Holomorphic maps on complex tori

Classical complex analysis tells us that every non-constant holomorphic map from the Riemann sphere to itself is a rational map $p(z)/q(z)$ for some coprime polynomials $p$ and $q$. Let’s increase the genus and replace the Riemann sphere with a complex torus. It turns out that the space of holomorphic maps from a complex torus to itself is much simpler that what we can imagine.

Let’s recall that a complex torus $\mathbb{T}$ is the quotient of the complex plane $\mathbb{C}$ by a lattice $\Lambda = a\mathbb{Z} + b\mathbb{Z}$ generated by periods $a,b \in \mathbb{C} \backslash {0}$ such that $a/b \not\in \mathbb{R}$. Complex tori are the only Riemann surfaces which are both compact and parabolic. (Have a peek at my post on classification of Riemann surfaces.)

Every holomorphic map $f: \mathbb{T} \to \mathbb{T}$ of a complex torus $\mathbb{T}$ to itself comes from its lift $\tilde{f} :\mathbb{C} \to \mathbb{C}$, an entire function which respects the lattice, i.e. for all $z \in \mathbb{C}$,

Basic covering space theory tells us that when $f$ is non-constant, it must be a branched covering of some finite degree $d$. Structure of the map $f$ is completely known.

Theorem: Every non-constant holomorphic self map $f: \mathbb{T} \to \mathbb{T}$ of a complex torus $\mathbb{T}$ is

• an affine map of the form $f(z)=\alpha z + c (\text{mod }\Lambda)$,
• a covering map of degree $\vert \alpha \vert^2$.

As $\Lambda$ is discrete, there must be unique constants $\alpha$ and $\beta$ in $\Lambda$ such that $\tilde{f}(z+a) - \tilde{f}(z) \equiv \alpha$ and $\tilde{f}(z+b) - \tilde{f}(z) \equiv \beta$. Let’s look at the function $\tilde{g}(z):=\tilde{f}(z)-\alpha z$. It is surely $a$-periodic and for all $z \in \mathbb{C}$,

Therefore, $\tilde{g}$ induces a holomorphic map $g: \mathbb{T} \to \mathbb{C}/(\beta - \alpha b)\mathbb{Z}$. Regardless of the value of $\beta - \alpha b$, the quotient $\mathbb{C}/(\beta - \alpha b)\mathbb{Z}$ must be a non-compact Riemann surface. Any holomorphic map from a compact Riemann surface to a non-compact Riemann surface must attain a “maximum” and by the maximum principle, this map must be a constant map. Therefore, $g(z) \equiv c$ for some constant $c \in \mathbb{C}$ and it turns out that $\tilde{f}$ is the affine map $\alpha z + c$.

The affine map $\alpha z+c$ expands length and area by a factor of $\vert \alpha \vert$ and $\vert \alpha \vert^2$ respectively. Therefore, the degree of $f$ is equal to $\vert \alpha \vert^2$. This gives rise to a restriction for the possible values of $\vert \alpha\vert$. What else can we say about $\alpha$?

• When $f$ is non-constant, $\vert \alpha \vert \geq 1$.
• When $f$ is not a translation, i.e. $\alpha \neq 1$, we can also solve the equation $\alpha z + c = z (\text{mod } \Lambda)$ and find that we must have $\vert \alpha - 1\vert^2$ fixed points uniformly distributed throughout $\mathbb{T}$. In particular, $\vert \alpha - 1\vert^2$ must be a positive integer.
• When $f$ is a conformal automorphism of $\mathbb{T}$, then $\vert \alpha \vert = 1$ and so $f$ must be a composition of the rotation $\alpha z$ which fixes the lattice and a translation.
• When $f$ is not a conformal automorphism, i.e. $\vert \alpha\vert>1$, then the $n^{\text{th}}$ iterate $f^n$ is an affine map of the form $\alpha^n z + \gamma$ and it has $\vert \alpha^n -1\vert^2$ fixed points ($n$-periodic points of $f$) uniformly distributed throughout $\mathbb{T}$. In particular, $\vert \alpha^n - 1\vert^2$ must be a positive integer.

The dynamics of these holomorphic maps are simple. The last point above highlights that when $\vert \alpha \vert > 1$, every periodic point must be repelling (since the multiplier is $\alpha^n$ for some positive integer $n$) and the set $S$ of all periodic points must be dense throughout the complex torus. Recall that the closure of $S$ must coincide with the Julia set $J(f)$. (See my post on Julia sets.) This yields the following result.

Proposition: Let $f(z) = \alpha z + c$ be a holomorphic self map of a complex torus $\mathbb{T}$. When $\vert \alpha \vert \leq 1$, the Julia set of $f$ is empty. When $\vert \alpha \vert > 1$, the Julia set of $f$ is the whole torus $\mathbb{T}$.

References

1: J. Milnor. Dynamics in one complex variable. Princeton University Press, third edition, 2006.