Images of compact subsets in the plane

Question

Let $K$ be an infinite compact subset of $\mathbb{C}$. Is it true that there exists a sequence $(f_n)_{n>0}$ of functions holomorphic in some neighborhood of $K$, such that the images $f_n(K)$ are pairwise non-homeomorphic? (Motivation for this question comes from: consider an element in a unital Banach algebra; how does the topological type of its spectrum change under holomorphic functional calculus?)

Answer 1

Here are a few thoughts, not really an answer.

I think you want to loosen the definition up a little bit, or maybe you don't. At any rate if you consider $K=\{0,1,1/2,1/3,\dots\}$ and fix an open neighborhood $U$ of $K$ which has $n$ connected components which meet $K$ non-trivially then if $g: U \rightarrow \mathbb C$ is holomorphic $g(K)$ is either finite of cardinality less than or equal to $n$ or infinite with one limit point. Let $U_0$ be the connected component of $U$ containing $0$ it follows that all but finitely many points of $K$ are contained in $U_0$. In particular $K_0=U_0 \cap K$ has a limit point in $U_0$ so if $g$ takes finitely many values on $U_0$ it is constant, hence $g(K_0)$ is either a single point or an infinite set with one limit point.

I don't really think this example is interesting to you, so you're probably better off considering a sequence of functions $f_n$ such that each is holomorphic in a neighborhood of $K$ (and maybe that's what you originally meant).

We can reduce to the case that $K$ is a perfect set fairly easily as well. Write $K=K^\prime \sqcup I$ with $I$ the isolated points of $K$. If $|I| < \infty$ then note that if $V \subset \mathbb C$ is compact and $x \in \mathbb C$ then either $x \in V$ or $d(x,V)>0$ so we see that if we can't find infinitely many up-to-homeomorphism distinct images of $K^\prime$ then $I$ won't help. In the case that $I$ is infinite we can find $f_n$ such that $|f_n(K)|=n$, by taking suitable neighborhoods.

That's about as far as I can make it, I was trying to figure out the answer to the question for a simple set such as $K=[0,1]$. But I don't have a good repertoire of poorly behaving holomorphic functions so I couldn't get anything more than a loop or a line segment. Oddly enough I think I can do it for the Cantor set, it should be similar to the case of infinitely many isolated points.

As a final though if $K$ is actually the spectrum of an element of a unital Banach algebra, then I imagine there are some restrictions we could place on $K$ to make the task easier.