Hartog's Theorem and Entire Functions

Question

I'm interested in multivariable Complex Analysis, and I have two questions:

My first question is as follows: after reading about Hartog's Extension Theorem I started wondering about the following problem - assume $B_r (0)$ is an open ball of a positive radius $r>0$ in $\mathbb{C}^n$, and let $f:B_r(0)\to\mathbb{C}$ be holomorphic in some neighborhood of the closed ball. Can $f$ be extended to an entire function in $\mathbb{C}^n$?

On the one hand, my reasoning says that yes, it is possible - after all, we can "invert" the ball and observe $f(\frac1z), z\in (B_r(0)^c)$ the complement of the ball, with $\frac1z$ denoting a holomorphic rotation of the sphere $\widehat{\mathbb{C}^n}$ interchanging $0,\infty$. This would make $f(\frac1z)$ holomorphic in the complement of some compact set of $\mathbb{C}^n$, thus by Hartog's Theorem it can be extended to $\mathbb{C}^n$, hence so does $f$.

Edit - by $\frac1z$ I mean the mapping $z=(v,w)\rightarrow{(\bar{v}/||v||^2,\bar{w}/||w||^2)}$ in, say, $\mathbb{C}^2$.It seems to me that it should be holomorphic as it is holomorphic in every component separately. Am I missing something about it?

On the other hand, I also know that any convex set is a domain of holomorphy. Therefore, given a holomorphic function defined on $B_r (0)$ that is not extendable to $\mathbb{C}^n$, I could always choose $r_1<r$. $f$ would of course be holomorphic on some neighborhood of $B_{r_1}(0)$, and by previous argument could be extended uniquely and holomorphically to $\mathbb{C}^n$, which is impossible as $B_r (0)$ is a domain of holomorphy.

Am I missing something here? I mean, I probably am, but I can't see what exactly.

My second question is, is there a theory for the dynamics of multivariable holomorphic functions? In particular, do the nice properties of the Julia and Fatou sets carry over to several complex variables? I haven't found much about it, and I'd like to read more...

Thanks :)

Answer 1

Your "inversion" of the mapping is not holomorphic. There is no biholomorphic map between a punctured ball and the complement of the ball. It is precisely Hartogs's theorem that says that there is no analogue of $\frac{1}{z}$ in several variables, and your procedure is the essentially one way to prove that no such mapping exists.

Another way to think about it is, if you had any such analogue it would have to have some isolated singularity, and holomorphic functions of several variables do not have any isolated singularities (again Hartogs's theorem).

As for your second question, there is some work on dynamics in several variables, but it is far more complicated. It is also a much younger field.

Answer 2

To answer your second question: there is indeed a theory of the dynamics of holomorphic maps in several complex variables, but some aspects are fundamentally different from one dimensional complex dynamics. Very broadly speaking, what works well in higher dimension is the ergodic theory part, and everything based on complex analysis (notably Montel's theorem) tends to fail in higher dimension. There is still a notion of Fatou and Julia sets, except that there are now several different Julia sets, forming a stratification. The smallest of those Julia sets is the most dynamically interesting.

You can read e.g. this text by Dinh and Sibony: https://arxiv.org/abs/0810.0811 to get a thorough but reasonnably self-contained introduction to the field.