SPEAKER 1: On behalf of the festival organizers, let me welcome you here. I also want to welcome you to the last Oliver Club of the year. The Oliver Club is the Cornell Mathematics Department weekly colloquium and this is the final one.
The Topology Festival was founded we believe by Paul Olum, Peter Hilton, Bill Browder, Roger Livesey, and Israel Berstein. And we think it's one of the-- perhaps the earliest of the subject concentration conferences in the US. And thank you for coming to help us celebrate the 50th anniversary.
A couple of practical things. Congratulations on finding this location. I'd like to point out that tomorrow and Sunday, the conference events will be elsewhere. The lecture theater is Baker 200, which is-- Baker Labs is a couple of buildings down in that direction. You all have-- if you registered, you have maps inside the envelopes you were given. Please consult those. Or you can find a map on the conference website.
There will be breakfast and coffee served ahead of the lectures tomorrow morning from 8:00 AM, and those will be in the physical sciences building atrium, which is right next door to Baker. Again, look at the maps please.
Also, you might find useful this evening in your packets, in your envelopes, there are lists of recommended Ithaca restaurants.
I'd like to introduce Laurent Saloff-Coste who's the chair of the mathematics department here. And we'll introduce our speaker.
LAURENT SALOFF-COSTE: Thank you very much for coming. Welcome to Cornell. It's an honor for me to have you here and for this edition of the 50th Topology Festival. It's pretty rare that we see the 50th something in mathematics these days, and so I wanted to tell you a little bit about the Cornell Math Department and Topology Festival. In 1945, Cornell Math Department consisted of about seven professors. And in 1949 Paul Olum was hired as a young assistant professor and he started-- at least that's the way I remember it-- he started the topology group at Cornell.
And so 12-13 years later, the Topology Festival was created in 1962. So I want to thank all of you for all the many visits you have made to the Topology Festival. I would like to encourage you to not start your talk by referring to your previous one at the 28th Topology Festival.
And I also would like to thank the topologists of Cornell for having created this absolutely wonderful tradition.
Finally, you are all invited to the reception that will follow the talk, in the Big Red Barn which is in the adjacent garden. And finally, it's obviously an honor and great pleasure to introduce our speaker today. I think John Milnor doesn't require an introduction in general, and he certainly doesn't require an introduction at the Topology Festival. So Professor Milnor.
JOHN MILNOR: Well, it's wonderful to come back to the Topology Festival after a hiatus of many, many years. The talk I'm giving today, however, is not about topology. It will be divided into two quite different parts. The first half I'll be giving an historical introduction as to what small denominators are about. And so that will be very general. And then in the second half, I'll be much more explicit, describing work in progress.
So well, OK, what is the small denominator problem? Maybe the most conspicuous example of small denominators is the gaps in the ring of Saturn. These occur when in the period of a small particle of ice orbiting Saturn is synchronized somehow with the orbit of a nearby moon so that the ratio of their periods is a fraction with small denominator. Or this is more or less equivalent to saying the ratio of their frequencies is a fraction with smaller denominator.
So this is a case where nature abhors a small denominator. The small denominators seem to be unstable so that there are gaps in the rings. But I'll be more concerned with the opposite situation where nature loves small denominators.
So the most familiar example is the orbit of the moon, which the moon travels once around the Earth every month, and it rotates once around its axis every month. So we have a ratio of 1 over 1 which is the smallest possible denominator.
So this is the classical observation, but much more surprising was the discovery that Mercury has a similar phenomenon, but its rotation period-- it goes around the sun faster than it rotates around its axis. So this is almost unique as far as I know, that the ratio is less than 1, but still it's a small-- three, which is considered a small denominator.
Now, by way of contrast, the Earth's orbital period divided by its rotational period is-- if you express it as a fraction, it does not seem to have a small denominator. First of all, I should explain that 366 may look like a typo here, but this is because I'm considering the rotation period in sidereal days measured with respect to the stars rather than measured with respect to the sun.
So the standard way of taking a real number and deciding whether it's close to a fraction with small denominator is to take its continued fraction expansion. If you do have a fraction with small denominator this will terminate very quickly. If it's really irrational it will go on forever. Of course, it doesn't make any sense to ask whether it's rational or not because it's only determined up to some finite accuracy. But in any case, it's not close to any number with small denominator.
And that means that you don't have the phase locking phenomenon where for example the orbit of Mercury and the rotational period are fixed with respect to each other in a kind of stable situation. So there will of course be small fluctuations, but they are self-correcting. Whereas there is no reason that the ratio of orbit period to rotational period of the Earth is locked in anyway, so this presumably will gradually shift over the centuries.
Well, I want to describe a very simple mathematical model for phase locking due to Kolmogorov and Arnold. So we consider circle homeomorphisms which are-- I think of r modulo z is a circle, and first of all just take a translation or rotation if you prefer, adding a constant to the angle, but then add a small nonlinear perturbation. And you get a picture like this, where the horizontal coordinate is the t-coordinate and the vertical coordinate as epsilon.
I've stretched out the horizontal coordinate to try to get as much detail as I could. So what you have are these colored regions called Arnold Tongues where the ratio-- where there is a periodic orbit of which attracts all points of the circle. So for example, on the left you have period 0 which is the same thing as period 1. I'm sorry, rotation number 0 which is the same thing as rotation number 1. It's an element of r mod z.
So these purple regions are places where there's an attracting fixed point, and every orbit in the circle will converge to this attracting fixed point.
In the middle we have the rotation number 1/2, so there's a period 2 orbit which is attracting. And 1/4, there's a period-- no that's, I'm sorry, that's 1/3. 1/3 there's a period 3 orbit. 1/4 there's a period 2 orbit. 1/5 there's a period 5 orbit and so on.
These Arnold Tongues all extend down to the epsilon equals 0 axis, to the t axis, but they become very skinny so it's hard to represent them without working very hard on the computation.
now, what I'm primarily interested in is the holomorphic dynamics. Dynamics of complex analytic functions.
So here is a classical result from almost 100 years ago. Suppose that you have a holomorphic map carrying a bounded region in a plane to itself, then it must be one of four possibilities. So attracting domain, there's a-- draw some pictures. An Attracting domain there might be an attracting fixed point in the center so that all orbits converge towards it. So that's the attracting case.
Parabolic case-- there is an attracting fixed point on the boundary so that all orbits converge towards it.
This case was classical. It's very easy to understand. This case was already understood in the late 19th century. But the other two cases are much more difficult. There's a possibility that you have a rotation domain, which is a simply connected region such that all orbits travel in circles.
And by definition there should be a change of coordinate taking this to a unit disk so that this just becomes a rigid rotation of the disk, where lambda some number on the unit circle which is not a root of unity.
And finally, there's the fourth possibility, a ring domain. In this case there is a some kind of region like this, where again, points move in circles, but it's not a full disk. There's something missing in the middle. And again, if you can map this onto a round annulus. Well, that is supposed to be a round annulus, then it becomes a rigid rotation where lambda is a number on the unit circle not a root of unity.
Well, so the attracting case is classically easy to find examples. Leopold Leau, in the late 19th century, studied parabolic fixed points. That is, they can be described as fixed points such that the first derivative at the fixed point of the mapping is a root of unity. So if you iterate the map a finite number of times it will be tangent to the identity. And this is a typical picture illustrating the situation.
So here, there is the parabolic fixed point, and if you start say in this largest component, which contains the critical point, this will map to here to here to here to here. So if we iterate seven times, we'll get back to where we started. But the map will be closer to the parabolic point. And if we iterate seven times repeatedly it converges to the attracting point.
And this is a very unstable situation. So we're starting here with a seventh root of unity. So it's a number precisely on the unit circle. Suppose we just push it inside a tiny little bit so that it becomes inside, then this parabolic situation will switch to an attracting situation.
Now we'll still have the fixed point in about the same place, somewhere up here, but now everything inside in the gray region will be attracted towards that fixed point. And notice now the Julia set, the bifurcations, that boundary between things attracted to 0 and things attracted to infinity now has become a simple closed curve.
Well, the next case, the case of rotation domain, is much harder. The first progress came with a theorem of Hubert Cramer who made the following assumption. Suppose you take a number t which can be approximated very closely by rational numbers. I won't try to make a precise statement but just hope you have some feeling for what that means. His condition is true for a generic point in the sense of it's true for a density [INAUDIBLE]. All points in a countable intersection of dense open sets. A density [INAUDIBLE] set on the real or on the unit circle.
Then he proved that this map has periodic points arbitrarily close to 0. So if we have such a point here you would fund their periodic orbits arbitrarily close, and therefore it can't be isomorphic to an irrational rotation which has no periodic points near the origin except for the origin itself.
So he proved that under this condition he found infinitely many examples of t for which there is no such rotation domain. And we now know that the situation is associated with very nasty dynamics, with very complicated sets which are hard to understand, but let me pass over that.
So the generic case, what happens for most real numbers t was first solved by Carl Ludwig Siegel in 1942. And he gave a very explicit criterion. Suppose that t bound-- see in the Cramer case we assumed that t was very close to many rational numbers. Here, we're assuming that it's not too close to any rational number. So you need a specific estimate that the distance to any rational number is greater than 1 over q to n for some fixed n. And for all rational numbers.
This is true for most real numbers. This condition is satisfied for some fairly small n. And then if you take any holomorphic map which vanishes at the origin and such that the derivative at the origin is e to the [? duplex ?] i t, where t satisfies this condition, then he found that it's contained in a rotation domain. We can always pick a maximal such domain, it's then called a Siegel disk. And this was sharpened by Bruno in 1971 who gave a precise classification of which numbers this is true for.
So just to make a very schematic picture, if we take the set of all irrational numbers, you can divide them up into two classes, there are the Bruno numbers and the non-Bruno numbers. These are the ones for which this theorem in blue is true. They contain the Siegel numbers as a subset. And these are the ones for which it's false like-- contain the Cramer numbers as a subset of this part. But there is a clear boundary between the two.
Of course, this is a very schematic picture. All of these numbers are jumbled up together on the circle. All of these sets are dense.
Well, here's an example of a Siegel disk. So here the fixed point is here, and you can-- the coloring sort of indicates various circles around it which map to themselves. The boundary of the Siegel disk is outlined in black, and all of these other bubbles you see eventually map to the Siegel disk. For example, this one maps over to this one, and this folds over and maps to here. And then once an orbit gets here it just goes around in circles.
The algorithm for making coloring like this is very simple minded. You have a Siegel disk centered at the origin, and given any point, you just take the-- say you have an orbit z0 maps to z1 maps to z2 and so on. You just take the average of zj squared, say sum from 0 to n or n minus 1 and divide by n, and that-- if n is very large, that will be roughly constant along each orbit. And then they choose a color depending on this average, which will be-- this average is small near the origin and larger for things farther away from the origin.
Well, this particular example is the golden mean. t is root 5 minus 1 over 2. Well, now get to the case, Herman rings, these-- it wasn't known for many more years whether these existed or not, but they were finally constructed by Michael Herman in 1979.
So this more or less concludes the historical part of my talk. Now I want to-- well, I'll provide examples of Herman rings later. That's part of the reason for the motivation is that it will-- I mean the historical introduction is that Herman rings will play a very big part in what I'm going to talk about now.
But I'm going to start with something which seems to be totally irrelevant.
You notice that the photographs I've shown you all have been people who look very solemn and studious. Now this is clearly a topologist. He looks happy. So this is Karol Borsuk, and the theorem I want to quote says, there exists a map from the sphere to itself of degree d which carries antipodal points to antipodal points if and only if d is odd.
So I don't really know how Borsuk proved this, but if you know a little cohomology theory then the proof is quite easy.
So you have to a map from a sphere to a sphere, and if it carries antipodal points to antipodal points, then you can identify the antipodal points to form the real projective space, and get an induced map of the predicted space.
And the cohomology of the projected space with coefficients in the field with two elements is just a polynomial ring with-- or truncated polynomial ring-- x to the n plus 1 equals 0. So you see that such a map induces isomorphisms and one dimensional cohomology if and only if it induces isomorphisms and n dimensional cohomology which translates-- so the isomorphism in n dimensional cohomology translates to this degree being odd, and the other condition, if you use the relation between the fundamental group and the antipodal map, the fundamental group if a projective plane is just generated by taking a path from a point to its antipode, and then collapse identifying antipodes. So you easily see the proof of his theorem.
Now, a side remark which has nothing to do with anything-- I wasn't aware that the word Borsuk has a meaning in Polish. So I was very confused when I Googled for a picture of Borsuk and got a picture of a badger. But in any case, as I say that's not relevant to anything.
Well, I want to describe work in progress. And we want to study this theorem of Borsuk as applied to the Riemann sphere. So we take the standard picture of the Riemann sphere. It's being identified so the c union infinity is identified with a sphere, but under stereographic projection. Then the antipodal map on the sphere corresponds to a map a of z is equal to minus 1 over z bar on the complex numbers union infinity.
So for example, the antipode of 0 is infinity. The antipode of plus 1 is minus 1, and so on. It's a fixed point free mapping.
Now, the theorem that there is-- Borsuk's theorem applied to rational maps is rather trivial. Or even more trivial, because if you look at a-- if you look at a rational map of the sphere to itself-- so it's f of z is a quotient of two polynomials, and you can look for fixed points.
Well, if you multiply through by q you got a polynomial equation of the degree n plus 1, and paying a little attention to possibility of fixed points at infinity and so on, you find in all cases there are exactly n plus 1 fixed points counted with multiplicity.
Now, if the map computes to the antipodal map then the antipode of any fixed point is a fixed point. So the fixed points occur in pairs, which implies that n plus 1 is even, or in other words that n is odd.
So to look for examples we have to look at rational maps of odd degree. And since degree 1 is very boring, the place to start is obviously degree 3. Now, if we look at the set of all cubic maps which compute with the antipodal map it's rather big, so to just narrow things down to something we can study explicitly, we just looked at mappings which had a critical fixed point at the origin. So the origin is a fixed point, and the derivative also vanishes there.
This narrows things down to a one complex parameter family, and it's an easy exercise to show that any mapping satisfies these conditions can be written in this form, with a cubic numerator and a linear denominator. And the parameter q is just the only non-zero number which maps to 0, the unique 0. And its antipode, minus 1 over q bar, is the unique finite pole. So this is an explicit family of mappings. We can look at the parameter plane consisting of the set of all possible values of q and try to study the dynamics in each one. And that gives us a picture like this.
So let's see. So I have to explain what these various colors mean. Well, the white or gray so the-- where are we? So the white in the center and also little white regions here and so on in the gray regions like this, these correspond to cases where almost all orbits converge either to 0 or infinity. And we can't in an obvious way distinguish between 0 and infinity because if one orbit converges to 0, then its antipodal orbit must converge to infinity. So the two are naturally paired.
And now but the central reason is especially interesting. The center point of course if you remember the formula is just f sub 0 of z equals z cubed.
And these are maps-- the central white region are maps for which the Julia set is a simple closed curve. So we have some simple closed curve with a attracting fixed point inside. All orbits inside converge to this. And all outside orbits diverge to infinity.
OK, the black region is what I'm principally interested in. That is the ring locus. The locus of parameters q for which there is a Herman ring. Now, it looks very solid in the picture, but if you look closely you'll see there are all sorts of things sticking out into that region, so most points in that region have Herman rings but there's certainly a very thin subset consisting of points which do not have Herman rings.
One of the colored regions are the analog of Arnold Tongues. There is the region where we have phase locking, which in this case means a periodic orbit. But the curious thing is that these occur only for rational numbers with even denominator. Whereas in the original Arnold Tongue picture, you had tongues for every rational number. And what happens here, for odd denominator, is that instead of having a tongue going out to infinity, we have instead a channel from the main hyperbolic component going out to infinity.
And this is something which still mystifies us. We don't have a real gut feeling for just why this should be true. But, oh, I didn't explain the coding the colors according to rotation numbers. So that varies from 0 down here up to 1, then from 0 up to 1 again.
Oh, and a little more explanation. So a cubic map has four critical points. In this case they're 0, infinity, and then two others which are called free critical points. And you can work out that they always lie in a straight line. So I'll label the one which lies between 0 and q as c plus of q. And now more precisely, the white is the region where c plus of q converges to 0, and the light gray is the region where c plus of q converges to infinity.
OK, I've talked about Herman rings. Now here's an example. This is just sort of typically chosen point in that black region in the parameter slide.
So first of all, 0 is in the center of the white region and the white region is the mediate basin of 0. The gray region out here is the basin of infinity. And the Herman ring is colored using the same number of them I described before. You follow an orbit, figure out the average of its square distance from the origin, and choose a color accordingly.
So if the number is relatively small you get this red region, and in between you get blue, green, light blue, and finally dark blue out here. And the Herman ring has two boundaries, both colored black. So here is the outer region, critical point is out here. The inner region-- coloring is a little harder to see, but there is a critical point at the antipode of this so if I go through it's about there.
Oh also, the unit circle has been colored black just to fix the scale, just so you can see where 0 is for example. OK. Now there are two invariants associated with this, the rotation number and the modulus. So I should explain what they are.
Well, you remember that we have a conformal isomorphism from the disk to a annulus between two circles, and this corresponds to w maps to either the 2 pi i t times w. So t is called the rotation number, and the other parameter measures how thick this is. So if you think of the annulus as being conformally isomorphic to a cylinder, it's the ratio of height to circumference.
So the modulus is defined to be the height over the circumference. So having a big modulus means a fat ring and having a narrow modulus means a very skinny ring.
Well, the other things in the picture were tongues where the rotation number is a rational number with even denominator. Here you still have a rotation number but it's become a combinatorial one because you have an attracting orbit, say here mapping to here to here to here. So this has period 6 and it's a invariant. That means that the third iterate maps this attracting point to its antipode, and then three more iterations take it back to there. And then the other olive colored regions eventually map to this cycle.
And again, white is the basin of 0 which is right here, and dark gray the basin of infinity.
And finally, there are these fjords which don't correspond to hyperbolic components, things with distinctive behavior, they just correspond to special points in the main hyperbolic component. So the bifurcation locus, the Julia set here is a simple closed curve again. I haven't drawn it in, but it's the boundary between the white region and gray region. The unit circle is again marked.
And this is in some sense appears to have a rotation number. If you start here it will map to here to here to here to here to here back again. So it seems to have a rotation number of 4/5, but as you keep iterating it will closer and closer and finally converge towards the origin.
So where is 4/5 here? That's 2/3, so 4/5 is somewhere out in here.
Well, there are a couple of problems with this-- this is again the parameter picture, but now instead of painting the Herman rings black they've just been marked according to the rotation number. So it's prettier in a way, but also you can't see the distinction between Herman rings and tongues anymore.
But in any case, there's some ambiguity with this. Well not ambiguity, there's some redundancy with this picture because points up here have the same dynamics as points down here.
They just correspond to 180 degree rotation of the plane but they have isomorphic dynamics with isomorphic properties. So to get rid of this redundancy, it would be convenient to identify q with minus q. And easy way to do that is to pass to the q squared plane. And then just for convenience I toss in a minus sign and get the corresponding picture in the minus q squared plane. The minus sign means that we have rotation number 0 going up to 1/4, 1/2, then up to 1.
But this is still a bit inconvenient because you can't take in the whole plane at one glance. It would be nice to have a picture where we can see the whole picture. So we can do that by just shrinking the whole plane down to a unit disk.
So this is the model I want to work with. We can-- so it has the same information, it's just all visible and without redundancy.
And one important thing we can do with this model is add a circle of points at infinity. Now if we add a circle of points at infinity to the complex numbers, it becomes a rather abstract object. But if we shrink down to the unit disk then we just have the boundary of the unit disk which is a more familiar and comfortable object. So we just choose some homeomorphism which takes infinite half raised through the origin to finite half raised through the origin and we shrink the picture down.
And this is what we get if we do the return to the original coloring with the Herman rings colored black. And now we can see the-- here's the 0 fjord. There is the 1/4 fourth tongue, the 1/3 tongue, the 1/2 tongue, and so on.
And this unit circle now has a real meaning because as the image of minus q squared tends to unit circle, the map converges uniformly to the uniform rotation through the angle 2 pi t, or at least it's uniform except near the origin.
Well obviously, you can't have a cubic map converging uniformly to a degree 1 map over the whole sphere. Something has to go wrong. So something goes wrong because you have a pole crashing into 0 and a 0 crashing into infinity, but if you stay away from 0 and infinity then you have uniform convergence.
OK, now using this model, I can give a more explicit description of the ring locus. So the set of all points in the minus q square plane, such that f sub q has a Herman ring. Any connected component of this set will be called a hair, and the theorem is that there exists a hair if and only if the rotation number is a Bruno number.
And for each Bruno number, there is a unique hair with that rotation number.
Each hair is a smooth connected real analytic curve parameterized by the modulus of its Herman ring. And this modulus takes all values between 0 and infinity.
And finally, as mu tends to infinity the modulus also tends to infinity, and we're converging to the corresponding point with angle 2 pi beta and the unit disks. So in other words, the Herman ring of rotation number theta is filling out until it fills the whole plane, c minus the origin, and it converges just to a rotation through that angle.
So here is to indicate how this works. Here's a rough plot of many hairs-- rotation number of just taking a little segment of the disk so that we can see more detail.
Now, just to see if you're awake, there's a typo in this slide. So this-- here we see the 2/3 fjord, the 5/8 tongue, the 3/4 tongue, and evenly spaced sample of hairs there. And the remarkable fact about the picture is that they look so uniform out near the boundary of the disk. In fact, that's sort of a consequence of the way the theorem is proved. But it's still striking. And they also seem to remain amazingly uniform even as you go in.
Of course, these are all real analytic curves, and they can't intersect. There are uncountably many of them and they can't intersect each other so they're pretty tightly constrained, but still, I find that sort of amazing. But you can notice some things here. For example, if you look what happens near this 2/3 fjord, you see that things curved in towards it. So that if you have a rational number which is very close-- or an irrational number rather which is very close to 2/3, then it will be sort of hemmed in between-- let me just try to describe it-- describe the situation.
So here is the fjord with a very fractal boundary and disk and you find that you have things like this. So if you take a rational number which is very close to 2/3, then it will be very short.
It's bounded by these other things so that it can't go very far. OK, I would dearly like to prove that, but this is just a picture. It doesn't prove anything.
On the other hand, if you look at what happens near the tongues, then they don't seem to get any shorter, they just get squeezed tightly together. Squashed.
OK, I said that the-- perhaps going back to this picture-- I said that the modulus goes from infinity down here up to 0 at the other end. Just here to give some examples, here's a relatively fat Herman ring. So as we go out to infinity, this part in the middle would get smaller and smaller and the basin of infinity would be pushed out until it looks more and more like just a uniform rotation through any finite part of the plane.
On the other hand, here is a very skinny Herman ring. So as you can see-- it may look fishy to you in here. This looks-- you can't tell what's happening there, so let me blow up this little region just so you see that it's still an honest Herman ring even in there.
And according to the theory, you can make an arbitrarily small. There's a mystery as to what happens is the modulus goes to 0. Is there a well-defined limit? If so, what is its dynamics? But that is a completely open question.
Now I want to look from a different point of view. So let's look at this central hyperbolic component, the white region in the middle where the dynamics just looks like z goes to z cubed for a first approximation.
You can show that there's a canonical way of parameterizing this by the unit disk, which is just defined by the dynamics. And we can then for any-- so but as a picture we have this central region, h0, and we have the unit disk, and to just take any ray of angle t in here, it will correspond to some sort of thing here, which who knows what.
But the theorem is that for every fjord you can have a ray, a unique ray, which lands that its end.
So for any t with odd denominator there is one and only one internal angle such that internal ray of angle theta extends out to infinity landing at the corresponding point with angle t. Call this ray gamma sub t. Then we'll say that gamma sub t passes through to infinity through the t fjord. So here is the picture giving some examples.
So if there is one ray which passes to infinity through the 1/5 fjord and another which passes to infinity through the 1/3 fjord-- we have one which passes through infinity through the zero fjord going straight out here and so on. And you can see how much the disk region is fragmented, because the rational numbers with odd denominator are everywhere dense so we have a-- for a dense set of points on the boundary, we have these rays going out there.
And this gives us a partition of the parameter plane. So suppose we take-- well I can probably describe it better in terms of the picture here. Suppose we take any two rays like this and this. They cut out a wedge-shaped region in the disk. And as the rays get-- angles get closer to each other there would get a narrower and narrower region.
We could also take a region which cuts out a region around the origin. So by doing that and just taking the intersections we get a set x sub t for every real number, or every real number modulo 1.
And this set is an intersection of a nested sequence of compact connected sets. So it's obviously compact and connected.
And by the construction they're disjoined except at the origin. They all intersect at the origin. And the union is the entire minus q squared plane. Or to put it differently, suppose we remove the origin then we have a partition of what's left, c with a circle at infinity and is strictly disjoint complex-- connected compact connected sets. Well, I'm sorry, they're no longer compact because 0 is missing, but they're closed.
So each hair, r sub beta is contained in the corresponding x sub beta. Each tongue is contained in the corresponding x with the same labeling. Number with even denominator.
I would like to say that each ray through a fjord is equal to the corresponding x, but the proof is missing. And any two sets are strictly separated. In fact, given any two distinct sets of this form, there are infinitely many fjords between them.
So it follows for example that no two hairs can have a common limit point, because between any two hairs there is a strict separation by fjords.
So this gives what's an amazing picture to me. We have this enormously complicated structure, and yet we have this way of cutting it up into pieces with a relatively simple description.
Well, the mystery here is, why is there such a difference between the even and odd? In the case of Arnold Tongues they behaved exactly the same way. And I'm sure you could explain this on many different levels. But here is one way of explaining it algebraically.
So this involves another change of coordinate. I want to take the unit disk model here and unfold it by removing the center point and passing to the universal covering. And then we'll get a picture like this, which is more convenient for what I want to do.
Now, so more explicitly I'll call this the ts plane, where t is just the usual angular coordinate, which varies from 0 to 1, and s will be 1 over q squared. So as the center point it will correspond to s equals infinity.
And then you see a little calculation so that in these coordinates, the map takes this form, which is similar to the original but different. But the remarkable thing is that this makes perfect sense even if we were to put in a negative value for s.
So this means that we can study dynamics in the full ts plane, not only for positive values of s but also for negative values of s. And you've got the picture we've been looking at above this line and a strange picture below this line.
What is the meaning of these things below the line? Well, if you look at the change of coordinates we've made, you'll see that the antipodal map takes the form minus s over w bar. So for s equals 0, minus s is just the absolute value of s, this is just the formula for inversion in a circle of radius root absolute value of s.
Now, maps which commute with inversion in the circle have been studied for almost 100 years. They're called Blaschke products, and they're very thoroughly developed. As an example, Herman's first example of a Herman ring was constructed using such a Blaschke product.
So it makes me think that we have two different worlds. We have the Blaschke product world which has been well-studied, and then you pass through the line s equals 0 and you come out in a different antipode preserving world.
So if you look at this picture, you see that in the Blaschke product world you have a tongue for every rational number. There's the tongue for 0, tongue for 1/2, tongue for 1/4, 1/3-- every rational number has its appropriate-- has its appropriate analog of the Arnold tongue. But only half of these extend above the line s equals 0.
So to explain the difference, let's take a look at the detailed geometry nearer the point-- at a point s equals 0. What is it that makes this different from this? Well in fact, one can make an explicit analytic description of what the boundary of the tongue is in this plane, and we see that for s greater than 0 and odd there are no solutions.
Here is some typical pictures. So maybe I'll show the two together. So for example, here is the s equals 0. So we get a parabola shaped thing which is indicated here. This doesn't go above the s equal 0 axis. For n equals 2 you got 2 crossed lines like this. And this goes easily above.
For n equals 3, that's here, you've got a sort of a 2/3 cusp which stops here. But for n equals 1/4, if you can see it, it goes above and below.
So we have a simple explanation in some sense that for rationals with odd denominator we have a nice tongue below, but for algebraic reasons it can't go above. So what happens is instead we get a fjord above. Whereas for even denominator, we get an algebraic expression which extends easily to above. And so we have tongues both above and below.
OK, well of course this subject has been studied for many years and many people have contributed. So just want to mention especially Shishikura who showed that if you have a Siegel disk you can construct a Herman ring, or if you have a Herman ring you can construct a Siegel disk.
So there's a close relationship between them. Jean-Christophe Yoccoz gave a study of quadratic Siegel disks showing that the rotation numbers that you can get are precisely the Bruno numbers. So comparing this with Shishikura, we see that the Herman numbers for-- the rotation numbers for our Herman rings are precisely Bruno numbers.
Risler made a careful study of what happens to Herman rings under permutation-- under perturbation. So if we start with the line s equals 0 we're just looking at rigid rotations and then make a tiny perturbation he was-- his work shows that the rings persist in a clearly defined way.
Buff, Fagella, Geyer, and Henriksen made a study of what I think is essentially the thing below the line s equals 0 in my picture. So this work is just rather an analytic continuation of that. The description of the geometry of the tongues near s equals 0 is based on a thesis of Bannergy.
And finally, the theorem that everything is well-behaved, smooth, real analytic curves, is work in progress by Booth and Epstein.
SPEAKER 1: Thank you. Are there any questions?
JOHN MILNOR: Well, let's see, I can describe what Risler's theorem is.
He starts with any given Herman ring, and let me just choose new coordinates so that it's a rigid rotation. So start with a rigid rotation. And then you take a-- so this is a map from call it annulus a, a map from a to c which happens to map the annulus to itself.
Then he takes a small perturbation of that and asks whether within a there is a small annulus which maps to itself as a Herman ring. And he shows that that defines a-- that condition defines a complex codimension one subset under perturbation.
So if you have a generic family and perturb you get a complex codimension one family of Herman rings.
SPEAKER 1: Any further questions? Well, let me reiterate that invitation to a reception that will start right away in the Big Red Barn. If you leave this building in that direction you will see the Big Red Barn and recognize it from its name. Let's thank Professor Milnor again.
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John Milnor of Stony Brook University gives a brief historical survey of small denominator problems, followed by a discussion of cubic rational maps, at the 50th annual Cornell Topology Festival, May 4, 2012.
The festival was organized by the Cornell University Department of Mathematics with support from the National Science Foundation.