known that the Bruno condition is optimal for quadratic polynomials but it is not known for in any other setting okay? so that's an open question interesting open question for rational mapping on P1 the question of linearizability depends only on the angle okay? so it's not known if this property of being linearizable or not depends only on theta it is known for quadratic polynomials but not for general rational mapping so that's an important open question in the field okay? I'm going to slow here one theorem by Fatou and Julia which was proved in simultaneously basically the Julia set is the closure of the set of periodic points of repelling points so you look at the set of repelling periodic points and take the closure you get the Julia set so again this is a property which looks like the corresponding one for client groups in one dimension another theorem due to another theorem of a lemma or whatever is that every attracting point every attracting point attracts a critical point and the proof is two lines proof look at the basin so let's look at the the immediate basin of A so you have an attracting point A it has a certain basin so the basin has many components but you look at the component containing A which is called the immediate basin and now F so the proof excuse me okay yes just F is a self-map from the immediate basin to itself and assume that F is a covering so assume that there is no so here there's an implication I'm not completely sure of okay assume there's no critical point then F is a covering so it's true but it requires a few lines of justification so if it's a covering it's an isometry for the Poincaré metric and then you contradict the fact that A is attracting okay so it's not a covering so it must have a critical point okay so it is not a covering so you just conclude by the short slimmer finite so a corollary of that the number of attracting points periodic points is finite just because there's a finite number of critical points okay there's a slightly more complicated theorem that the number of non-repelling points is also finite so actually when you look at periodic points for rational mappings almost all of them are repelling actually the bound is 2d-2 so this bound is due to Shishikura so it's optimal and there's a list of previous bound and I think the first one was due to Fatou it was 6d-6 and a proof looks like that one proof looks like that you take your rational mapping for proof that's not you can make it work in the show take f a rational map okay so you know that it has finite many attracting cycles perhaps it has a certain number of neutral ones now if you perturb f in the space of rational map what you expect is that half of the neutral points will go inside the unit circle and half of them will go outside the unit circle so you know that the number of f is less than 2d-2 number of critical points and the number by perturbation so for nearby f you get that the number of attracting points of f plus half of the neutral points is less than 2d-2 okay so again if you perturb for generic perturbation what you should expect and this is true after some work half of the neutral point becomes repelling and half of them become attracting this is a matter of saying that there are enough parameters and also that the multiplier map so if you want to prove something like that you need to show that the multiplier map is a submersion because the space of rational map is big it's 2d-2 and also if you look at the space okay you write that you need some work at least that's an idea of proof I don't remember if you can make this work or not I don't remember but in any case the idea is that half of them will become neutral and half of them will become repelling I totally agree that you can imagine that some bad situation happens at least from this idea from a graphical analysis from this you get the number of attracting plus neutral point is less than 4d-4 so you still need more work to get this 2d-2 is optimal 2d-2 is optimal and it's done by quasi-conformal techniques okay that's it for periodic points so I want to talk about the fatu set now this item fatu components I should have made a remark I forgot one property of the Julia set so let me add it somewhere so one property of the Julia set is that the interior of the Julia set is empty unless it's the whole sphere okay again this is just mental theorem if not if you take a component of the interior it the iterates must be contained in the interior of the Julia set and must avoid at most two points so it must be written so it's just a montage here so now remember the basic problem of dynamical systems is describe the behavior of orbits or at least describe the behavior of most orbits so since the Julia set is empty interior in first approximation you can try to describe the orbits in the fatu set if you're able to describe the orbits in the fatu set you've done almost all points in topological sense so a remark here so that's the theorem by Buff and Sherita Buff and Sherita it can happen that the the big measure is positive actually in the yes it may happen that the measure of the Julia set is positive but j is not the whole sphere okay so it means that if you want to describe the behavior of almost all points in the Lebesgue sense you need a little bit more but at least in a topological sense you can restrict the fatu components and the good news is that you can describe everything okay so what are the so if you if you want to study the dynamics in the fatu set so the dynamics is locally it's a normal family so in a sense the dynamics is locally constant in fatu components so it's enough to describe the dynamics in the whole fatu components so a fatu component it can be periodic so you have a certain iterate which maps the component to itself let me remove this center here it can be a pre periodic the picture is your component which is mapped after a few iterates to something which becomes periodic and the last possibility is that it is wandering so these are the three possible cases this is so in a few minutes I will say that wandering components do not exist for rational mappings but it is easy to cook up examples of transcendental mappings wandering fatu components so example is something like that if you look at this mapping f of z is z minus the sine of z plus 2 pi so that's an entire mapping from c to c and which satisfies f of z plus 2 pi is equal to f of z so it induces a mapping from c mod 2 pi z to itself so now if you look at 0 so 0 is mapped to 2 pi so on c mod 2 pi z 0 is a fixed point and if you look at the multiplier z minus sine z it's a super attracting fixed point so in the quotient so the quotient is just a cylinder c mod 2 pi z you have 0 here and it has a certain attracting basin now if you look in c what happens you have this attracting basin which comes from the cylinder and you have the same around every multiple of 2 pi but now since I added 2 pi in the formula what is the point here it will be mapped to the next copy of the basin and so good question good question good question I guess this is because you can only have one attracting point because this point is fixed no because this point is mapped to the next one and what and what and yes I agree with that but it's not possible because what the image of 0 is 2 pi so no the argument is not this one yes yes and the same basin okay okay okay okay thanks for the argument I forgot this one okay so the argument is that the picture must be like that because if you look at this mapping you get the same picture in the quotient okay so and in that case each of these points is a fixed point so they must correspond to distinct basins okay and now if you add the 2 pi you see the same picture but now now you have this point okay thanks for the argument so in that case you get a wandering for two components okay and associated to a transcendental map and the theorem is that you can completely describe all possible types of for two components so theorem is like that so it's due to a number of people that the last step and most important one is Solivan something like 95 or 84 85 or 84 or something every so there are no wandering for two components so let f be rational so the first thing is that there are no wandering for two components so every component is pre-periodic so if you want to describe all of them it is enough to describe the periodic ones and every periodic component belongs to fixed list of types periodic factor component is of the following form usually it's written as a five item five item theorem so item one and two are basically the same that's a super attracting or an attracting basin so it means that it's the basin of attraction of some attracting point so we have type three so omega so how do we want to so there exists a certain so let omega be a fixed for two components so another possibility is that there exists a certain point z not at the boundary of omega whose multiplier is one which is fixed with multiplier one and omega is the basin of attraction of this parabolic point and omega and for every z in omega the iterates of z converge to z not so we have a parabolic point and omega is just the attracting basin of this parabolic point and the multiplier must be equal to one if the component is fixed a fourth type so that's the parabolic case and then you have two more types which are called rotation domain omega is bihomophic to a disc so omega associated with dynamics is bihomophic to a disc with irrational rotation right so that's called the Ziegle disc so a picture is like that you have a certain disc with fractal boundary inside you have a linearizable fixed point and the dynamics around is that of a rotation and type five omega f is bihomophic to an annulus with an irrational rotation so same picture so these are called rotation domains ok so this annulus cannot be a punctured disc otherwise you would get an isolated point in the Julia set so it must be a natural annulus so have an annulus like that and it is foliated by smooth circles and dynamics is a rotation on them ok so let me try to prove a theorem ok I will not finish today but let me prove it let me at least give ideas of proof so proof well so I will not give I will not explain the proof of the non wandering factor component so the proof of at least the idea of the proof of the this non wandering theorem is something like that the argument is like if you had a wandering factor component then you would be able to construct by quasi conformal deformation an arbitrary large space of deformations of your rational map ok but now the space of rational maps of degree d is finite dimensional and then you get a contradiction ok perhaps I should add a complement here I should have a complement to this theorem not trivial and a complement there are only finitely many cycles of periodic components ok so we already know that there are only finitely many non-repelling points so it is said that you have finitely many attracting basins, parabolic basins and also z-hole disks but if you want finiteness for the number of hermene... so this is called a hermene ring so the finiteness number of hermene rings is more subtle and you need again some quasi conformal ideas to prove it and so you see that this complement so this morning we heard about the alphos-finiteness theorem ok so it is the exact analog of the alphos-finiteness theorem and actually it is the clue to the proof so Sullivan saw this as the analog of the alphos-finiteness theorem and used the tools to get the theorem here basically the same ideas ok so I don't want to say more about that now let me take so let omega be a fixed fatu component fixed fatu component so there are two cases there are two cases the recurrent case and the transient case so what transient means that for every z in omega the iterates of z go to the boundary point escape to infinity in omega and the recurrent case is just the contraposite there exists some z in omega such that z has infinitely many iterates in a compacted in a compacted so you see that in the transient case all these cases are recurrent except the parabolic one so what we something that we have to prove is that every transient fatu component is a parabolic basin so let's prove that there's one statement so there's a fact statement one transient fatu component invariant fatu component fatu component is a parabolic basin so first argument in the proof so you know that for every z in omega if we iterate go to the boundary now remember that the f is weakly contracting for the poincare metric in omega so now f so of course omega is hyperbolic because it avoids at least two points is weakly contracting for the poincare metric so it means that let me call it z n z n is the nth iterator of z and z n for the poincare metric in omega is less than or equal to the distance from z0 to z1 so it's bounded but now we know that these points go to infinity in omega so it is well known that the poincare metric at the boundary of a poincare metric is proper so when you go to the boundary of a domain the poincare metric diverges with respect to the spherical metric so this implies that z n goes to the boundary implies that the spherical distance between z n plus 1 and z n must go to 0 now if you look at the cluster values of this sequence so every cluster value so the set of cluster value of this sequence set of cluster values is a connected set so that's a classical lemma when you have a sequence which says the set of cluster values is connected it's a connected set of fixed points you have finitely fixed points for your homomorphic map so it is one fixed point so z n converges to some fixed point p in the boundary you know that there must be a fixed point at the boundary and z n converges too and now why the multiplier is equal to 1 so there's something called a snail lemma which is due to Fatou and which does exactly that so it's a local station so let f be a germ of homomorphic map at the origin fixing 0 assume that there exists a certain open set such that f of u is contained in u and for every z in u f n of z converges to 0 and the conclusion is that the derivative is 1 and the proof goes like that so you have your... perhaps it's not a neighborhood of the origin just an open set which is mapped into itself so u is something like that and f of u perhaps is something like that okay it's a domain it's a connected set it's an open connected set so that's my u so z0 is here z1 is here I will consider a certain path joining z0 to z1 and I just take the iterates of this path so let gamma0 be a path joining z0 to z1 okay and let gamma be the union of iterates of this gamma0 okay so this is a path going to the origin because so 0 is somewhere under boundary perhaps so you know that all these points here must converge to 0 under iteration okay so this is a path converging to the origin under iteration okay now excuse me I missed something here the derivative at the origin must be of modulus 1 okay otherwise of course it can be attracting and it's exactly the case here because my fixed point at the boundary cannot be an attracting point and cannot be a repenting point either so it's exactly the situation which I have in my simplification here okay so what does this path look like actually this path must look like a snail because the dominant effect of f so assume so it's approved by contradiction assume that the derivative at 0 is different from 1 so it's a certain rotating complex number so the dominating effect is the rotation so if you look at this path gamma it must converge to the origin like a snail right so gamma must look like that then the picture is that gamma is a certain path going slowly to the origin now you take this snail here so that's the origin take this snail and you cut a piece you close it the domain so the domain is here so I make a little slit here to cut my snail to get the domain and this domain is mapped into itself by the dynamics okay so it should be a contraction for the Poincaré metric okay and again you contradict you contradict the assumption on the multiplier okay so I don't write the details here but you make a certain slit to close the snail and you cut that into itself so that's the proof of the snail lemma okay so what we have done here is prove that any transient component must be the basing of attraction of a certain parabolic point with multiplier one now you have the recurrent case because it's a normal family the distance Poincaré distance is constant well it is decreased under iteration okay another argument is just normal families okay there's a certain limiting mapping and this limit mapping is constant because it values in the boundary okay so the second case is the recurrent case so there exists a certain z in omega such that fn of z accumulates a certain compact set okay again there are a number of sub cases so sub case one f is not a covering okay so in that case if you work a little bit using the contraction of the Poincaré metric the fact that the Poincaré metric must be a strict contraction then you get there's a little bit of work here but it's not very difficult then you get that everybody must convert to a certain point and this point must be attracting so in that case attracting basing okay second case is a covering is an isometry isometry for the Poincaré metric then you have to prove that so it's clear that I will not go to higher dimensions today but at least you have the details of the classification it was the end of the part one dimension so I will do higher dimension tomorrow anyway so f is a isometry for the Poincaré metric so in that case if it's an isometry if you take cluster so it's a factor component so you can always take cluster values of f converging to some mapping g on omega and now since f is a local isometry g cannot be degenerate okay so g is also an isometry it's a local it's a local isometry so g cannot be degenerate g is not degenerate degenerate okay so if you think a little bit this implies that if you look at f and i plus 1 minus n i so composed with f and i of z of course you get f and i plus 1 this is going to g of z this is going to g of z and g is not degenerate so this implies that f and i plus 1 minus n i converges to the identity okay so in your domain there is a certain subsequence of iterates of f and n which converges to the identity and there's a lemma here lemma let me not state the lemma so lemma let omega be a hyperbolic freeman surface and f be a covering of omega okay assume there exists a sequence n i such that f and i goes to identity so it changed the name of my sequence then omega must be a disk or a punctured disk or an annulus or something c star excuse me c is not hyperbolic and f is conjugate to an irrational rotation so the convergence is a strict and f and i is distinct from the identity irrational rotation okay and the proof is not very complicated assume that omega is not simply connected so in our case this does not happen because again the Julia set has no iterated point okay so in our case omega is not simply connected then it must have a hyperbolic geodesic okay and now we have this isometry which is close to the identity it has no other choice of it must send the geodesic to itself so if it must send the geodesic to itself and it's an isometry so i mean you have a holomorphic mapping locally you have a holomorphic mapping mapping a circle to itself which is a certain isometry for a certain metric it has no choice it should be a rotation on that circle so the geodesic is just a question of a geodesic by a luxodromic element right so the mapping must be a question of some translation on the geodesic okay and then locally it should look like an annulus with an irrational rotation and then you propagate using a unique continuation and the domain itself must be an annulus that's a little bit of work here it has no geodesic meaning that it's a disc and then again an automorphism a covering of the disc you don't have many choices and then you get this rotation okay so let me stop here and so tomorrow i will start by comparing that to the higher dimensional case and i will go to this equidistribution business thank you


Arzan of the eight brought to you from out the pages of Edgar Rice furrows intense books The inky blackness of the jungle depth is lighted only by the pale yellow beams of a waning moon Long endless foreboding tunnels that are animal trails by day echo to the soft sinister padding feet the frowling jungles Are in the distance dangled all the hyena lips his head to the heavens and tells his repulsive slinking brothers that new mother lion has fed that the king of beasts has left the remnant of a kill The moonbeams glint on the shiny stalks of a clump of bamboo by the water hole a pair of greenish gold eyes gleam Evil as they stare from behind the bamboo screen a spotted tail twitchers a chilling rock of screams Wakes the jungle to instant life she does pray startle gleeps to one side She just stops momentarily in midair It seems drops to the ground and crashes into the underbrush and go to hyena forgets his meal and slinks tail between his legs into the Deeper shadows in her leafy alcove Jane bought her wakes she crawls out onto the platform and finds Tarzan standing Listening what is it like him? What's going to happen? It sounded like an explosion Tarzan motions to Jane to keep quiet He leans forward listening intently Tarzan knows that the sound came from the direction of the cannibal crawl But he also knows that in all his many dealings with the goman gunny. He's never heard such a noise as this He turns to Jane as she touches his arm man many man Boom boom gun gun many bang bang. Oh, I think I don't know. I can't explain But that sound convinces me that they must be near that's all on the different lander or father father Tarzan looks at Jane incredulously He had thought that father was one of the parties that occupied his hut But to associate the word father with the noise he has just heard that was him I know something has happened to them. They were out in the jungle searching for me I heard their voices you remember my skin and now something has happened to them Tarzan looks helplessly at Jane his command of this new language is not sufficient for him to understand Jane's excited talk But he knows she's upset over something more than the noise itself Why skin go no, no, you can't leave me now and yet if you don't go something may happen to daddy I can't be selfish. I'll be all right. Yes, my skin. Yes, go I would never forget myself if I kept you back. Go quickly and come back quickly Yes, why skin go quickly come back quickly swiftly Tarzan grass the branching with one last look at Jane The eight-man swings himself into the trees and is gone Meanwhile at the cannibal crawl Professor Porter Philander Clayton and darno picked themselves up from the ground where they've been thrown by the force of the explosion Great Scott, whatever that was it certainly was powerful. I was completely stunned for several moments And I have been watching the natives. They are certainly mystified. They are more than mystified Don't you think that while they're lying there we might try to escape? Yes, let's move toward the gate as long as we have a chance before they recover their senses And they may be lying not on the ground, but they are watching out every move And may we and did you see that party of warriors move slowly to the gate just as we started forward? Well when we meet them it would be time enough to think about what we'll do I think we're sure that we had better stand our ground and let them make the first move Anyway, I too am in favor of that. They don't seem to want me to fear over this, but they watch us closely Well how much longer are we going to stand here doing nothing? Frankly, Monsieur Clayton, we have not much choice, have we? Every time we take a step towards the gate, the natives get between it and us or head us off I have quite the right now, no quite right While the shock of the explosion seems to have stupefied them and they show no inclination to attack us Yet they apparently are determined not to let us get away Evidently the explosion was to them supernatural They promptly threw themselves on their faces before the place where the idol used to be I'd like to know how your sabers are and we're able to get to the dynamite Because I'm sure now that it was a stick of dynamite that did the trick We'll probably find that out later Monsieur Say, these blighters are coming towards us again Yes Clayton, yes But with an entirely different attitude Quite worshipful I should say See that one in front with his hands outstretched But yes, Monsieur Magesture of peace Quickly, mes amis, we must make up our minds It is my belief that these cannibals think that we are miracle workers, medicine men To make them believe that is our one chance I joke Don, I believe you're right You say they believe in black magic And let's give them an overdose of their own medicine A very excellent suggestion As much as I, a man of science, dislike the idea of imposing on credulous savages Professor, I consider that my skin is more important than the beliefs of these cannibals If we can black magic ourselves out of this situation Then I'm solidly behind the idea All right, on, on, on, do I shall act as interpreter Monsieur Professor, you will be the evanderable worker of magic But Don, no, with your knowledge of meaty psychology, hadn't you better Don't argue, Professor, do as Don O'Shea says That chief, or whatever he is, is coming closer Don't you think, Don, that some definite sign of our importance would impress them? Oui, Monsieur has right Over there are their drums Make some sort of signal I know, I'll do my best And, Monsieur, Professor, can you pretend to work some magic Go through a lot of signs, et cetera Look, every eye is on Clayton As long as we have them spellbound Don't you think we ought to really do something that will appeal to their superstitious natures? Ah, mes amis, I have it I shall get the hand full of cartridges from the chief's hut When Arquito said they were stored Et puis, Monsieur le professeur, you walk toward the sacrificial altar And when I give you this signal, you raise your hands as though commanding something to happen Here, Clayton has their attention Walk forward, Professor, that'll draw their attention from Don O I am little like Professor, with speaking hold, in a very few moments, we'll have them completely in our control Yes, but this is most un-dignified, most un-scientific So is being put to death by cannibals un-dignified Archimedes But there, Donald's ready Don't get too close to the fire when Donald throws those cartridges into it Now, I shall go forward to talk to the chief You won't have to Here he comes now I'm going to go forward I'm going to go forward I'm going to go forward I'm going to go forward I'm going to go forward I'm going to go forward I'm going to go forward I'm going to go forward Monsieur, I have arranged a consul meeting And I have also warned them of the danger of making any attack upon us They seem thoroughly cowed, don't they? At least we will have a respect to formally plans for the future The meeting will take place shortly Meanwhile, they will perform their ceremony welcoming us to their village Everything seems to have come all right I have told the chief that no harm will come to them while we are with them Well, they are busy with their ritualistic act Why don't we join your sailor, Donald, and decide what to do?


Good afternoon and welcome back to the private property show house edition. We start off in the absolutely gorgeous Durbanville central two-bedroom a Loft apartment with 24-hour security. It is a luxury at its best Of course, it's got the open-plan living area and two parking bays just for you Move on over still in the northern suburbs of Cape Town to a gorgeous three-bedroom home with two bathrooms Absolutely beautiful kitchen as you can see that leads out to an extensive living room area Over to a two-bedroom apartment in Rosendale This is absolutely the perfect buy for those first-time home buyers Yet another one in Buran estate that estate living right over here three-bedroom two bathrooms and two garages This is a family home pit friendly as you could see from the entrance It is welcoming as you take that first step into this gorgeous home Every single room has built-in closets and some of course have an unsuitable bathroom We're in Parkland still in the gorgeous Cape Town three-bedroom home two bathrooms and two garages This is a low maintenance property not much to be done It is the perfect perfect home for those who love gardening and of course It's as you can see it's got the stunning finishes and the kitchen and bathrooms are fully equipped Move on to a bigger home a seven-bedroom house in Golden Mile with six bathrooms and one garages This is St. Helena Bay literally meters away from your white sandy beaches that ocean living right over here It is fantastically decorated and it is currently used as a holiday home But perfect for your own family home back to the northern suburbs of Cape Town a three-bedroom Townhouse in Durbanville literally minutes away from your wine lends your wine farms a Three-bedroom two-barthroom and one garage Townhouse in the heart of Durbanville as you can see it's got that gorgeous open-plan Settings as well as a gorgeous while lit kitchen Family home right over here with of course security as well Edmied a Three-bedroom home in Edmied two bathrooms and one garage This is a popular suburb in the northern suburbs of Cape Town a cozy home and absolutely inviting space Family-orientated home right over here Over to that a state-loving kind of lifestyle six-bedroom house in the water pool Equestrian a state six bathrooms and eight garages of course a Electric-styled home over here a masterpiece on an award-winning estate Innovation class and of course you have an 180 degree views from your garden This is absolutely the entertainer's dream as you can see a Special feature about this home is that it has a soft cleaning swimming pool How's that for a state living? And then of course last but not least still on waterfall country estate a five-bedroom five bathroom and three garages This is elegance at its best Please look at this masterpiece luxurious master suite in this absolutely gorgeous home over here Time for today, but all of these homes that you've seen today is definitely a must-see So please contact private property for more details and come and view these amazing homes


The Nobel Prize Awards explain simply. Meet Swedish inventor, Alfred Nobel. He's known for inventing dynamite, a breakthrough in chemistry. However, having mostly invented explosives, he worried about how people would remember him. So when he died, he left most of his wealth in a trust to fund what would later be known as the Nobel Prizes. These prizes are a set of annual international awards given for landmark achievements or discoveries made during the preceding year. They are widely regarded as the most prestigious honors in the fields of literature, medicine, physics, chemistry, peace, and economics. A separate committee awards each prize, but the processes are similar. About 3,000 people, usually academics, have the right to nominate candidates. From these, the Nobel committees choose around 300 potential recipients and prepare a report reflecting the advice of experts. The prize-awarding institutions then select the laureates by a majority vote. Each winner receives a gold medal, a diploma, and a sum of money. There is a maximum of three laureates per award, except for the Peace Prize, which can be awarded to institutions. You surely know some of the most famous laureates. Ernest Hemingway won for writing The Old Man in the Sea. What would humanity look like without Alexander Fleming, who discovered penicillin? Albert Einstein won for discovering the law of the photoelectric effect. Martin Luther King, Jr. showed the Western world that struggle can be won without violence. The only person to win two Nobel Prizes in different fields is Marie Curie, one for her work on radiation, and another discovering radium and polonium. And the Red Cross won a record three Nobel Prizes, mostly for their work protecting the rights of prisoners of war. These are the Nobel Prizes. Make a positive impact and get yours.