 Thank you, Philip. And the other two organizers, of course, for inviting me to speak. So just a couple of words before I start. I mean, so the title of this talk, hopefully you can see my slide here, is a transcendental birational dynamical degree. If you have been around NTWeb for a while, then you probably saw a talk with a very similar title from my co-author Jason Bell a couple of years ago, except it did not include this word. And so what I'm going to tell you about today is, well, first of all, first of all, joint work with Jason and his co-authors from that previous work, so Jason Bell, Jeff Diller, and Matthias Jonsson, where we construct a transcendental number related to algebraic dynamical systems, in fact, a family of them. And so the talk is really, I mean, from a number-theoretic perspective, what you should think of this talk as being about is this relatively rare object in number theory, which is a transcendental number that arises naturally, somehow mathematically, that we can also prove is transcendental. So with that said, the first portion of the talk will essentially be me trying to convince you that this number arises naturally. So to understand a question about some integer sequences and their growths, which are related to dynamical systems, and then the latter part of the talk, I will discuss how we actually approach the transcendence of these numbers. All right, so the integer sequences that are the fundamental objects that I'm going to talk about today are these so-called dynamical degree sequences, which is to say that they will arise from algebraic dynamical systems, that is, dynamical systems, which are discrete and arising from polynomial maps. So here is a basic example of a dynamical system in two variables, formed by iteration of the map, which sends xy to yxy, the product. So iterating this map, we obtain a collection, so accountable family, of polynomial maps of the following form, so xy maps to yxy, which then maps to xy, xy squared, and then to xy squared, x squared, y cubed, and so on. And the feature that we're going to, sorry, the feature that we're going to be interested in for these is the degree of the iterates of these systems as polynomial maps, which is to say the maximum of the polynomial degrees that occur. So in this particular example, of course, before we iterate, we have degree one, and then here we have degree two, degree three, I don't know why that keeps happening, I think maybe my phone is too close, I'll throw it away, sorry. Degree five and degree eight and so on, and you can see exactly or at least predict what you hope will happen, which is that the sequence of degrees that we obtain here is the Fibonacci sequence. Okay. And that's not hard to see, by the way, by definition of the map, when we apply an iterate, we take the degree of this component and add it to the degree of the component before it. And since we started with one one, we obtain the Fibonacci sequence for our degrees. Okay. So that's my claim that they're interesting integer sequences that arise this way. What is actually happening geometrically, what is this this degree actually encoding? Well, it's actually telling us what the dynamical system does to curves or more accurately how it pulls back curves. So here, for example, if we look at the iteration of the line say x equals one, then we can start looking at the iterates of x equals one under this map and we see similar degree. I'm really sorry, I don't know what to do about that or why it's happening. I'm just going to keep getting rid of it when it pops up. The degrees that occur precisely the degrees of those iterated maps, right? And so geometrically, what is the interpretation of the degree of one of these maps, let's say a map of C2 or more generally affine two space, it's the degree of the pullback of a generally chosen line, right? So we take some line and we pull back under our map in this example. We have the same one from the previous slide and this preimage, of course, then if we look at the collection of points whose image satisfies some linear relation will be something of degree two. Okay. All right. So if this is the geometric interpretation, the degree of some curve in this surface, then in fact, we had better work with compact geometry. And if anyone knows what is happening with my iPad and wants to tell me how to make it stop, then please feel free to put that into the chat. Okay. All right. So let's continue on with this example, but now viewing it as a rational map of projective space so that we actually have some nice compact space to live in and compute degrees. So if we homogenize the map from the previous slide, here's what we get. And computing the degree sequence from this map and projective coordinates, well, of course, composing two degree two polynomials will give us a degree four polynomial. But once we look at the second iterate, for example, we see that these polynomials have a common factor, right? And so as a map of projective space, we need to remove this common factor. And what we're left with is that the second iterate has degree three. Okay. And so these degrees then will precisely match the degree sequence from our first slide. But now somehow we're working in this nice compact space. And geometrically, what's happening here when we fail to have full degree, which is to say we have a cancellation here in the second iterate, for example, is that some positive dimensional thing like a line is mapping to eventually to some point in the indeterminacy of the map. Okay. So this map, for example, is not defined at the point zero one zero. Since all of my polynomials take the value zero there. And this line z equals zero maps to that indeterminate point and causes the degree of that iterate to be lower than say expected. Okay. And so the punchline here is that the degrees, the degree sequence. So the degree of these iterates is telling me something about the geometry of the iterated pre images of the indeterminacy set of the map. Okay. All right. Now the good news in projective space is that of course when we compose to morphism or to rational maps more generally, we're composing polynomials and the degree cannot increase by anything more than, sorry, rather the degree of the composition can't be anything more than the product of the degrees. Okay. And that means that the following quantity is going to be well defined for any dominant rational map, namely the first dynamical degree of a dominant rational map defined by say n plus one polynomials f not through f n will be the limit as n goes to infinity of the degree of the nth iterate taken to the one over nth power. Okay. So we have this degree sequence associated to our rational map. And all I'm doing is pulling out a properly normalized growth rate for that degree sequence. Okay. So just a comment in the literature, often the logarithm of this number is referred to as the algebraic entropy of this system. And we'll see in a moment why that might be an appropriate name. For those who are comfortable with the algebraic geometry involved here, of course, there's a higher tech perspective here, which is not about degrees of polynomials per se. But we don't have to just look at the geometry say in the surface case of what happens to curves or more generally, in a projective space of higher dimension, what happens to co-dimension one sub varieties, right? We could certainly try to understand how the map pulls back linear sub varieties of any co-dimension. Indeed, we do do that. One can construct exactly the same definition and obtain these things which are known as higher or intermediate dynamical degrees. Okay. We're not going to use these in the talk. But of course, it's important to know they exist. And I will mention them just in one brief comment about entropy a little bit later. Okay. All right. So let me make a few comments first. Okay. So because of course, we only have a failure of multiplicativity for the degrees of the iterates when we have indeterminacy, if we have a map which is actually a morphism, so has no indeterminacy on projective space, then this sequence, the degree sequence will just be of the form, say a degree of F to the N. So this D is supposed to be degree of F. Okay. And so taking the 1 over Nth power, of course, we obtained just the constant sequence in that limiting definition for dynamical degree. And so we have dynamical degree equal to D. Okay. So we can obtain integer values of dynamical degrees in this way. The example we were carrying over in the first few slides with the Fibonacci degree sequence, of course, has dynamical degree equal to the golden ratio. Okay. Okay. More generally, if instead of just taking these particular choices of polynomials, we consider a general polynomial map of the plane, then it's known due to work of Favre Mjönsson that in fact, so long as these are polynomials, so we have this sort of plain polynomial transformation, then this dynamical degree will be not only algebraic, but a quadratic integer. Okay. So this golden ratio is of course one example, but we have control over these dynamical degrees over these growth rates in this more general setting. Even though we have an upper bound of exponential growth on the degree sequence, other growth rates can occur. Okay. So here is an example to illustrate that for you. So this is a very straightforward map. It's a map which sends xyz to xz, xyz squared. And if you go ahead and write it down, which is quite easy to do, you can compute the degree of the nth iterative of this map, and it's in fact linear in n. So this is n plus one. And so of course, because this dynamical degree is measuring exponential growth, we do not have that. And so this dynamical degree is just equal to one. Okay. So this loses some information, of course, when you don't have exponential growth. All right. And then here's one of particular interest I think is quite fun, actually, and is only known in the bi-rational surface case. So if you take the projective two space and you take a bi-rational map, so a rational map with rational inverse on the projective plane, so long as it's defined by integer polynomials, say, then we can consider the reduction of the map at any prime. Right. Now, at all but finitely many primes, we will again obtain a bi-rational map. If we view this as, say, a bi-rational map of P2 for Fp bar, and we can again consider what is the dynamical degree of that reduction at any prime P. And what Shunya proved is that the limit as P goes to infinity of this dynamical degree of the reduction actually agrees with the dynamical degree. In fact, he proved a more general lower semi-continuity statement for bi-rational surface maps where the surface is over an integral scheme. Okay. And the reason I find this example particularly interesting, or this theorem particularly interesting, is that this is not a sequence which is eventually stabilizing. Okay. These these lambda of sps are actually limiting and can be strictly limiting to the, say, complex or characteristic zero dynamical degree. And so here is an interesting example that he provides of when that can happen. So this is an example with dynamical degree two as a map of, say, complex projective space. But for every prime P, the reduction has dynamical degree strictly less than two. Okay. So I found that quite interesting. All right. And I should say it's also interesting to me because I'm unaware of any generalizations, right, higher dimensional or removing the assumption of bi-rationality. Okay. All right. So why are we interested in this quantity beyond it being very sort of straightforward and natural once you encounter a polynomial dynamical system? A lot of the initial interest in this actually came from integrable systems. And without getting into what integrability is, I mean, a sort of algebraic integrability of a dynamical system would be something like preserving a vibration or, well, you can see these examples in the bi-rational surface case. So here in this chart, which I have stolen from the paper of Chef Diller and Chauffin, is a characterization of possible growth for that degree sequence in the case when we have a bi-rational map of a surface. Okay. So two dimensional invertible maps. And there's three situations where we have dynamical degree one, which is to say sub exponential growth of that sequence. And two situations where we have dynamical degree, which is greater than one, so exponential growth. And what I wanted to point out, the reason I've included this table for you, is that in a dynamical degree one case, there is in fact some preserved vibration for the map, unless okay, maybe it's dynamically trivial. But in the sense that we have an iterate, which is essentially the identity. But dynamical degree one, in the case of bi-rational surface maps, characterizes this type of algebraic integrability, which is to say that the map actually preserves some kind of vibration. Okay. And all of these things can happen. It's worth saying, it's not so easy actually, to construct this example. I already constructed for you an example with linear growth, right? This one is a bit tougher, but they can all happen. Okay. Without commenting much on what topological entropy is, it's simply a dynamical way to measure how this system separates orbits of nearby points. Okay. I just want to say that it's controlled, in a sense, by the dynamical degree, or rather, this generalization to all co-dimensions of the dynamical degree. Okay. So the dynamical degree, at least in some cases, being greater than one or not tells us something about integrability. It also controls, so gives an upper bound for the topological complexity of the dynamical system. To more perhaps of greater interest to this crowd, so the dynamical degree also controls the arithmetic complexity of orbits. Okay. So in the situation, I guess maybe I've written this so that I'm still over projective space, so of course you could just take some projective variety and all the machinery goes through. So in the situation when your map is actually defined over some number field, okay, then if you take an algebraic point inside of projective space, the set of iterates will form a collection of algebraic points, so long as the orbit is well defined, right? You don't run into the indeterminacy. Okay. And if you consider the height of those points, I've got a little plus here because I'm taking the nth root and I don't want to do that to zero, so the plus just means the maximum of the height and one. Okay. Maybe that's worth writing down instead of just saying quickly. Okay. That if you look at this height growth, which will have, which is properly normalized by taking nth roots here, that in fact this height growth is controlled, so bounded above again by this dynamical degree. Okay. So first of all, I should say it's not known in full generality that the soup is unnecessary, that in fact, so the expectation would be that this sequence, if say the orbit is the risky dense, should form a limit, should approach a limit. Okay. But we don't know that in full generality and that's why the limb soup is here. In fact, it's also conjectured in this work of Kawaguchi and Silverman that not only does this sequence, or rather, sorry, this sequence have a well-defined limit when say we have a risky dense orbit, but in fact that that well-defined limit is equal on the nose to the dynamical degree, that there's a connection between the arithmetic of orbits and this notion of algebraic complexity for the map given by the dynamical degree. Okay. So this is known in a number of cases, but not in full generality. Okay. The other connection that I particularly like, not because I know so many applications of it, but I just find it very nice, is a theorem due to Blanc and Conta, which is that again, restricting my attention to the birational surface setting, I'll say a word about why all these theorems are easier to prove there in a moment. This dynamical degree controls say a minimal degree representative in the birational conjugacy class. Okay. Which is to say that if you look at the degree of any conjugate of the map that you're working with, and you allow your your conjugacies to vary over all birational maps of P2, then this cannot be any larger than a relatively explicit polynomial in this dynamical degree. Okay. So again, there's some notion of complexity here. I suppose a more geometric notion of complexity is controlled by this algebraic dynamical degree. Okay. And maybe I'll make the comment. So, so why is this, why these assumptions? The reason, of course, is that this this group, the group of birational transformations of P2 is something that we have reasonable control over in the sense that we understand a normal set, a good set of generators, which is to say the Cremona involution, which we'll see on the next page, and all the linear transformations. That statement is wildly false, as soon as we take something of dimension greater than two. Okay. And so the control over the structure of this group is what allows all of almost all of these of these cases where it's known for birational maps of P2, almost all of those to go through. Okay. All right. So why else might we care about this dynamical degree? So it's interacting with the geometry in an important way. So first of all, the dynamical degree is a birational invariant. Okay, if you take your map and you conjugate it by something birational, it will not change the dynamical degree. It does, of course, they could, of course, change the actual degree of the map, but the dynamical degree is a birational invariant. Okay. And if you think back to what I said about that Fibonacci example, the reason we had that cancellation in the second iterate is that we had a line, say Z equals zero, which mapped to a point of indeterminacy. Okay. In the surface case, this is essentially what always goes wrong when your degree feels to be multiplicative. And so here's an example. So this Cremona involution, which if you wanted to, you could write as say one by X, one by Y, one by Z. And then you really believe me that it's an involution. This Cremona involution has indeterminacy at the intersection points of the coordinate axes. Okay. And in fact, it maps each coordinate axis to one of those indeterminate points. And so its dynamical degree does not behave well with respect to iteration, right? However, we can blow these three points up to obtain a new surface on which we no longer have just a rational map, but in fact, a regular map of morphism, right? So we can resolve the indeterminacy of the Cremona involution by blowing up these indeterminate points and obtain something which does not have any indeterminacy. As soon as we do that, we know what the dynamical degree has to do, because as soon as we fail to have any indeterminacy, then, well, maybe let me wait till I get to the next slide. But as soon as we don't have indeterminacy, this is no longer projective space, right? And so I'm not necessarily saying that the dynamical degree is an integer, although in this case it is. But as soon as we can resolve indeterminacy, we can compute the dynamical degree, okay? All right. And so as I'm sort of heavily laying on here, the general strategy then for trying to understand the growth rate of this sequence of integers is to try to bi-rationally conjugate the map to one that doesn't have any indeterminacy or whose indeterminacy you can control. All right. So the basic question then, at least from our perspective, is, well, okay, which integer sequences arise as degree sequences for rational maps of projective space? I have here a note that this is a countable set of sequences, okay? Because the set of rational maps of a given degree of projective space of a given dimension is a finite dimensional thing, right? So this is some finite dimensional quasi-projective variety. If I want to say, hey, the degree is not multiplicative at the fourth iterate, right? The degree drops at the fourth iterate. That's imposing a Zersky closed condition, okay? And you can only do that finitely many times. All right. So there's only countably many integer sequences which arise this way. We could ask a weaker question of what are their possible growth rates, right? Which real numbers arise as dynamical degrees for these kinds of maps? In fact, historically, sort of the original question and what tends to be of a bit more interest is imposing the restriction of invertibility. So if we require our map be invertible, we saw this example of the Cremona involution where we were able to resolve the indeterminacy. If we make this assumption that we have by rationality, then how do these sets change, right? So which integer sequences arise for by rational maps and which real numbers arise as dynamical degrees for them? All right. So let me give you some more interesting examples. The first one will generalize the Fibonacci example. So just a definition. So to any n by n integer matrix, if it's not singular, I can associate a rational map, which I'll call the monomial map associated to the matrix, which sends coordinates t1 up to tn to the product of those coordinates where the powers here are coming from the matrix entries. Okay, so this Toric map. This, of course, can be homogenized to a rational map. So it defines a rational map of pn to itself. I'll call it h sub a, where a is my matrix. And it has the useful property that the nth iterate of this monomial map is easily expressed in terms of the matrix A, right? So it's just the monomial map associated to the nth power of the matrix that I started with. And the dynamical degrees of these things because of this property turn out to be relatively straightforward to understand. So first, just noting here that that Fibonacci example, of course, is one of these monomial maps. Okay. This does not happen to have any negative entries, but of course, I'm allowing that just so you know. So if I have one of these monomial maps, then in fact, the dynamical degree of the map is equal to the spectral radius of the matrix that we started with. Okay, we'll see a little bit on the details of why that's the case, but I'll just say a word. Given one of these matrices, the degree of the monomial map can be written as an expression in the entries of the matrix. Okay. And of course, the entries of the matrix have say the nth power of the matrix have a growth rate, which is a big O of the spectral radius to the nth power, right? And so you can utilize that argument to show that the growth rate of the degree sequence will also be exponential with growth rate equal to the spectral radius. This is the approach that Hasselblad and Prop took. Okay. But as a corollary, in particular, the dynamical degree for any of these monomial maps will always be an algebraic integer. Okay. And so in a partial answer to this question, the sort of basic class of maps that we have to work with produce only algebraic integers for dynamical degrees. So the actual statement that Diller and Favre proved is that in fact, we still get algebraic integers if we consider any birational map in two dimensions. Okay. So if we take a projective surface and a birational map of that projective surface, then the dynamical degree is either one or the degree sequence satisfies an integer linear recurrence. And consequently, well, first of all, the dynamical degree is then certainly algebraic. But in fact, by the Hodge-Index theorem, it follows that it's either a solemn number or a piezo number. Okay. So recall my sort of shoddy picture here of solemn numbers and piezo numbers that we have a unique real conjugate amongst all the Galois conjugates of this algebraic number, which is on the real axis and outside the unit disk. In the case of solemn numbers, the requirement is then that the reciprocal is the only other conjugate lying off the unit circle and the rest lie on the unit circle. And for a piezo number, we require that all the other conjugates lie strictly inside of the unit disk. This is supposed to be even though my graphic doesn't quite show it. Okay. So the special class of algebraic numbers are precisely the ones which arise from birational maps of projective surfaces. We'll see in a moment this is in stark contrast with the rational case, which is why we tend to separate out the two possibilities. Okay. All right. So the strategy that Diller and Favve successfully implemented was to try to find a better model for the dynamical system to conjugate the map to a new birational map on a new surface, which had the property that the behavior on co-dimension one sub radius, so curves in this case, was functorial. So which is to say that the pullback on the Picard group commutes with iteration. Okay. And if you can prove this, then you can show that the degree sequence actually satisfies a linear recurrence arising from the linear operation on cohomology and that the dynamical degree is actually the spectral radius of that linear operator. Okay. And so this was viewed, I think, as evidence toward a conjecture that had been made a few years earlier in the context of studying these things from the point of view of integrable systems, which is that in fact, if you take a birational map of projective space, so they didn't quite say that way, then the degree sequence always has to be a linear recurrence sequence, so finite linear recurrence sequence. And consequently, the dynamical degree is always an algebraic number. Okay. But the thing is that this Diller-Favve strategy is valid for birational maps in the surface case, but in fact is not always possible if you remove either of those two assumptions. Okay. So if you move the assumption, remove the assumption of birationality and you're just dealing with a rational surface map, or if you increase the dimension and you allow yourself to look at birational maps in higher dimension, then this conjecture is false or this part of the conjecture is false. Okay. And this was shown in the same papers that dealt with those monomial maps, because in fact, it's false even for the most basic, most basic maps you can think of, which are these monomial maps. Okay. So there exists monomial rational maps of projective space in two dimensions, or if you want birational maps, you need to go up to at least three dimensions whose integer sequences of degrees of iterates do not satisfy a finite linear recurrence. So the work of Jason's that he spoke about a couple of years ago, so the work of Bell, Diller, and Jonson back in 2020 showed that in fact, removing the assumption of birationality and dimension two, so allowing yourselves rational maps. So if you're birational, of course, then you have to be one of these piezo or solemn numbers for your dynamical degree. But as soon as you allow yourself noninvertible maps, they show that you can obtain maps with transcendental dynamical degree. Okay. And their construction is very similar to what we ended up doing for this generalization, but just to give you a sort of brief sort of hook to hook on to their construction was related to monomial maps. So recall that it's possible to construct monomial maps whose degree sequences do not satisfy a finite linear recurrence. What Bell Diller and Jonson realized is that it was possible to twist those monomial maps to obtain dynamical degrees, which satisfy a functional equation like this or an equation of this form. Okay. That we have this degree sequence, which is not a finite linear recurrence. And the dynamical degree of a related map of a twisted map satisfies this kind of equality. Okay. And what they were able to prove is not only that they could do this construction, but then in fact, the resulting number has to be transcendental. And I'll talk a little bit more about that in just a moment. Okay. So I saw an early version of this paper and I thought it was really something spectacular. And I was interested in in the birational case because well, first of all, because of the initial motivation was really about invertible systems from the from the integral integrable systems perspective, but also because the transcendence question involved turned out to be much more intricate and general. All right. So a short time later, well, it was during COVID, it felt like a very long time later. We proved that there in fact exists a birational map of projective space, in fact, in any dimension higher than three, sorry, higher than two, with transcendental dynamical degree. So there's really this breaking point in dimension two, birational maps of projective surfaces, must be piezoarous alum numbers, birational maps of higher dimensional projective space can include ones with transcendental dynamical degree. Okay. All right. So concretely, the construction mimicked their previous work, which was to construct a map related to a monomial map, which had this property that the degree sequence of this monomial map, if you look at the monomial map associated to this particular matrix, I'll tell you a little bit about how we came up with this example, that that degree sequence doesn't satisfy any finite linear recurrence. And the map that we work with has some relationship to that degree sequence. And so we can deduce transcendence. But the relationship is more complicated than what I have up here. All right. So what does the dynamical degree of our construction satisfy? So let's take a brief, sort of very brief historical diversion here, on transcendental values of power series. Okay. So I've sort of combined three pieces of work here into a single theorem. So I recall that Mahler was interested in the case when theta was a quadratic irrational. I think Bamer was able to deal with a case where theta was an irrational, which was well-approximable. And both of them were assuming that we were just taking an identity polynomial here. And then the full generality here is due to Loxton-Randford. Okay. And the statement is this, that if you take an irrational number and an algebraic say B, base B is what you should think of here, and P a polynomial with algebraic coefficients, assuming that the series converges, then the series which evaluates the polynomial at the fractional part of n times theta, so fractional part of n times theta times B to the n here. So if you like, you can think of this as my base, although of course it's strictly less than one. So I'm in, you know, sort of the small parts here. And taking coefficients, which are the values of this polynomial at these fractional parts, then this series will evaluate to something transcendental. Okay. And I think that, yes, B not equal to zero. Thank you. Very good point. And I think that the somehow most believable description of this and one that's also related to our own approach is probably the setting of theta being well-approximable. Okay. If theta is well-approximable, so we have rational numbers, which are very good approximations of theta, then those rational numbers here, if we approximate theta by those rational numbers, then what we will get is a series here, which is very, very close to the series that we're interested in for a large number of n. So the partial sums will be very close for a long time. And being a little vague here, what you want to take is, you know, if say theta is approximately equal to say P over Q, what you'd want to look at is, overloading P a little bit here, my apologies, n mod Q theta. So this series will provide you with an algebraic approximation to your power series, which is good enough that, well, in Bramer's case, a Leaville type argument was enough to produce transcendence. Okay. In general, of course, you need some stronger statements about the failure of well-approximability for algebraic numbers by algebraic numbers, which we'll get into the details of in a moment. All right. So what Bell Diller and Jonson showed is that this theorem also holds not just for polynomial values of P, but they were interested in a particular choice of continuous piecewise linear function in, you know, this value modulo one. Okay, sorry, I wrote it on the circle, but equivalently, this value modulo one. And this piecewise function really arises from this degree computation I mentioned earlier that the degree of a monomial map can be written in terms of the entries of the matrix. Here it is. Okay. And so dealing with this function, they were in the two-dimensional case. You can really write down very explicitly the particular, going back a moment here, the particular power series here, which the dynamical degree that they constructed satisfies. And so the transcendence theorem they approved then argues that this value, if lambda f is algebraic, is necessarily transcendental. Since one is not transcendental, you obtain a contradiction to the algebraic city of lambda. Okay. All right. And again, sort of similarly to Bamer here, they were using the convergence of theta to provide these high-quality algebraic approximations. Okay. So what is the transcendence statement in a similar vein that we approved? So let me just sort of digest it for a moment here out loud. So again, we take some irrational number, and we are interested in a function of the fractional part of multiples of that number. Okay. So those are the coefficients of our power series. But now, rather than working in one dimension, we are going to take a vector of algebraic numbers. Okay. And so this is replacing our B in the previous page. And replacing the P of the previous page will be some piecewise constant function with finite discontinuity set. Okay. So it takes finitely many constant values, say, modulo one. Okay. And there's a technical assumption here that can probably be removed. Let me say this is about the interaction of theta with the discontinuity set of gamma. Okay. As I say, I believe this can probably be removed. Maybe let me take a moment before I deal with the question in the chat. And if you know that your function has the property and your algebraic numbers have the property that this map is real valued. So I'm allowing general algebraic numbers here, so it wouldn't necessarily be, but I'm going to force it to be real valued, non-constant and maximized by, say, J theta. Okay. This is for sufficiently large values of J, of course. Then the sum here of this is just the vector product of this piecewise function evaluated at the fractional part against this vector of powers of the rho i is actually going to produce a transcendental number. Okay. I see there's a question in the chat. Philip, do you mind if I take a quick look? Yes. Yeah, go ahead. Yeah. So it's maybe honest, maybe just don't mute and ask the question. Yeah. I was wondering if it's possible to estimate the gap between the transcendental number and the closest algebraic number. I would be curious how that... So the answer is sort of yes. It's yes up to the ability to estimate how well approximable theta is, right? So you can get a bound, but it's going to depend on, so if you have small... Sorry, what am I trying to say here? It's going to depend on the continued fraction expansion for theta, I think, but now that I'm saying it out loud, probably you could get rid of that dependence because having unbounded continued fraction expansion is actually the good case, is the easy case somehow, and having bounded continued fraction expansion is the tough case. So actually probably the answer is just generally yes. Yeah. Is that enough of an answer? Is it possible to construct that or from what I see, this cannot be... It's not a constructable estimate. So the sense in which it's a constructive estimate is that the mechanism of the proof is to take convergence of theta and to produce analogs of this object that I mentioned on the previous page. So looking at instead these periodic things, which of course evaluate to algebraic numbers and controlling the quality of the estimate between those two things. Now that comes at the end of the day, down to finding how far out these sequences actually agree with each other, because our gamma, because we're not dealing with a polynomial, we're dealing with a piecewise constant function. And so if I take something which approximates theta very well, then the fractional part of n theta and the fractional part of that approximation times n will evaluate to the same thing under gamma. So in fact, it's not just in this case, because we're dealing with piecewise constant functions, it's not just that the values are close to the approximating series, which takes an algebraic value, but in fact, it's on the nose in agreement up to some n for the partial sums of the two series. And finding that n where they start to diverge, yes, I think that we do in the course of the proof control the size of the tail of that difference. Yeah. Okay, thank you. Yeah. All right. So I think I have said enough about this, I mean, right, so why is this so much more difficult than the previous slide? Well, the very obvious issue that could arise is cancellation, right? I mean, I'm taking a sum of three things here. Okay. And of course, just because you know that one of those sums evaluates or each of those sums evaluates to a transcendental number doesn't mean they don't cancel each other out into something algebraic. Okay. And so some kind of assumption here is certainly necessary. Really, the key one is this maximization property. Okay, well, and then I guess by default also the real value so that maximization is meaningful. Okay. And just another comment that in fact, this argument is simpler when we have unbounded continued fraction expansion when theta is actually well approximable, just as it is sort of historically the case with these transcendence of power series statements. Okay. All right. So the key tool of course, some deep theorem in defunding approximation, the one that is particularly well suited for us is ever to form this bound on sum of S, well S integers, but it's particularly powerful in the case of S units. Okay. So this is a consequence as he showed of the so-called Piatic subspace theorem. And the statement is here, what I want to point out, you can read it if you happen to not be familiar. So the statement is about approximability of these sort of non-degenerate sums of S integers in terms of the various input data. Okay. The important thing of course is that all that matters is the length of the sum once you've fixed your base field in your set of primes that you're integral outside of. The useful thing in our setting, because we were dealing again with a piecewise constant function taking only finitely many values, those coefficients in our power series are then S integers for some choice of primes S. In fact, they're S units for some choice of primes S. And in the case when you're dealing with S units, this statement will lose this term. So by the product formula, this term will evaluate to one. And so the lower bound that you get on sums of S units, and you should think here that we're applying this to partial sums in the power series from the previous slide, will give you a lower bound assuming you can prove non-degeneracy, which is the tough part, will give you a lower bound, which is what we use to prove the transcendence, which conflicts with the good approximation from the previous slide. Okay. All right. So where does this fit into the dynamical degree story? So it fits into the dynamical degree story with that construction that I mentioned of twisting a monomial map. So in fact, what we show, well, in fact, I've only written down the three dimensional case, but there's higher, it works in any dimension with proper assumptions, that if you take an invertible three by three integral matrix, and it satisfies some say geometric assumptions, which is to say that it has strictly leading complex conjugate eigenvalues, no power of which land inside the reels. Okay. This is precisely how you construct a monomial map, which doesn't have degree sequence, which is a finite linear recurrence. Okay. Then we can cook up a birational involution of P three, so that, okay, maybe not just twisting the original matrix monomial map, but some power of that original matrix, matrix or some iterate of that original monomial map, will satisfy that the twist here has dynamical degree, which is the unique positive solution to a power series equation of this form. And my function here is in fact precisely the type of function that we dealt with on the previous page. Okay. And so the transcendence theorem of the previous page tells us that the value on the left hand side here is necessarily transcendental whenever x is algebraic. But of course, then my dynamical degree cannot be algebraic because the value it takes, the series takes as algebraic. Okay. All right. So just to comment here on where this all comes from. So what is the map doing geometrically? Well, as I said, we have this monomial map with this sort of irregular degree sequence, and we're composing it with an involution. Okay. The point is that this involution is very carefully chosen so that the indeterminacy of this map actually provides a recursive relation between the quantities that we're interested in, this function which I understand, for lack of a better way to say that, okay, which takes the form of one of those functions in my power series from the previous slide. And the degrees of the monomial maps, which I also understand. Okay. Here is the relation. By the way, this is a picture. Sorry, I should have said this is a picture of the indeterminacy of this map is that you would expect it to be some kind of general line. But in fact, it's very carefully chosen so that the lines that lie in the indeterminacy locus here are intersections of particular planes inside of p3. Okay. And that's enough to get you a relation of the following form, which notice is almost is basically an infinite linear recurrence sequence in the degree of the nth iterate of the map. Okay, almost not quite. But this is sufficient to then yield the power series formula that we use to prove the transcendence of the dynamical degree. All right, so just some final questions. We're dealing with these birational maps in p3, of course. Is it possible to construct a map which has transcendental dynamical degree whose inverse also has transcendental dynamical degree? If you recall, the monomial map constraint, the monomial map we sort of built these things out of, had to have complex conjugate eigenvalues, leading complex conjugate eigenvalues with no power that landed in the reals. Okay, the very silly reason why you cannot do this for birational maps of p2 is that you cannot put two of those on the unit circle in degree 2 and obtain, what am I trying to say here, that a matrix with complex conjugate eigenvalues that satisfy this will never be invertible. That's what I'm trying to say. Okay, if it's two-dimensional, you have to go into three dimensions at least. And so I suspect you can do this easily with our construction in even dimensions, or maybe even just dimension four or higher. But it's not obvious to me whether or not this can happen in dimension three. Okay, certainly our construction will not suffice to do this in dimension three. The second question is, in the case when you do have transcendental dynamical degree, do we still hope that that conjecture of Kawaguchi and Silverman is true? Is height growth along orbits still controlled by that dynamical degree, even though it's transcendental? And somewhat surprisingly, so Matsuzawa recently has shown that not in our birational p3 case, but in the rational p2 case, the original case of Beldiller and Jonson, that yes, the maps with transcendental dynamical degree, which are bi-rational maps of, sorry, rational maps of p2, do admit orbits whose arithmetic degrees have transcendental growth, okay, which agrees then with that transcendental dynamical degree, which is a bit surprising. In fact, Kawaguchi and Silverman had initially made two conjectures, sort of as sub-conjectures of the same conjecture. One was that the arithmetic degree is always an algebraic integer, and the other is that the arithmetic degree always agrees with the dynamical degree. And so now we know that we knew from the transcendental dynamical degree construction that one of those parts was false. Matsuzawa told us that, in fact, it's the algebraicity. Okay. The third point here is to look for finer information about the collection of dynamical degrees. This is understood somewhat for bi-rational maps of p2. Okay, so first of all, that there's a lower bound with the property that all bi-rational maps of p2, if they're not equal to one, are greater than or equal to this lower bound. L is lemur, by the way. So this is related to some interesting number theoretic questions. Nothing is known outside of the p2 setting there. Okay, essentially the same question here. And then finally, from a dynamical perspective, we use this degree to construct invariant dynamical tools like invariant currents and variant measures, things that help us study these objects. And just recently, Diller and Reuter have made some progress on those constructions in the case when the dynamical degree is transcendental, which is somewhat surprising. Okay. And I think with that, maybe I will thank you for your attention.