1/2 Diffeomorphisms of the Circle
This is a modal window.
Das Video konnte nicht geladen werden, da entweder ein Server- oder Netzwerkfehler auftrat oder das Format nicht unterstützt wird.
Formale Metadaten
| Titel |
| |
| Serientitel | ||
| Teil | 34 | |
| Anzahl der Teile | 34 | |
| Autor | ||
| Mitwirkende | ||
| Lizenz | CC-Namensnennung 3.0 Unported: Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. | |
| Identifikatoren | 10.5446/46184 (DOI) | |
| Herausgeber | ||
| Erscheinungsjahr | ||
| Sprache |
Inhaltliche Metadaten
| Fachgebiet | ||
| Genre | ||
| Abstract |
|
5
6
7
8
10
12
14
17
18
23
24
25
26
27
28
29
30
31
32
33
00:00
GruppenoperationSortierte LogikGruppe <Mathematik>AutomorphismusFlächentheorieRechter WinkelKreisflächeAlgebraische StrukturMinkowski-MetrikHyperbolischer RaumHomöomorphismusKreisbewegungMaß <Mathematik>TheoremVektorraumAnalogieschlussLinearisierungÄußere Algebra eines ModulsAbelsche GruppeWahrscheinlichkeitsmaßKommutatorgruppeGammafunktionOrientierung <Mathematik>InvarianteDichotomieOrdnung <Mathematik>MultiplikationsoperatorEinflussgrößeObjekt <Kategorie>UntergruppeDreiecksfreier GraphMultigraphIndexberechnungEbeneFundamentalgruppeKombinatorikHyperbolische GruppeTransformation <Mathematik>Physikalische TheoriePunktFreie GruppeFundamentalsatz der AlgebraZahlensystemMorphismusGeometrieRandwertKomplex <Algebra>WürfelAmenable GruppeKurveTopologieSigma-AlgebraGruppentheorieVorlesung/Konferenz
08:50
SechseckInverseFreie GruppeTheoremDickeMathematische LogikHomöomorphismusPunktVollständige InduktionSortierte LogikEinsMereologieÜbergangTranslation <Mathematik>MengenlehreÄquivalenzklasseAbgeschlossene MengeGeradeReelle ZahlBeweistheorieKreisflächeStabilitätstheorie <Logik>Ordnung <Mathematik>UnendlichkeitGruppe <Mathematik>Zyklische GruppeRechter WinkelArithmetisches MittelStatistische SchlussweiseHyperbelverfahrenParametersystemDifferenteFaserbündelEinfache GruppeVorlesung/Konferenz
17:39
KreisflächeTheoremInnerer AutomorphismusVertauschungsrelationKreisbewegungGeradePunktHomöomorphismusTranslation <Mathematik>InvarianteOrientierung <Mathematik>Gruppe <Mathematik>MengenlehreIterationNumerische MathematikMultiplikationsoperatorGanze ZahlWinkelRechter WinkelTransformation <Mathematik>ZahlensystemVollständigkeitDezimalzahlAbgeschlossene MengeOrdnung <Mathematik>AuswahlaxiomBeweistheorieMathematikFlächeninhaltDiffeomorphismusFreie GruppeUntergruppeGruppenoperationÜberlagerung <Mathematik>AnalogieschlussVorlesung/Konferenz
23:46
VertauschungsrelationHomöomorphismusExakte SequenzSurjektivitätRechter WinkelÜbergangswahrscheinlichkeitMultiplikationsoperatorFolge <Mathematik>QuotientOrientierung <Mathematik>KreisflächeTranslation <Mathematik>Vorlesung/Konferenz
24:38
MittelwertGeradeHomöomorphismusPunktMultiplikationsoperatorKlasse <Mathematik>InverseKonditionszahlTeilbarkeitKreisbewegungMinkowski-MetrikInverser LimesDickeNumerische MathematikGüte der AnpassungLokales MinimumFolge <Mathematik>GruppenoperationElement <Gruppentheorie>ZahlensystemUnendlichkeitDivisionVorlesung/Konferenz
25:58
KreisbewegungAuswahlaxiomFolge <Mathematik>Numerische MathematikElement <Gruppentheorie>DickePunktHomöomorphismusInverser LimesReelle ZahlKreisflächeMultiplikationsoperatorÜberlagerung <Mathematik>FunktionalVorlesung/Konferenz
26:55
GeometriePunktGruppe <Mathematik>Leistung <Physik>KreisflächeRechter WinkelNumerische MathematikGanze ZahlVertauschungsrelationDifferenzkernKreisbewegungIrrationale ZahlScherbeanspruchungTranslation <Mathematik>Kategorie <Mathematik>Algebraische StrukturUmkehrung <Mathematik>OptimierungGraphUntergruppeInnerer AutomorphismusAbelsche GruppeSortierte LogikProdukt <Mathematik>MultiplikationsoperatorTorusJensen-MaßSummierbarkeitArithmetisches MittelMinimumKreisringHomomorphismusDivisionReelle ZahlÄquivalenzklasseNichtlinearer OperatorOrdnung <Mathematik>FunktionalBetrag <Mathematik>Inverser LimesDifferenteMatrizenrechnungAmenable GruppeAbstandStichprobenfehlerMittelwertGruppentheorieFinitismusCantor-DiskontinuumAggregatzustandFlächeninhaltSelbstähnlichkeitFraktalgeometrieHyperbolischer RaumEinflussgrößeHomöomorphismusMorphismusQuotientAdditionParametersystemStetige FunktionNichtunterscheidbarkeitFreie GruppeGüte der AnpassungGruppenoperationTransformation <Mathematik>Element <Gruppentheorie>GammafunktionMengenlehreVorlesung/Konferenz
35:42
Iteriertes FunktionensystemFlächentheorieRechter WinkelMengenlehreGegenbeispielSortierte LogikOrbit <Mathematik>Algebraisch abgeschlossener KörperIrrationale ZahlTeilmengeKreisflächeGruppe <Mathematik>KreisbewegungGammafunktionOrdnung <Mathematik>HomöomorphismusInvarianteDichte <Physik>Translation <Mathematik>PunktAbgeschlossene MengeGruppenoperationUnendlichkeitKategorie <Mathematik>Element <Gruppentheorie>FinitismusFunktionalGeradeJensen-MaßNumerische MathematikLokales MinimumResultanteEindeutigkeitDynamisches SystemBeweistheorieTheoremTorusKompakter RaumCantor-DiskontinuumVorlesung/Konferenz
44:29
Chi-Quadrat-VerteilungDimension 6Algebraische StrukturGruppe <Mathematik>UntergruppeTheoremParametersystemSortierte LogikKreisflächeEinflussgrößeRechter WinkelInvarianteHomöomorphismusWahrscheinlichkeitsmaßCantor-DiskontinuumLebesgue-MaßPunktGammafunktionGruppenoperationMengenlehreVerschiebungsoperatorAlgebraisch abgeschlossener KörperMultiplikationsoperatorOrbit <Mathematik>BeweistheorieTopologieNumerische MathematikÄußere Algebra eines ModulsFreie GruppeElement <Gruppentheorie>KreisbewegungOrdnung <Mathematik>Lokales MinimumFamilie <Mathematik>Innerer AutomorphismusKategorie <Mathematik>Innerer PunktAnalysisRuhmassePermutationEndlichkeitExistenzsatzKompakter RaumVorlesung/Konferenz
53:17
Rechter WinkelKreisbewegungVertauschungsrelationÜberlagerung <Mathematik>Projektive EbeneAlgebraisch abgeschlossener KörperOrdnung <Mathematik>Freie GruppeGruppe <Mathematik>GammafunktionTeilmengeKreisflächeQuotientGruppenoperationInnerer AutomorphismusDickeInnerer PunktUntergruppeMultiplikationsoperatorKonfigurationsraumGeradeElement <Gruppentheorie>PunktGebundener ZustandFolge <Mathematik>Cantor-DiskontinuumGrenzwertberechnungBeweistheorieZeitbereichParametersystemSchnitt <Mathematik>MomentenproblemLokales MinimumOrbit <Mathematik>FunktionalMengenlehreNichtunterscheidbarkeitOrientierung <Mathematik>InverseKonditionszahlKategorie <Mathematik>FinitismusArithmetisches MittelHomöomorphismusAggregatzustandEinflussgrößeStetige FunktionTransformation <Mathematik>SpieltheorieSupremum <Mathematik>Vorlesung/Konferenz
Transkript: Englisch(automatisch erzeugt)
00:17
it's fun to be here. So I think in geometric group theory we enjoy, we like to talk about groups
00:25
that act nicely on nice spaces where the nice spaces are usually kind of combinatorial objects, you know, cube complexes, curve complexes, graphs, things that kind of have a combinatorial flavor. I'm going to talk about groups that act on the circle and this is not a kind of
00:42
discrete object at all, except secretly it is. And although I won't kind of highlight why secretly it is in my talk because that involves co-cycles and things like this, I do want to highlight some parallels between groups of homeomorphisms of the circle and sort of some of the other groups
01:03
that we've seen so far and approaches to talking about both. So my plan is, here's the plan, in the first half, so that's this hour, I'm going to talk about some basics. Let's call them
01:20
fundamentals. To prove a theorem, that would be the end of this half of it, I'm going to prove in quotation marks a Tietz alternative for groups that act on the circle or for the whole
01:49
group of homeomorphisms of the circle. This says orientation preserving. I like oriented things so this is a group of orientation preserving homeomorphisms, no flips. And I'll
02:04
say in a second what that is and then in the next hour I'm going to prove a theorem about a particular action of a particular group, about the group of
02:22
automorphisms of the fundamental group of a surface. So I want G to be at least two here, so a hyperbolic surface, a nice surface group. You can look at the group of all automorphisms of that group. Great, this is an object we're all good with. Oh look, and this is even a hyperbolic group so there's some geometry going on there. So this group,
02:45
being a hyperbolic group, has a boundary and the group of automorphisms acts on this boundary. Hopefully that's something you can dig out of your mind from last week. And what is this
03:00
boundary? It's a circle. Topologically it doesn't have any further structure than just being a topological circle so this action is by homeomorphisms. And so I'm going to prove a rigidity theorem about this action. It turns out for this particular group any action of that group on the circle looks like this one exactly. So that's going to be an exciting
03:26
theorem and a way to teach you some tools for talking about these objects and hopefully relate it to things that we've thought about last week. The Titts alternative, many of you or some of you might know what this theorem is for linear groups. It says that Titts'
03:47
theorem is that if you take a finitely generated subgroup of GLN, some linear group, then there's a dichotomy. Either that group contains a non-abelian free group or virtually
04:01
up to finite index it's solvable. So it sort of looks free or it looks billion. And so generally we say that a group G satisfies a Titts' alternative if something like this is true. Finally generated subgroups are either looking like solvable or contain free groups. This isn't quite true for groups acting on the circle generally but something very close is due to
04:27
Margulis. So he showed that any, not even finitely generated, but any subgroup of homeomorphism of the circle either contains a non-abelian free group, either contains an F2, or the
04:52
analog of being solvable or looking kind of like your abelian or in this case amenable. The best analog you can have here is that there is a measure on the circle invariant under your group.
05:04
Okay so or there is a gamma invariant probability measure on the circle. This isn't a dichotomy,
05:22
it's possible that you could have both happening at the same time. But I'll explain sort of what all these words mean when we get there. But let's start right at the beginning. How about some examples of groups of homeomorphisms of the circle? And I give you some examples to
05:46
convince you that this big group is a rich and interesting object. Okay so we saw one already odd pi 1 sigma g. Wow that's a pretty complicated group. Let's maybe try and go for other things.
06:06
How about like just the group of rotations? So I can think of the circle as like, I don't know, you know, unit vectors in the plane I can rotate. Another thing we've already seen so far, let's see what have we seen. Last week we saw PSL 2 R acts on RP 1 by Mobius transformations,
06:38
and I mean RP 1 is just a circle. So that gives you a nice action of this group on the
06:44
circle and the SO 2 subgroup inside here is acting by rotations. What else have we already seen? Oh this morning we saw Thompson's group, group F. This acts on the interval by piecewise
07:07
linear homeomorphisms. And if I take my interval and I just glue the two sides together, I get a circle. So this acts on the circle with a fixed point. But there's even that there's
07:26
a there's a slightly bigger Thompson's group that acts on the circle. This is contained in a group P which is in between F and the group V advertised before. T is what happens,
07:44
it's a subgroup generated by F acting on S1 which you think of as this this interval with the two endpoints identified by piecewise linear, that gives you a nice linear structure, by piecewise linear homeomorphisms plus, now I'm going to allow you to move the endpoint that
08:06
I glued. So you can take this group to be generated by this and there's an order to rigid rotation. So a silly way to write this if you like this kind of notation is the transformation
08:21
X goes to X plus a half mod 1. Where I think of the circle is this interval mod mod 1 or R mod Z if you like. And this group is interesting and special because we saw that
08:45
this group was not simple although its commutator subgroup is. This group is another example of a simple group. Good, so I've got some particular examples. Let's do a nice
09:04
very hands-on example for a couple. Three groups fit inside of here in lots of different ways. In fact you can prove if you decide you want a challenge that the generic two in the sense
09:20
of bear even, generic two homeomorphisms of the circle generate a free group. But I want a particular example that we should hold in our minds for later. You can do this even in PSL2R but I'm going to forget that I know anything about PSL2R. I'm going to produce a free group generated by two homeomorphisms say F and G. So what do
09:48
I want these to do? Here's my circle. I'll give some instructions to F. Sorry I want F to be anything that takes, let's take a little interval here and stretches it out. So I
10:04
want F to take this interval and stretch it out to being something like this. Okay so F of I is going to be really long like this. And so what's left over I wanted to take the complement and shrink it. Okay so this interval here I can call F of circle minus I.
10:28
And G I'm gonna have to do the same thing to two other intervals. I know maybe I'll call this J. And I'll do the same kind of stretch and shrink. Just take some your like elastic circle and like pull it this way and smoosh it that way. So that this interval over here is G
10:45
of the complement of J. Okay I claim that these two things generate a free group. Okay just as I've described them I don't need to give you any more information. And there's only one way that we really know how to prove. Okay the only one widely applicable way that
11:05
we know how to prove that things generate a free group and this is the ping-pong argument. Right so actually what I claim is that if I take a point I know say this point X that is not in any of these colored in regions that I drew I haven't definitely I have a map
11:27
that takes a point that takes a word in my free group okay and sends this to so it's
11:40
some abstract word and F and G and F of inverse and G inverse I just take that word and I apply it to my point X that's this is now a homeomorphism it sends X somewhere else okay I claim that this map is injective so not only is this group free at this point has trivial stabilizer.
12:07
So for some of you there's a one line proof it or there's a one word proof and it's the word ping-pong and for others of you here's how you do it the proof is by induction on word length okay and I'll do that okay the base case word length zero that's true okay I don't
12:31
know let's do the case of word length one I want to show that F and G and F inverse and G inverse all send X to different points so let's check if I apply F I'm supposed to
12:42
take this guy and shrink it into there so I would go here somewhere in this region if I apply G I'm supposed to land in I'm not in J so I'm supposed to land in that region okay so that these are different if I apply the inverse of F the inverse of F what takes this and stretches it down so it'll send X into here so F inverse sends X into this region
13:06
and G inverse same kind of logic sends it into that region and so these regions the four of them distinguish the image of X under words of length one and okay so that's the base case and
13:21
to do this longer you assume that you sort of have this already and I can actually take it further nested intervals right inside of this picture of this interval if I pull if I draw the image of these guys under F I'll see some little sub intervals in here I'm supposed to take this and smoosh it so I'll see a picture like that and you can use these
13:45
kind of refinements these level two to distinguish the image of X under words of length two etc etc okay so this is a great exercise to do if you haven't had to do one of these before and I'll leave it as proof by audience great so that's a nice example
14:14
of how to make a free group and actually if you you know if you if you are an expert at this there's there's one thing here that people don't often notice if you use this very
14:25
by hand induction proof to do this you don't only recover that this is injective you specify the the or the location of these intervals I J and then F of the complement G of the complement is enough to determine the cyclic order of this point X meaning if I want to
14:48
know if you know some first word of X if I start reading around to the right do I encounter a word or a second word first or second in what order that's recovered just from this combinatorial data and that will come out of the proof I mentioned that because that secretly
15:06
lives in the proof of a theorem I'll quote later but but it's not necessary for understanding everything today okay so there's free groups maybe just one more example of very pervasive
15:22
things all right so I went from complicated groups down to simpler ones let's go for maybe one of the easiest groups how about just an infinite cyclic group you're very easy to produce infinite cyclic groups here is one way to do it where for example F could
15:46
be specified as follows okay so I take my circle I'm gonna choose some closed set okay
16:02
so it might be finite or it might contain some intervals or it might have a part that looks like a canter set or some accumulation points I don't care take any closed set there's K and on every complementary region okay so every interval in the complement like here's
16:31
an interval in the complement okay I can identify this interval with you know the real line it's topologically the same and under whatever identification I I looked I I chose
16:47
I can have F in these coordinates I can make it look like X goes to X plus 1 just the translation okay so this closed set is going to specify the set of points fixed by F and the
17:09
complementary intervals are going to be the non fixed points and for each of them I have a choice of whether I thought infinity was this way or infinity was that way in the real line which way I thought was positive or negative and that'll determine whether F is shifting
17:24
points to the right or to the left so I could choose you know a function positive or negative to choose right or left maybe positive there on this complementary interval maybe I wanted it to go the other way minus okay so this data is really an assignment or a little more
17:45
specific than an assignment of complementary intervals to pluses or minuses telling me orientation of the line which way points are moving clockwise or anticlockwise so what's a fancy way to write that I don't know pi not of the circle minus K all right point regions
18:05
in the complement to the set containing plus or minus so that gives you lots and lots of choices of ways to produce homeomorphisms all of these will be infinite order so they'll generate a Z subgroup and the fact is that this data the closed set and the plus or minus
18:29
assignment is a conjugacy invariant okay so the pair K and this assignment up to conjugacy so
18:49
obviously I could rotate my set K and like rotate all the places I put pluses and minuses and I'd have the same kind of looking thing I'd have a conjugate homeomorphism is a complete
19:06
conjugacy invariant so it distinguishes homeomorphisms up to conjugacy in homeomorphisms of the circle if K is not empty so provided you know you have some fixed points you can
19:28
always write it down using this recipe and the recipe the topological data of this is enough to specify up to conjugacy so why I like homeomorphisms rather than diffeomorphism you
19:41
can kind of draw a picture of them no like here's a here's a bad example what if I take rotations okay so here's a remark not generally okay for example how am I supposed to draw a
20:09
picture distinguishing a rotation by you know angle pi and one of angle I don't know pi over three or something like this both of them are just like saying move all your points around some amount okay so I don't know what to do for example rotations non-trivial rotations I guess
20:40
I'm sorry does it mean that every analogism of R would you gave to the transition yes every
20:47
with no fixed points every fixed point free homeomorphism of R is conjugate to translation I mean that is the proof of this fact basically is conjugate to translation by an orientation preserving or reversing homeomorphism and that you can actually do I hand you don't need a hard
21:01
a sophisticated theorem to build a conjugacy so that's a nice didn't put it on the sheet but that's a nice exercise to do okay so actually let's um solve this problem here and try and write down something that looks like I don't know a conjugacy invariant that will pick out
21:23
different rotations my goal being to like come up with like a number or something you know easy to write down that will solve my problem and I don't have to invent this this is secretly hidden in work that goes all the way back to Poincare this is Poincare
21:45
is rotation number and it's something that will play a giant role in the second half of this in our two okay so let me set up some definition so what is the rotation number do what I want
22:05
to do is I want to pick a point on the circle I have a homeomorphism I want to capture on average if I as I iterate how far does this point go around okay so you want to capture the average amount you move around the circle under iterates of F okay so to do that rather than you know
22:27
this guy doesn't have a fixed point imagining I'm really like rotating a circle rather than remembering every time I crossed over and went around a time and then another time and then a third time an efficient way to do that is to lift to the universal cover and unwind the circle
22:43
okay so I'm going to start by working there okay so let's let's set some notation here's the definition slash notation if I think of my circle as r mod z I'm going to lift it
23:01
up and get all of r but if I remembered the circle is there right I have this deck transformation which is translating by integers so we'll say homeo of r superscript z okay just like in many areas of math you write this as the invariance this is the set of
23:22
homeomorphisms of the real line which commuting which commute with x maps to x plus one integer translation and hence they commute with all translation by
23:44
integer amounts okay if I have something that commutes with this well then it defines a map on the quotient of r by this map right so there's a surjection from here to the homeomorphisms
24:01
of r mod z I want everything to preserve orientation I think you do automatically if you commute with this but this will make it clear which is just homeomorphisms of the circle so this surjects and the kernel of this map is just this translation so this actually lies in
24:26
short exact sequence like this where this one is generated by x goes to x plus one so lifting to the line lets me think of elements there up here and it counts how far
24:44
I move around or how many times I'm wound around okay I'm going to define the rotation number upstairs first so for what's good notation for a lift maybe with a little squiggle
25:05
for one of these homeomorphisms of the line we'll define the lifted rotation number of f to b what do I want I want the average amount it translates a point okay so I'll take
25:25
a point let's choose zero that's a good point in the line I'll take f I'll iterate it n times and to get an average I'll divide by n and I want the limit I didn't leave myself as much
25:41
space as I hoped for I want the limit as n goes to infinity of this quantity one of your exercises is to prove that this limit exists and in fact zero wasn't a special choice I could have taken any point x and I'd still get the same number okay so this is
26:06
it's it's annoying to do on the back blackboard but I've given it to you with some hints this is a well-defined number it's just some real number that is the rotation number of f and if I looked downstairs everything makes sense mod z now so for a homeomorphism of the circle
26:34
I'll say the rotation number of f is I'll take a same kind of limit
26:45
I'm going to choose any lift I want apply it to any point I want I picked zero apply it n times divide by n and this will give me a number but if I took different lifts that's like composing this with a translation
27:01
by one okay that will change my average translation amount by one or by two or by whatever power I chose so this is only well-defined mod integers and that is the
27:20
rotation number or the lifted rotation number okay so let's do a little mental check I really if I have a rotation by I don't know maybe that one that's like x goes to x plus a half mod one that order to rotation hopefully I would get a half out of this well let's see let's pick the lift that's really x goes to x plus a half if I do that n times I'll add
27:45
n times a half to zero I'll divide by n I'll get a half I don't even have to pass to a limit this is great so the rotation number of a half mod z is a half and indeed that gets rotation number a half so in general the rotation number of rotations is exactly what you think they
28:03
should be rigid rotations this has lots and lots and lots and lots of nice properties so let's write some of them down to use and again these are things that I'm asking you as exercises to check I'm going to clear on the definition okay so properties okay so I already
28:33
said that rotations or their lifts translations if I take any real number this is alpha that's
28:43
pretty immediate from the definition it has a homogeneity property meaning that the rotation number the lifted version of a power of f k would be any integer is k times the rotation number
29:07
of that okay including for inverses better than this so this is a particular abelian subgroup right it's a subgroup generated by f and I'm saying it looks like a homomorphism an additive
29:24
homomorphism to r on this abelian subgroup right this is true in general for things that commute that's all you need to prove this so it's a homomorphism to some subgroup of real numbers
29:43
as an additive group when restricted to abelian subgroups not in general all right as you may
30:03
even know from say multiplying matrices in psl2r you can write a product of two things with six points two hyperbolic matrices you can write a rotation a non-trivial rotation is a product of two hyperbolic things the hyperbolic guys will get rotation number zero they have fixed points your rotation is whatever you wanted it to be okay so not in general but it is generally
30:27
a conjugacy invariant meaning that the rotation number of conjugate of f is the same as the rotation number of f and although it's not a homomorphism in general it's up to bounded
30:47
error which is all we care about in geometric group theory right a homomorphism namely it's a what's called a quasi-morphism so what does quasi-morphism mean it means something that is
31:04
bounded distance away from a homomorphism in the following sense it satisfies that if I compare the rotation number of a product here my product my my operation is like function composition right if I compare this to the rotation number of f and the rotation number of g well if it really
31:28
was a homomorphism if the sum of these should be the same as that right so let's see what happens if I take this and I subtract off these guys okay well the claim is that this
31:41
is uniformly bounded and in fact in this case you can show that it's bounded by one an absolute value yes are there any non-trivial obedient subgroups of the group of
32:11
just some little interval of the circle you could identify with the real line and make a group of translation okay but if there are no fixed points if there are no fixed points
32:21
then there are weirder examples than things in so2 yeah and I'll actually how about I show you one right now making sure that in a second once I make sure I've told you all the properties I'm going to use it later on yeah when does equality hold good question if f and g don't
32:47
commute typically not okay but you can also construct examples where f and g don't commute and equality does hold so oh sorry when does oh when does equality hold sorry I meant I was answering some version of this question where I thought when when are these when is this
33:04
zero do you mean when are these when is this one or when is this zero when is this one yeah when is it zero okay so I answered the zero question actually both questions are interesting and in general if I give you f and I give you g and I ask and I tell you like
33:22
what these are and I'm like what are the possible values can this guy take that is that is an interesting problem and in certain cases for this particular question there's a complete answer which I can show you a picture of there is a graph you can draw like what are the possible values it has this crazy self-similar fractal structure this was done algorithmically by
33:46
Danny Caligari and Alden Walker and they have a computer program that sort of answers this question in many cases so yes you asked actually a very subtle and interesting question what values does this take when based on what data
34:02
okay let me take a slight detour for a minute which might help us over there I want to talk about an example we saw later on last week Jair Minsky had a had a picture where he's trying
34:20
to describe how you make a lamination that's not a foliation on the torus or on the punctured torus and I'm going to be just very slightly less sketchy than he was you know he started with if I take you want to understand that either torus or an annulus with foliation of irrational
34:41
slope so one way to make an irrational slope foliation on a torus is I'm going to take here's my circle so I'm gluing these two sides together cross interval all right and I'm going to mod out by an equivalence relation where I'm going to glue top and bottom of the interval by
35:11
an irrational rotation or translation if I do that well that'll take this like vertical
35:20
stripes under my gluing what am I supposed to do I'm supposed to shift them all over before I think of them being attached back down right so that's like I've really identified my taken like a slope like this and I've used this to identify my stripes okay and if I sort of shear this back over to say hey just glue them up and down normally
35:44
you'll get an irrational slope foliation of your torus okay that means like well here instead of doing that I want one where it's not all dense and in fact I only want these sort of going on a canter set of points I'm going to like take one of these leaves that
36:03
goes wrong and I'm going to thicken it up a little bit and then thicken it but not so much over here and something like this if instead you glue by not a rigid rotation or translation
36:24
but by some homeomorphism f that say has rotation number alpha but not conjugate to a real rotation okay the result is that you'll produce something that
36:54
has a canter set somewhere here identified with itself by kind of a shifting over
37:01
and this picture well will look exactly like the picture we saw before okay where there's stretched out and thinned out lines following some canter set so if you did the if you got interested in his lecture and looked up danjua counter example as was suggested then the
37:25
danjua counter example is exactly constructing a function that satisfies this property that lets you build one of these strange foliation okay I want to explain sort of what's actually going
37:41
on there but I wanted it to tie it to his lecture first okay so consider that maybe
38:04
motivation for what's about to what's about to come now and this is a very useful kind of
38:25
dynamical trichotomy that says that pictures either look like this or look normal so here in the theorem which I'll need for what I want to do next okay it says the following if you take
38:54
any group acting on the circle so any group of homeomorphisms of the circle then either
39:04
this is an exclusive either or okay one of three things can happen okay one it could have like a fixed point or more generally a finite orbit okay a finite orbit that's one possibility
39:29
right another possibility is I don't you have like a very rich large group like psl2r or something like this or like a surface co-compact service group in there another thing that
39:41
could happen is that all orbits are dense this also happens if your group is just like a single infinite order rotation and the claim is if neither of these two things happens then something weird goes on
40:05
there is an invariant so a gamma invariant closed subset
40:21
how do I want to state this so that I don't let's give it a name okay it's contained in the closure of every orbit and so that the restriction decay has all orbits dense
40:48
all orbits on k dense okay so that doesn't tell you very much what k looks like but I
41:10
can say a lot more k in case three is homeomorphic to a canter set and it's unique
41:30
it's a unique set with the properties listed here closed set contained in closure every orbit where the action has all orbits dense this is often called the action being minimal there is a gamma advanced closed subset k
41:49
containing the closure of every orbit with all orbits on k dense in k so well let's say this in English words with the action on k is minimal
42:06
yeah so I get a I can forget the rest of the circle ever existed I now have an action on something I claim as a canter set and all of the orbits of this are dense so I could make
42:30
it bigger and not have something like this happen if I took it take a minimal such example I'll get like a smallest possible set that satisfies these I'll get this property but it's not necessarily for example the whole circle is a gamma invariant closed set contained
42:48
in the oh contained in this ah yeah you're right you're right if I put it this way contained in closure every over it yes that's true this is a constant I didn't realize what I had wrote you are correct let me give you a quick proof of this because it's good to prove
43:06
some things I mean one way to approach this is you say well suppose neither of these holds and
43:21
like let's find k but very generally we could start right from the beginning by being like okay what am I aiming for let's look at I'm trying to understand closures of orbits okay so let's look at the set of subsets of the circle consisting of orbit closures
43:44
okay so this is a set of things that look like gamma orbit of gamma on a point the closure of this where x is some point in the circle okay this is partially ordered by inclusion okay so I can take a minimal element all right so you
44:11
show that it has the finite intersection property or whatever you want take a minimal element
44:24
okay under this ordering uh if that's finite you just showed that some orbit is finite great okay so you're if your minimal element is a finite set you get case one
44:44
if your minimal element is the whole circle what does that mean that means that the closure of every orbit is the whole circle so we're in case two okay and if it's not well what is a minimal element of this in general it will be
45:08
some gamma invariant set because orbit closures are gamma invariant it's going to be a closed subset okay and I claim it will look like a canter set so it's every point should be
45:30
an accumulation point it's not finite so it has some accumulation points and the set of accumulation points will be invariant on your group action your group is acting by homeomorphisms
45:41
so if I didn't want this if this didn't if if there were some non-accumulation points I could throw them out and I'd get something smaller okay so every point is an accumulation point that's perfect and the same kind of argument says it has empty interior otherwise
46:09
I'd throw out the interior and I'd get some smaller closed invariants under my group set so the closure and orbit in that point would be contained in it would be smaller so
46:25
must have empty interior okay so the last two just come from the fact that I took a minimal element okay so that says oh I already proved sort of my addenda that this is homeomorphic to a canter set and the contained in the closure of every orbit is just because I could
46:48
take an intersection right if I had something that was that was smaller okay so that was a little quick and sketchy at the end but I promise there's nothing too complicated going on
47:04
you just check that this has the properties that you want is the existence of a minimal element guaranteed by compactness yeah you just check the descending things yeah yeah okay
47:24
nope and you can do one of your examples if you believe that this really works to make fat canter sets on which you do a shift kind of thing all right so things are pretty pathological if you don't if you just have actions by homeomorphisms okay let me in the remaining 16 minutes or so tell you about what I really wanted to kind of get to
47:50
this proof of the of the of the teets alternative the rotation number I need
48:02
maybe next time but not right now so now I'm going kind of quickly my aim is to give you
48:27
a flavor for sort of how some of these kind of arguments go so that you can play with them and imitate them and sort of have a toolkit to play with some problems in the problem session
48:40
okay so let's prove the most general kind of structure theorem that we have for this group this one I have a subgroup of homeomorphisms I want to find a free group
49:04
or an invariant measure right so for instance if I was acting literally by rotations my invariant measure would be like lebesgue measure on the circle that would be nice more generally maybe I could I won't write the lebesgue measure one down but if I'm in
49:25
case one of there if I have a finite orbit I can just put point masses on all those
49:42
finitely many points my orbit has cardinality five I'll put a point mass of one over fifth on each point those points will get cyclically permuted around that will be an invariant measure under my group action okay so you you know so I don't need it to act by literal rotations
50:01
all I need to know is that this point these three these five points are always they'll stay in their order right all elements my group permute these I can put point masses there to produce an invariant measure okay in particular if I have a single fixed point
50:30
then this titz alternative is very easily satisfied it could have a billion fixed points or uncountable many I don't care just put a point mass at the single fixed point it never moves that's an invariant measure so we're really interested in what happens in two
50:45
or three all right so when this kind of thing doesn't happen I'm expecting it to look like rotations with lebesgue measure some version of that or it to what if I want to prove there's a free group I better imitate my ping pong picture oh which got erased but from before
51:07
okay so I'm looking for one of the other well one thing that would put me in the invariant probability measure would be if I had a group that acted by homeomorphisms
51:23
where those were equicontinuous right if the action is by some it's some family of homeomorphisms these are all equicontinuous I'll remind you what that means in a second if you forgot your
51:43
analysis then the arzela schooley theorem means that your group is compact and it's a fact or an exercise
52:07
which you can't do and I put on the sheet it's a fun one any compact group acting on the circle is conjugate to a group of rotations so gamma is conjugate so compact implies gamma conjugate
52:29
to a subgroup of SO2 okay so then that conjugate of your lebesgue measure pulling it back under that conjugacy gives you an invariant measure for the action
52:45
yes you're just acting by home yeah gamma could be countable uncountable oh maybe I want it to be countable in what I'm thinking here
53:04
no this is fine this is I think fine just in the contact of topology do you have a sort of a example that's bothering you it's not just um you mean that the uh the closure of it should become I see um right well wait so so
53:35
the if the closure of gamma is compact then its closure is conjugate into SO2 and so gamma itself is a subgroup of SO2 being a subset of its closure so that's good enough right
53:45
then gamma yeah so that's fine then the closure of gamma is uh relatively compact is fine so its closure is some subgroup conjugate into SO2 that'll do it thanks okay so if you fail to
54:01
be equicontinuous if not okay so by the book definition says what does it mean that means that there's a bunch of intervals of length going arbitrarily small so that elements of your okay so let me write that down so there exists some
54:23
bound some epsilon there exists intervals where the length of i n goes to zero and there exists some sequence of elements of your group where the length
54:43
of g n of i n is at least epsilon all right so this is the case that we're worried about and we want to hopefully produce some ping pong out of this and it's looking pretty good because i have like tiny little intervals get stretched and that's what i was using in
55:02
this ping pong picture stretching and shrinking lengths of intervals all right so to make my
55:22
life easier i'm going to assume for a little bit that i don't have one of these crazy cantor sets that seem like a pathological thing i'm gonna assume that my action is minimal all orbits are
55:41
dense and then we'll deal with what happens if not else elsewise so let's assume for a moment a little bit of time that gamma acts minimally and what i want is to run my ping pong argument
56:16
let's draw this picture of what i want i want to find something like this and some f that takes this interval to like this big one and then some g that has something
56:27
like this and it takes this thing and squishes it down or it takes this one and stretches it over like i had that before and now i'll have a free roof and i'll feel happy but the problem is that this picture the way i drew it doesn't always happen
56:45
all right i might have to draw a more complicated picture all right so here's a problem this picture doesn't happen the way i drew it even in like i don't know like sl2r acting by
57:02
acting on like raised the origin or like the the twofold cover of psl2r if you like right if i wanted to try and do this picture and sl2r what the thing about sl2r is is everything looks the same on both sides of the circle right everything commutes with a order to rotation it's like an action on lines not the projectivized version so i can't have
57:24
something that takes this giant interval and swishes it down whatever i do up here a squish is also done down here i'd have to have another squish so i wanted to do this in sl2r
57:44
i'd have to draw the following picture i wouldn't you know i maybe need more of these and where my f is taking the these two points like here and taking these two points there and so my domains that i was stretching or squishing are
58:01
no longer connected okay and i would want it g to do something like this okay and then i could run the same argument but it would look different but this is just silly you know what did i draw i drew the like twofold cover of this picture okay everything commutes with the rotation of order two in sl2r here's the projectivized version it's the
58:26
picture i had before okay and more generally if you have some kind of group that acts on the listen the circle happens to be a k-fold cover of itself so i could like look at all the lifts
58:42
to a k-fold cover and i'd get a more complicated looking picture okay so this is just to say that i might have this silly issue where i should i was i had a kind of a complicated action i could have just taken a quotient and then i could draw my original ping pong game from before so this is nice picture and this is more complicated so i want to simplify my life
59:13
right off the bat and be able to detect if secretly your whole group actually just came from one of these lifting tricks if i could have just been like listen you were really acting
59:23
on this circle and then you took all the lifts to this cover all right so the outline of the is to first uh produce that covering map if there was one and then say all right uh if i didn't have this covering map going on then i have exactly this picture going on i can find intervals in the
59:43
configuration i wanted before from my step zero of this lecture example of free group uh and and and show that there's a there's a free group in it um and i don't have much time left in this hour so i will maybe out
01:00:00
how this works and we can either take it up next time by request or you can sort of fill in details as you like. Okay so let's see if I act minimally I can massage this statement into something else let me massage it right away put
01:00:24
it right there beside, remind us. So if I pass to a subsequence these GNINs it's a sequence of intervals in the circle they'll Hausdorff converge to some set after passing to a subsequence. So I could pass to a subsequence so that
01:00:49
this condition will say that you know all of these contain a very slightly smaller interval maybe call that J and then the inverse images of J under GN
01:01:03
are getting really small. Okay so this says that there is some interval J of length more than epsilon over 2 I can certainly guarantee with the
01:01:25
length of after passing to my subsequence GN of J, the inverse of J I guess going to 0. So I have a fixed guy that gets shrunken. Okay so this
01:01:42
let's call this interval a contractible one. If my action is minimal I can move any point to any point okay so minimal implies that I
01:02:02
could put J wherever I want it that says that for any point in the circle there exists some point Y okay so that the oriented interval between X and Y is contractible in the same sense that J was there exists a sequence of
01:02:23
elements that shrink it to arbitrarily small length such that there exists some sequence HN will depend on X with a length of HN of this guy XY going to
01:02:42
0. I will pick out if I was secretly covered by a cover by the following function so I'm going to define a function on my circle that assigns a
01:03:10
point X to the biggest possible Y where I can do this. So the supremum
01:03:22
preceding clockwise around the circle so that XY has this property. For example if you're in PSL2R acting by Mobius transformations this function
01:03:43
happens to be the identity. If you give me a point X you give me an arbitrarily large interval there's some very very very strong hyperbolic element that takes this to being a very short interval as you want and the claim is
01:04:00
that in general this might not be the identity I'll state it as a fact that you can check this guy is a finite order rotation or a finite order homeomorphism and I've cooked it up so that it commutes with the action of
01:04:29
your whole group meaning that Phi of GX is G times Phi of X. So these are easy to check from the definition it's saying when are you in this picture
01:04:45
instead of that picture here Phi is order 2 okay but it commutes with your group so I might as well look downstairs and to say that Phi is the identity map says that at every point I can take as big an interval as I want
01:05:01
maybe I took this really giant one here and I can contract it to intervals as small as I wish okay and similarly for one containing some point over here or bound with a left endpoint like this I can contract it as small as I wish that is exactly what I need to make the ping-pong argument work all
01:05:23
right so I'll just summarize and say so passing to a quotient Phi is the identity in which case you can play the ping-pong argument okay that's a good
01:05:57
place to stop perhaps that's too much I should say if you weren't acting
01:06:03
minimally then we have this canter set picture and I can just pretend my canter set was the original circle right you imagine you know that like canter staircase function that takes your interval with a canter set in it and
01:06:22
collapses each complementary region to a point this is some continuous function that makes your canter set disappear right my canter set is invariant so I could apply the collapsing function and I'll get a new circle on which my group acts by homeomorphisms I can run this case all right either I was
01:06:41
equicontinuous on this new circle and so I can pull back an invariant measure to my old one or on this new circle I could play ping-pong and that must mean I contain a free group okay so in very short the fact that I'm assuming that gamma acts minimally is no big deal my other option is this
01:07:06
canter set and I think that's a that's a good place to end this first bit