I have been reading Joseph Silverman’s new book on arithmetic dynamics lately. There’s a lot of really fascinating stuff in there, including a large number of potential research problems that are currently way beyond me, but I’ll continue thinking about them! Most interesting so far is the Uniform Boundedness Conjecture:

Let $d\ge 2$, $N\ge 1$, and $D\ge 1$ be integers. Then there exists a constant $C=C(d,N,D)$ such that for any number field $K$ with $D \ge [ K : \mathbb{Q} ]$ and any morphism $\phi:\mathbb{P}^N(K)\to\mathbb{P}^N(K)$ of degree $d$, the number of preperiodic points of $\phi$ in $\mathbb{P}^N(K)$ is at most $C$.

Not much is known about this conjectures; even the case $d=2$, $N=1$, and $D=1$ is open. It’s even open if we restrict to morphisms of the form $\phi_c(z)=z^2+c$. Bjorn Poonen has shown, however, that these maps have no rational periodic points of exact period 4 or 5; it is conjectured that they have no rational periodic points of exact period greater than 3.

However, there is a positive result of the above type that doesn’t depend that much on some of the above quantities:

Let $K$ be a number field and $\phi:\mathbb{P}^1\to\mathbb{P}^1$ be a rational map over $K$. Let $\mathfrak{p}$ and $\mathfrak{q}$ be prime ideals of $\mathfrak{o}_K$ so that $\phi$ has good reduction at $\mathfrak{p}$ and $\mathfrak{q}$ (meaning that when we reduce $\phi$ modulo $\mathfrak{p}$ and $\mathfrak{q}$, we end up with a map $\tilde\phi$ of the same degree as $\phi$) and whose residue characteristics are distinct. Then the period $n$ of any periodic point of $\phi$ in $\mathbb{P}^1(K)$ satisfies $n\le (N\mathfrak{p}^2-1)(N\mathfrak{q}^2-1)$, where $N$ denotes the (absolute) norm.

(See, for instance, my algebraic number theory notes for definitions of some of these terms.)

Anyway, that wasn’t really the point of this post, as you may have guessed from the title. I meant to talk about theorems that pretend not to be related to dynamical systems but actually are. First we need to discuss height functions a bit; there’s a lot more about them in Silverman’s book and in my elliptic curve notes.

We let $K$ be a number field and $M_K$ the set of standard absolute values on $K$ (These are the absolute values on $K$ whose restriction to $\mathbb{Q}$ is either the standard absolute value or one of the $p$-adic absolute values.) We write $n_v=[K_v:\mathbb{Q}_v]$ (where $F_v$ denotes the completion of $F$ with respect to the absolute value $v$). Suppose $P\in\mathbb{P}^N(K)$; we can then write $P=[x_0,\ldots,x_N]$ for some $x_0,\ldots,x_N\in K$. We then define the height of $P$ with respect to $K$ to be $H_K(P)=\prod_{v\in M_K} \max\{|x_0|_v,\ldots,|x_N|_v\}^{n_v}$. One can check that this is well-defined, and that if $L/K$ is a finite extension of number fields and $P\in K$, then $H_L(P)=H_K(P)^{[L:K]}$. Hence it is possible to define the absolute height of $P$ by $H(P)=H_K(P)^{1/[K:\mathbb{Q}]}$.

One of the key facts about heights is the following: If $B$ and $D$ are constants, then $\{P\in\mathbb{P}^N(\overline{\mathbb{Q}}):H(P)\le B \text{ and } [\mathbb{Q}(P):\mathbb{Q}]\le D\}$ is finite. A corollary is the following well-known result of Kronecker:

Let $\alpha\in\overline{\mathbb{Q}}$ be nonzero. Then $H(\alpha)=1$ if and only if $\alpha$ is a root of unity.

Proof: If $\alpha$ is a root of unity, then $H(\alpha)=1$ is clear. Now suppose that $H(\alpha)=1$. For any $\beta$ and $n$, we have $H(\beta^n)=H(\beta)^n$, so $H(\alpha^n)=H(\alpha)^n=1$, so $\{\alpha,\alpha^2,\alpha^3,\ldots\}$ is a set of bounded height and is therefore finite. Therefore there are integers $i>j>0$ such that $\alpha^i=\alpha^j$, so $\alpha$ is a root of unity.

And now for linear algebra. Sheldon Axler has a well-known book on linear algebra without determinants. He therefore uses dynamical systems to show the following familiar result:

Theorem: Every operator on a finite-dimensional nonzero complex vector space has an eigenvalue.

Proof: Let $V$ be such a vector space of dimension $n$. Let $T$ be an operator on $V$, and let $v\in V$ be nonzero. Then $\{v,Tv,T^2v,\ldots,T^nv\}$ cannot be a linearly independent set. Hence there exist $a_0,\ldots,a_n\in\mathbb{C}$, not all zero, so that $a_0v+a_1Tv+\cdots+a_nT^nv=0$. Suppose $m$ is maximal with respect to $a_m\neq 0$. Then $a_0v+a_1Tv+\cdots+a_mT^mv=0$. Since we’re working over $\mathbb{C}$, the polynomial $a_0+a_1z+\cdots+a_mz^m$ factors as $a_m(z-\lambda_1)\cdots(z-\lambda_m)$. We then have $0=a_0v+a_1Tv+\cdots+a_mT^mv=a_m(T-\lambda_1I)\cdots(T-\lambda_mI)v$. Hence some $T-\lambda_jI$ is not injective. This $\lambda_j$ is an eigenvalue for $T$.