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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 or