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Episode 80 - Kimberly Ayers Kevin Knudson & Evelyn Lamb Download Evelyn Lamb: Hello and welcome to My Favorite Theorem, the math podcast with no quiz at the end. My name is Evelyn Lamb. I'm a freelance math and science writer in beautiful Salt Lake City, Utah, where fall is just gorgeous and everyone who's on this recording, which means no one listening to it, gets to see this cute zoom background I have from this fall hike I did recently with this mountain goat, like, posing for me in the back. It kind of looks like a bodybuilder, honestly, like really beefy. But yeah, super cute mountain goat. So yeah, that really helpful for everyone at home. Here is our other host. Kevin Knudson: I’m Kevin Knudson, a professor of mathematics at the University of Florida on the internet, they would call him an absolute unit, right? EL: Definitely. At least they would have five years ago. Who knows these days? KK: Shows how out of touch I am. That's right. Yeah, here we are. Yeah, it's actually lovely in Gainesville. Like I've got short sleeves on, but it's like 75 and sunny and just everything you want it to be. EL: Perfect. KK: And tomorrow, tomorrow's homecoming at the university, which means that it's closed — this is bizarre — for a parade. But as it happens, tomorrow is also my birthday. EL: Wow! KK: So I get the day off, and it’s unclear what I'm going to do yet. EL: Well, just having a day off to lie in bed as long as he want, you know, drink your your coffee at a leisurely pace. KK: Absolutely. EL: It’ll be great. Yeah, and we are recording this shortly after Hurricane Ian. And so you're here, so you made it through okay. I actually don't know my Florida geography well enough to remember where Gainesville is. KK: Gainesville is north central. And weirdly, this cold front sort of pushed just south of town right before. It was about 65 degrees for three or four days, which is freakish. The hurricane, of course, took its very destructive path entering around Fort Myers, went across over Orlando, then to the Atlantic side. We got about a half inch of rain. It was — I mean, we were expecting, like, 10 inches, and then that that weird path happened. Of course, a lot of a lot of our students, you know, their homes have just been devastated. It's a rough time, but you know, the governor and the President are at least putting aside their differences temporarily and making some good progress. Well, we'll see. It's gonna be a long rebuild down there. EL: Yeah. KK: And it's a beautiful part of the state, and I feel bad for everyone down there. EL: Definitely. KK: But yeah, it's not the first time, you know? EL: Yeah. Well, yeah, I hope it it continues to progress on the cleanup and everything. And today, shifting gears entirely, we are very excited to have Kimberly Ayers on the show. Welcome, Kimberly, would you like to tell us a little bit about yourself? Kimberly Ayers: Hi, thank you. Yeah, I'm super excited to be here. And happy early birthday, Kevin. KK: Thanks. KA: So I am an assistant professor in the math department at California State University San Marcos, which is about half an hour north of San Diego, so for those of you who are less familiar with California geography, and my research is in dynamical systems and ergodic theory. KK: Cool. EL: Nice. And I said “shifting gears” because I know you're also a biking enthusiast like I am. KA: I am. Yes. I love to get out on my bike. And California weather is — San Diego weather, it’s hard to be outside. So I’m a big bike fan. EL: Yeah, I — the other day, someone on a local social media thread was posting like, you know, “We shouldn't have good bike infrastructure in Salt Lake because, you know, we're not San Diego, so there's so little time that you can bike here.” And I was like, well, first of all, that's just not true. But you do live that dream of the, like, San Diego biking weather all year. KA: Yeah, it’s — I can't complain about it. KK: Sure. Well, it doesn't really snow that much in Salt Lake right. I mean, it hits the mountains. But yeah, I mean, so when I was a postdoc in Chicago, I cycled a lot, but come November I was finished, right? EL: Yeah, because the roads just never get all the way clear, but here it's dry enough that they do get cleared. And so — you know, I am not an especially hardy person. But, you know, if you’ve got some layers on and the ice is off the road, it's actually doable. KK: It’s not a problem. EL: I discovered. I mean, this was a pandemic discovery because I grew up in Texas, and I would just put my bike away in, like, November here, but decided I mental health-wise that I really needed that during especially the height of that covid winter — our first covid winter. Anyway, lovely to have, I guess three people who enjoy biking on this show, but we are not here to talk about biking. We are here to talk about Kimberly's favorite theorem. So yeah, what is that? KA: So my favorite theorem is a theorem called Sharkovskii's theorem, which is a pretty famous theorem in dynamics. To back up a little bit, when I talk about dynamics, right now I'm talking about discrete dynamical systems, where the idea is if you have a function that has the same domain and codomain, you can think about compositions of that function with itself, right? You can take successive compositions over and over again. And so as a dynamicist, I'm interested in looking at these sequences, like if I start with the point x in my domain, and then I apply f, so I get f(x), and then apply f again, get f(f(x)), and then f(f(f(x))), and so on and so forth, right? This is a sequence. And so we can ask questions about sequences, right? We can ask, like, do they converge? Or maybe if they don't converge, do they have a convergent subsequence? Do they ever repeat themselves, and that repeating themselves is actually what Sharkovskii’s theorem is all about. So Sharkovskii's theorem is about continuous functions on the real line. So it's important to say that there is at the moment, no, like higher dimensional-analog to Sharkovskii’s theorem. This only applies to one-dimensional functions. But Sharkovskii’s theorem says, okay, so we have to take a really weird ordering on the natural numbers. So we're going to start with all of the odd numbers except for 1. So starting at three, we'll take all the odd numbers in a row, so 3, 5, 7, 9, 11, so on and so forth. And then, once you're “done” with all the odd numbers, yes, then you'll consider 2×3, 2×5, 2×7. And again, all of those, right, and then 22×3, 22×5, taking higher powers of to multiplied by 3, 5, 7,.… And then again, once you're “done” with all of those, the only numbers that you haven't included are the powers of 2, so then you take all of the powers of 2 in descending order, until you get back to 1. So it's not a well-ordering, right? You have to kind of wrap your mind around this fact of like, once you're “done” with the odd numbers, then… EL: Yeah, right. KA: But it is a total ordering. EL: Yeah, so you can take any two numbers, you can tell, like, which one is before the other one. KA: Exactly. KK: Right. EL: But you can't say this one is 17th. Well, you can actually say which one is 17th in the series, but, like, a number that isn't an odd number, you can’t say what position it has. KA: Exactly, exactly. If I were to count, like, the natural numbers in their usual ordering, you know that eventually, you're going to get— like, if you asked me are you eventually going to hit 571? Yes, I will. Right. But if I were to try to do this with the with Sharkovskii ordering, that’s not going to happen, right? So it's kind of weird that there's a minimal and a maximal element in the Sharkovskii ordering. So I haven't told you the punchline yet. EL: Yeah, we’re just wrapping our head around. KA: But yeah, exactly, right. So we start by taking this really strange ordering on the natural numbers. And then what Sharkovskii’s theorem says is if you have a period — so I guess I didn't quite define which way which the ordering goes, but let's say that 3 is the maximal element and 1 is the minimal element. So Sharkovskii’s theorem says that if you have a period N orbit for some discrete mapping on the real numbers, then for every M that's less than N, you also have a periodic orbit of that period. So, for instance, if you have a period 2 orbit, you have to have a period 1 orbit, which is what we just call a fixed point, right? If you have period 64, then you also have a period 32, 16, 8, 4, 2, 1, etcetera, and probably most excitingly, is that if you have a period 3 orbit, then you are guaranteed to have periodic orbits of any other period. KK: Right. KA: So sometimes people talk about Sharkovskii’s theorem, and what they say is period 3 implies chaos. EL: Yeah. KA: Now, I haven't told you what chaos is. And I kind of joke, actually, that chaos in the dynamics community is sort of a bit of a chaotic concept in and of itself. Because there is no really one universally accepted definition of chaos. There's several different types of chaos. There's what we call like Devaney chaos or Li–Yorke chaos. But this definition of chaos, which I believe is Li–Yorke, says that in order to have what we call chaos, you need periodic orbits of all periods. So there's that period 3 gives you that condition, you also need an orbit that is going to be dense in your space. So an orbit that kind of fills up your entire space. And then you need this other thing, which is probably the most famous aspect of chaos theory, which is the sensitive dependence on initial conditions, otherwise sometimes termed the butterfly effect, which basically says that if you have two points, no matter how close together, your initial points are, if you apply f enough times, their sequences, their orbits eventually grow some distance apart from each other, no matter how close together, they start. So there's essentially like no room for error if you have a chaotic system. KK: Right, right. KA: Sothat's why once you have period 3, you're guaranteed at least one of the requirements for a chaotic system, right? So my students asked me the other day, they were like, what's your favorite number? And I was like, Oh, that's a really hard question to answer. KK: Three! KA: But I think it has to be three because of this, like, you see a period 3 orbit, you kind of automatically get excited, because those are, like, pretty rare. So that is, yeah, I guess I have to say that three is my favorite number EL: So we get a favorite number for free on this episode. KA: A favorite number and a favorite theorem! EL: Okay, so I want — I'm dragging a little bit this morning. And so having a little trouble with, you know, putting these things together. So what I mean, so we have this weird order on the natural numbers. Like sure, I'll let you do that. Can't really stop you. So what kind of — what is our f? What? What kinds of dynamical systems are we talking about here? KA: Yeah, so let's talk about a couple of famous examples. So oftentimes, I'm talking, I've been talking about functions on the real line, but I guess really dynamicists really like studying functions on compact sets. Because if you have compactness, you're guaranteed things have convergent subsequences. So so we get what we call this limit behavior. So a lot of dynamicists study functions on, like, the closed unit interval. And so some examples are what's called the logistic map, which is a quadratic function. So it's some parameter R times x times 1−x. So it looks like an upside down parabola, right, it intersects the x-axis at 0 and 1. And then in order to make sure that you map back to things in between the 0 and 1 interval, we're going to require that R be between 0 and 4, right? KK: Right. KA: If R is bigger than 4, then the top of that parabola bumps up above 1, and, and there are actually cool things that you can study with that as well. But I'll talk about that another day. KK: Oh, the bifurcation diagrams. That's what you want to talk about. Right? KA: Yeah, once you vary that, so that's the cool thing, is like as you vary this parameter R, if you start at 0, and then kind of think about what happens is you increase R a little bit, you see these periodic orbits appear in exactly the order that Sharkovskii tells you, they're going to happen. So you start with a fixed point, so that's your period 1 orbit, right? And then I believe it as once R gets above 3, that's when that period 2 orbit shows up, right? And then if you bump up R a little bit more, then you see a period 4 orbit appear, and then a period 8, and we call this a period doubling cascade. And then, of course, as you can imagine, there's only finite room for R, right, we only consider up to 4, but there's infinitely many kinds of periodic orbits that have to appear, so they have to start coming at you faster and faster and faster. And then once R gets above — I want to say it's roughly 3.87, and it's only a number that we've sort of gotten that numerically, unfortunately — then you enter what we call the chaotic regime, which means that period 3 orbit shows up. And then we know we have out there somewhere periodic orbits of all periods. So people might be familiar with this bifurcation diagram, which shows kind of like, where the periodic orbits appear. And it starts off, like on the left hand side, looking pretty simple and smooth. But as you go to the right hand side, it gets more, like, fractally looking. And that's because, once again, those periodic orbits have to show up more and more and more quickly, right, as R increases in value. It’s very cool. EL: You are just blowing my mind, that there isn't just, like, what I thought was going to happen here was like, we would only get these powers of 2 ones. And then you're going to tell me a different dynamical system later that would have — but like, because it makes sense, it somehow makes sense to be like, at 3, we get this, at, you know, 3 1/2, we get this, at 3 3/4, we get that and, like, the fact that, that like, we some at some point do shift to this period 3 place? KA: Right! EL: And we can't even know what the number is? I’m sorry! KK: And these functions are so simple, right? It's just a quadratic. It's a tent map, basically. Right? KA: I know! It blows my mind every time I think about it. Yeah, so okay, so you're seeing the beauty in this theorem now. Like, just how cool it is. And I guess I'm gonna, you know, shift gears, pun completely intended there, once again, because not only is the statement of the theorem really cool, but the proof of the theorem is really incredibly beautiful. And so I'm going to do my best to explain it without being able to draw anything. But let's talk specifically about the period 3 implies periodic orbits of every other period, that kind of statement right there. So let's suppose that you have a period 3 orbit, so you've got three points on the real line, call them a, b, and c, such that f(a)=b, f(b)=c, f(c)=a, so they go in this cycle, right? You can label — so that's going to partition, at least between those, into two different intervals, right? So let's say that you have a, b, and c arranged from left to right. And I'm going to, for purposes, it's going to be a little bit weird. But there's a reason I'm going to call I1, the interval between b and c, and I2 the interval between a and b. So what you see happening, then, is that because of continuity, that interval I1, when you when y…
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