Applications of the Small and Great Theorems of Picard

Question

I just presented the two famous theorems of Picard (Small and Great) in a graduate course, but I have not managed to discover a good number of interesting applications.

List of applications (rather straight-forward though) found so far:

1. If a meromorphic function on $\mathbb C$ misses three values, then it is constant.

2. The equation $f^n+g^n=1$ has NO non-trivial meromorphic in $\mathbb C$ solutions if $n\ge 3$.

3. If $f$ is entire an 1-1, then it is linear.

4. If $f,g$ are entire and $g'=f(g)$, then $f$ is linear or $g$ is constant.

Could you provide any interesting applications of these theorem?

I have asked this question in Mathematics StackExchange, but I only received one response.

Answer 1

One of the reasons Picard's theorems are so famous is that they have many generalizations, and these generalizations have more applications. Let me mention 2 of the most useful generalizations, both good for a graduate course:

1. Montel's theorem. The set of meromorphic functions omitting 3 values is a normal family. (There is a short and neat derivation of this from Picard's theorem via the so-called Zalcman Lemma (also known as Brody's lemma). This derives Montel from the STSATEMENT of Picard's theorem, no matter how you prove Picard's theorem itself.). Montel theorem in turn has many applications, the most famous one is to iterations of holomorphic functions. The Chapter in Montel's own book on this gives some basic results of this theory. If you prefer a more modern source, Thurston, Combinatorics of rational maps, in the book MR1500163 Complex dynamics. Families and friends. Edited by Dierk Schleicher. A K Peters, Ltd., Wellesley, MA, 2009. The first part of this paper gives an excellent introduction to Montel's theorem and the subject of holomorphic dynamics.

2. Generalization of your item 2 says: if $F(x,y)=0$ is an algebraic curve of genus $>1$ and $f,g$ are two meromorphic functions such that $F(f,g)=0$, then $f$ and $g$ are constant. This is due to Picard himself, but does not have a standard name like "the third Picard's theorem".

For example, the irreducible differential equation $F(w',w)=0$, where $F$ is a polynomial can have meromorphic solutions only when $F(x,y)=0$ is of genus $0$ or $1$, and all solutions are either rational of trigonometric or elliptic functions (Weierstrass theorem).

Answer 2

In general, holomorphic maps $f: \mathbb{C} \rightarrow \mathbb{C}$ have no fixed points; but using Picard theorem we can show that $f\circ f$ always have fixed point.

Theorem (Fixed-point theorem) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be holomorphic. Then $f \circ f: \mathbb{C} \rightarrow \mathbb{C}$ always has a fixed point unless $f$ is a translation $z \mapsto z+b, b \neq 0 .$

Proof:

Suppose $f \circ f$ has no fixed points. Then $f$ also has no fixed points, and it follows that $$ g(z):=\frac{f(f(z))-z}{f(z)-z} $$ is entire function. This function omits the values 0 and 1 ; hence, by Picard, there exists a $c \in \mathbb{C} \backslash\{0,1\}$ with $$ f(f(z))-z=c(f(z)-z), \quad z \in \mathbb{C} $$

Differentiation gives $f^{\prime}(z)\left[f^{\prime}(f(z))-c\right]=1-c$. Since $c \neq 1, f^{\prime}$ has no zeros and $f^{\prime}(f(z))$ is never equal to $c$. Thus $f^{\prime} \circ f$ omits the values 0 and $c \neq 0$ by Picard, $f^{\prime} \circ f$ is therefore constant. It follows that $f^{\prime}=$ constant, hence that $f(z)=a z+b$. Since $f$ has no fixed points, $a=1$ and $b \neq 0$