 What is an object in this? And so action Satisfy G If I had a point in a movie already, that should be the same thing as just moving it by the composition And if I hit it with the identity element of the group it should be the identity element should be the identity Symmetry. So this is all a G flow. So we just have some, we use an object, we have some group of symmetries It's moving around Well, there's some examples. Let's make it a little concrete. Four Itch examples running at all times. We can like think about how these examples relate to later notions Probably the easiest group to wrap our heads around is the integers The integers are a really nice group. What's a really easy space to visualize? How about the circle? S1 is literally just the unit circle unit circle Pretend that's a circle Action is going to be given by irrational rotation So in this case since Z is generated by Added to be generated by 1, I'm going to tell you what a one-step transformation is one step of the transformation is going to Rotate the whole circle alpha-many radians, that'll be some irrational Rotation You could think of a rational one, but we'll think about the irrational ones for reasons yet to come So this is a flow Since Z is a discreet group to check the continuity. You just need to know that each Group element acts continuously on the circle and these rotations are clearly The same space can admit many different Flow structures so I could change the group for instance. Let's change the group G is going to be homeo plus What is the little plus here doing so homeo S1? That's just all of the homeomorphisms of The plus means no flips orientation preserving This is not a discreet group anymore. This is going to carry the compact open to follow in this case It compact open to quality has a general definition, but since G is just this nice How far away is a typical group element from Doing nothing Well, I look at every single point every single point moves by some amount I Look at the supermum of those amounts and that'll be the distance of this group element from the identity And that'll be a nice right in variant. You declare that to generate a right If I want to compare G and H, I just say where does G send this point X where does H send this point X? compare those distances look at the look for the largest such thing So that that's a topological group and then X is the BS one and a is going to be just the evaluation so Not this two prime This is going to be the same as two, but I could have just kept a discreet topology. That'll be fine Same as to be discreet So the contact open to policy to be this is actually the courses It's one way to define the compact open to Paul G on a homomorphism space It's the courses thing that makes this evaluation continuous So it's the courses thing we could possibly need to get this to be a flow But I could take finer things and so just just the discreet topology would be a finer thing So that's one two two prime and then oh, maybe I'll put three we all know an action And I'm gonna put three above Just to have everything on that part of the board to keep and example three This is just to show I mean so the circle is kind of a it's a manifold It's a kind of easy to visualize object These things can be a bit more abstract So let's let G be the integers again Now the space is going to be the set of all by infinite zero one strings Just zeroes and one's going to the left and right forever I Give this the product topology, so that's gonna be nice And then what is the action going to do again since it's the integers I just have to tell you what The generating transformation doesn't see the shift a string of zeros and ones going in both directions forever I can bump it to the left and right as I want to do So that's a bit more Maybe a bit more abstract that comes from symbolic dynamics. All right, so these are four examples of flows I wrote a remark here Do I want to say this remark? Yeah, I'll say this I'll say a remark So we saw a little bit of We can mess with the topology of the group a little bit in general What is the thing that we need to get a flow? so So X say that X is a compact space and if G is topological group a useful thing to keep in mind G-flows Are the same thing from the group so by a so say that a a G flow a Need the same thing as a continues from the morphism 5a that'll go from the group to homeo X Where this is a tell if something is a G flow it should give me a What is 5a each group element is doing something to the space just Output what it does to the space that will be a homeo And so this this makes it a bit clearer that if I have some topology on G where this is a Flow I can make it finer But anyways, this isn't terribly central to what I want to do So what I want to do is I want to Discuss some properties the question Discuss some properties of the flow might have And then we're going to zoom in on one of those properties. I'm gonna we're gonna construct a universal flow with that So what are some properties we might be interested in some properties of flows call this a zero Because it's pretty rare For any pair of points there is a group element of these four examples Who can tell me which ones are point? Yeah, so homeomorphisms of the unit circle I can easily here's a point on the unit circle Here's a point on the unit circle. I can just rotate to get Another It turns out yeah, neither one or three are going to be point transitive so the irrational rotation Looks kind of promising because I can I can sure get pretty close, but The orbits here can be countable in the circle is uncountable And up here, I guess same obstacle works Transitive, but yeah, home OS is huge Well point transitive almost never happens, so let's weaken it a little bit Minimal so Zero is asking for one Everything is in this orbit is weakening that a little bit. I no longer require Everything to be in the same orbit But just that every orbit Sees most of the space so do we get any so two to two prime will still be Minimal do we get any new members in the minimal club? I see I see some heads saying yes. Yes irrational rotation is a good example of a minimal flow Here's a point say that I want to get into this Open interval I just have to I Guess I should compute. I just need to figure out some number of multiples of alpha that get me into here mod 1 and You can do that So this is this is This is this is not minimal. What's an easy way of seeing that? There are some really stupid strings like the zeros like I could send zeros off in both directions forever and That will never if I have a neighborhood that says put a one in this spot. You're going to have trouble reaching that If you only think about the AP here, I'll do Oh, so the eight that those won't be That won't be closed in general But we're gonna do one further weakening which will which will save a One said every orbit was dense So it's clear that that's a weakening of one which is internal weakening of zero two is the weakest of these notion So one two and two prime will still all be topologically point transitive and Now I claim that three three is also a weak point transitive. So How might you need to find some string whose orbit is dense? How do you do that? Don't take some string that just contains every finite. Yeah Neighborhoods are neighborhoods are gonna be described by finite patterns of zeros and ones and we have infinitely many infinitely much So I just need to type every possible That's exactly All right, so we're gonna do So those are examples What we're gonna do for the rest of the talk is we're gonna focus on number two so goal for rest of the talk construct the maybe one part of this goal is a definition But a definition I guess there's a group involved the GM X is just some G flow X will be topologically Point transitive as witnessed by this little point that's not though. I declared a point of interest with dense orbit So what we're gonna show? theorem greatest G and so what this means is that there's going to be a So there is some in it. It's usually to know that SMG for historical reasons Pierre Samuel was the person responsible for This stuff originally And I this point So this is going to be a GM it and then given other GM it There's going to be a Unique which what are the things we want our map to satisfy? I really should have I should have defined some sort of category of G and it's an amic math, but an amic map is continuous It respects the G action It respects the point I told you about so it should send this point to that point The uniqueness actually just follows from that last fact. I sent this point to this other point We know that it has to be G at covariant and they both have dense orbits, so if I Want to know Where I have some other point P in here. I want to know where Phi sends it Well The orbit of one is going to cluster around P. I know where everything in the orbit of one goes So I know I know where that's going to cluster Let's spend the rest of today constructing this Okay I don't have the intuition for why Especially this one because I would think of the initial object for the category as being the least thing Because we're Given if I had some other why why not in some map here I can complete the diagram What the reason that's not the right thing to draw is because all these maps are unique There's unique map between any pair of objects, which kind of simplifies the picture So so the simplest thing to say is that there's just there is an arrow from this object In what sense is a minimal flow That's just in a In a Zorn's Lina sense it contains no proper sub flows And here's another Annoying one so the universal minimal one It is unique up to G flow isomorphism, but not unique up to a unique G flow isomorphism That is an annoying wrinkle if you care about these things and this is a good stopping point Oh Yeah Each one of these things is an ambit Want to take the whole thing I'm going to take the orbit closure acting on the designated point so I have a designated point in each Coordinate that gives me a designated part of the product. I want that point to have dense orbit So I'm just going to pass it down to a sub space where it does happen This is this behaves like a product and so What's kind of cool is Is a set this is a bit of a logical bugaboo There's a set of representatives What this really amounts to saying is that ambits can't get too too complicated Why is that? It's really this dense orbit so you have your in particular that provides a map of your group into this a priori it could be a huge compact space But it can't be too huge because your group has a fixed size Say that your group has size Kappa works half of some cardinal We get a map with dense image So what does that tell us so if x is the ambit? so x has dense Subset can't be too too big Cardinaly back to less than 2 I mean, it's pretty large, but it's it's bounded Your amids aren't going to get bigger than this so I can with enough of a Initial fragment of set theory you can see all the ambits So what that allows you to do Let's take a set of representatives for all the isomorphism types of ambit. You take the amic product that is by construction by Absolute overkill construction a universal and it you just smushed all the Yes, well you use the what? Do you like this this connects to anything other category? Yeah, it's the product in the category The category of ambits is a little bit It's a bit easier than a lot of other categories because there's at most one arrow between any that's one way of Exhibiting this greatest GM, but it's not very useful like I'm joining what it looks like I want to actually construct this thing so I At least in the case of a discrete group, let's try to construct and as I Indicated in the abstract filters are awesome. So Without further ado, let X be a set collection of some sets of X So that it's non-trivial, which is usually these two Statements for clothes for a notion of a largeness if I have something I'm declaring to be large I make it bigger and that's still the way large if I have two large things and they intercept So it satisfies one two three and four so Look, there's a purely combinatorial way You So ultra filters are a notion of largeness that are very opinionated ultra filters have an opinion about everything Everything is either large or small Equivalent definition for crime is that you Is maximal to satisfy one That's sort of the non Abstract way of saying so ultra filters are maximal filters Ultra filters are really opinionated filters You may have seen ultra filters before They show up in like functional analysis. They show up in model theory But most of the time you think about like one ultra filter letting you take some sort of Like it's a it's a particular way of like I have some Countable set of things I want to grab a cluster point an ultra filter is a good way of letting you grab a cluster point So what we're gonna do today? I'm not gonna worry about one ultra filter. We're gonna worry about all of them in the space that I should remark that identify X as a sunset By taking principle If I have a point in X the collection of all subsets containing that point is an ultra filter But they there's gonna be a lot of other stuff Apology on this thing What we're going to do is A basic camera open set are gonna be over for filters that contain Over all subsets This notation might be a little bit Confusing at first because a bar kind of looks like the closure of a and that's not an accident under this identification a bar is the closure of it, so this is a zero-dimensional space It has a basis of clothing This is a I Don't think I have time to prove it This is a compact easy Compactness probably the easiest way to do it is through the finite Intersection property characterization of the practice so if I have a collection of basic closed sets The basic this basis for the open sets is also a basis for the closed sets Because a is also By this identification So basis for closed sets also If you have some Collections and with the finite intersection property Then the sets that you're mentioning form a filter Extended to a filter that's all right. So here is this is the key property about me to X the magic triangle Those of you who have seen me give GSS talk before have probably heard me Pontificated length about the magic triangle, but it's it's the best triangle So say that let Y be House dwarf everything is house dwarf function Exit just a set remember so there's no It's a discrete topological space So so any function So what are we gonna do? Here's X? Here's why you have here's the inclusion And then here is the last side of the magic Let's actually look for this I'm going to erase my the uniqueness Just follows I should remark that the principal the principal members dense just because if I have a basic open set like this and a member of a That's principal ultra filter a is a set containing So if this dense We know what's going on a dense subset of beta X So we just have to argue that there is an extension and How do we do this? Let's let let Why you know why consider That's going to be back here in X and He has an opinion about it. So let's Suppose that it's This is just a definition so P Thinks it wants to go why if this happens For every P this it's suppose not Why is compact and every point in why is a little neighborhood around it where something bad happens? So let's just grab a finite cover of the bad things So the union Every point has to somewhere so that's I Didn't spell this out explicitly, but this is going to be a problem in regards to four I guess one X is in there you Repeatedly apply that I guess came any times One of these has to be inside Way to If they are different Let's put little disjoint neighborhoods around them because everything is house door These have disjoint pre-images and so you're never going to have Disjoint things that P thinks are both large. That's bad So that's all right You have to do a little bit of fiddling to check that it is continuous, but I'm just going to say yes, it's continuous Why is it continuous because it's sort of the obvious thing that you wrote down That's kind of the way a lot of these proofs go filter spaces Like there's just one way to write down the possible function Let's get an ambit out of this. So if I have a discreet group So say G is a discreet group For you the G1 G Is an ambit what the flow is we have a space but where's the flow I'm going to send G It's pretty easy. So if A is a subset of the group G is a group element P is an ultra filter I need to tell you When a is a member of G dot P And it's really the only thing that can happen P is a collection of subsets group G is just going to move around This this works this It's continuous because I can ask about this basic open condition by asking about this Condition that's continuity G is discreet so to check continuity of the whole thing you just need to check So why is it the greatest and it well because of the magic charming wolf let's Be another GM it so let F Go from G to X G I'm just gonna look at what G does to the distinguished point That's sort of the only thing we could possibly write down And that's the only part of the map we could possibly know how to define ahead. We know what's happening on this orbit So the orbit tells us to do that and then we're going to extend To all beta G You have to check the magic triangle does not buy you G So that is that's the greatest Which is still I mean it's kind of abstract still but it's maybe a bit less abstract than just saying multiply all the emits together This is at least an object. We know what it is So yeah, I will not have time to talk about G not discreet, but you can speculate what would you do What's the problem maybe I'll say If G is not discreet you have to worry about joint continuity of the action So if G had some topology You can write down the same action it's just Beta G is the wrong Compactification to use so you can think of this This beta operation is turning In this case a discreet space into a compact space and the freeest way possible And so when G Has some of a topology that's you don't just want to regard the set You need to remember that it started off with a topology. And then there's a few extra All right, that's it. So is that any questions or comments? How simple is the simplest example you can think of for G that's Apologized. Oh, it's still easy to know No, but the subs is a counter example to this. Oh, any Not sure no topological group. Any no true topological group breaks this One one case where it's easy say that G is a compact then SG will just be the group itself That's kind of on the other extreme end of like Being trivial compact groups are Really really simple dynamically Anytime it acts on another compact space It has to be like a group like a group of quotients Oh locally compact groups are disgusting. You have Veatch's theorem that local compact groups admit free flows So that's gonna tell you that SMG is not matrizable, so SG is not matrizable So SG is gonna be a huge So they still like to see concrete description is If you were in the logic seminar today, that's sort of the like The hint is to where things we consider like nearness As far as I know