Final remarks

The Picard theorem follows from Bloch's theorem in one variable. It is then interesting to ask whether the same is possible in the case of $\mathbb {T}$-holomorphic mappings in $\mathbb {C}^{2}$. However, here we can directly find a Picard theorem without invoking our Bloch theorem.

Theorem 10 (Picard)   Let $f\in TH(\mathbb {C}^{2})$. If there are two bicomplex numbers $\alpha,\beta$ such that $\alpha-\beta$ is invertible and for which the set

$$\{w\in\mathbb {C}_{2}:\mbox{ }w-\alpha\mbox{ is noninvertible... ...\{w\in\mathbb {C}_{2}:\mbox{ }w-\beta\mbox{ is noninvertible}\}$$

is not in the range of $f$, then $f$ is constant.

Proof. We have just to apply the so-called ``little Picard theorem" [15] to $f_{e1}$ and $f_{e2}$. The fact that $\alpha-\beta$ is invertible will insure us that $\alpha-\beta$ will be equal to a bicomplex number $s_{1}e_{1}+s_{2}e_{2}$ with $s_{1}$ and $s_{2}$ nonzero. Let $\alpha=\alpha_1e_1+\alpha_2e_2$, $\beta=\beta_1e_1+\beta_2e_2$. Suppose $f_{e1}$ takes the value $\alpha_1$ at $a_1$. There exist $z_1,z_2\mbox{ }\in\mathbb {C}_{1}$ such that $z_1-z_2i_1=a_1$. Thus,

$$f(z_1+z_2i_2)\in\{w\in\mathbb {C}_{2}:\mbox{ }w-\alpha\mbox{ is noninvertible}\}.$$

Contradiction. Hence, $f_{e1}$ omits $\alpha_1$. Similarly, $f_{e2}$ omits $\alpha_2$, $f_{e1}$ omits $\beta_1$ and $f_{e2}$ omits $\beta_2$. Since $\alpha_1-\beta_1=s_1\neq 0$, $\alpha_1$ and $\beta_1$ are distinct. Similarly $\alpha_2\neq\beta_2$.$\Box$

In the same way, it is possible to find also a Casorati-Weierstrass theorem:

Theorem 11 (Casorati-Weierstrass)   Let $f\in TH(\mathbb {C}^{2})$ with $f^\prime(w)$ not identically noninvertible. Then, $f(\mathbb {C}^{2})$ is dense in $\mathbb {C}^{2}$.

Proof. The hypotheses imply that we can write $f(z_{1}+z_{2}{i}_{2})=f_{e1}(z_{1}-z_{2}{i}_{1})e_{1}+f_{e2}(z_{1}+z_{2}{i}_{1})e_{2}$ with $f_{e1},f_{e2}\in H(\mathbb {C}_{1})$ and nonconstant. Then we can apply the Casorati-Weierstrass theorem for $\mathbb {C}_{1}$ to $f_{e1}$ and $f_{e2}$ in order to prove that $f(\mathbb {C}_{2})$ is dense in $\mathbb {C}_{2}$.$\Box$

A famous example of Fatou and Bieberbach (see [10]) shows that the usual formulation of the Picard theorem in $\mathbb {C}$ does not extend to holomorphic mappings in $\mathbb {C}^2$.

In this connection, we have some interesting consequences of Theorem 11 which can be interpreted as an other kind of little Picard theorem for bicomplex numbers:

Corollary 1   There is no nondegenerate $\mathbb {T}$-holomorphic mapping

$$f:\mathbb {C}^{2}\longrightarrow\mathbb {C}^{2}$$

such that $\mathbb {C}^{2}\backslash f(\mathbb {C}^{2})$ contains a ball.

Corollary 2   Fatou-Bieberbach examples cannot be $\mathbb {T}$-holomorphic mappings, i.e. they connot satisfy the complexified Cauchy-Riemann equations.

For a beautiful formulation of Picard's theorem which holds in higher dimensions, see [9]. Also, for a version of Picard's theorem for quasiregular mappings see Rickman [12].