Two topological properties in Renormalization Theory

Question

I am using McMullen-Complex Dynamics and Renormalization (p. 99). There you can find the definitions of renormalization and polynomial-like mapping.

Assume $f(z) := z^2 + c$, where $c\in\mathbb{C}$ is fixed. By hypothesis I know that $f^n:U\to V$ is a renormalization. Defining $V(i) := f^i(U)$, it can be written \begin{equation} U\overset{f}{\longrightarrow}V(1) \overset{f}{\longrightarrow} V(2) \overset{f}{\longrightarrow} \dots \overset{f}{\longrightarrow} V(n) = V, \end{equation}

where the first map $U\to V(1)$ has degree 2 (because the critical point $0$ is in $U$) and the remaining maps are univalent (because $f^n$ and the first map of the composition have degree $2$).

1. Here is the first doubt: it is also stated that the first map $U\to V(1)$ has to be proper (this means that the preimage of a compact set is compact). Is it trivial?
2. Moreover, on the next property stated in the book (p. 99), it refers to the connected component of $f^{i-n}(U)$ contained in $V(i)$. My question: Why do we know that $f^{i-n}(U)\cap V(i)\neq \emptyset$? And, once proved that $f^{i-n}(U)\cap V(i)\neq \emptyset$, why is this intersection a connected component of $f^{i-n}(U)$?

I have tried to imagine the sets $V(i)$ or at least to relate them. If I am not wrong, although $U\subset V$, $U\cap V(1)\neq U$ because if the critical point $0\in U$ is in some $V(i)\;(1\leq i < n)$ then the corresponding iteration should have degree $2$ and this is not possible as I argued above.

I hope someone can help me. Thank you.

Answer 1

1. By definition of renormalization, the map $f^n:U \to V$ is proper, so $f:U \to V(1)$ has to be proper, too. (If not, $f$ maps some boundary point of $U$ into $V(1)$, and then $f^n$ maps that same boundary point into $V$, so it would not be proper.)
2. $f^{n-i}$ maps $V(i)$ univalently onto $V$, and $U$ is compactly contained in $V$, so it maps some domain $U(i)$ compactly contained in $V(i)$ univalently onto $U$. This shows that $U(i)$ is the unique connected component of $f^{i-n}(U)$ which intersects (and is contained in) $V(i)$. (Here I am assuming that $1 \le i \le n$.)