This question is a particular take on the following theme. Suppose $A$ and $B$ are two notions of "large subset of $\omega^\omega$;" when is there a uniform method for turning an element of $A$ into an element of $B$?

We work in ZF (although results under strengthenings of ZF are also interesting).

For $U, V\subseteq\mathcal{P}(\omega^\omega)$, say $U$ *spreads onto* $V$ if there is some $F:\omega^\omega\rightarrow\omega^\omega$ such that $F(X)\in V$ for every $X\in U$, where $F(X)=\{F(x): x\in X\}$.

*The original motivation for this was in the context of working with cardinal characteristics of the continuum without choice - about which I have asked some questions before. This also was a natural, and hopefully less trivial, outgrowth of this question.*

*However, especially in light of question 2 below, it is unclear whether there's actually any relation. So now it's just a pure curiosity question.*

Here's a trivial example. If $A=\{$dominating families$\}$ and $B=\{$escaping families$\}$, then $A$ spreads onto $B$ via the identity map.

Here's a less trivial - maybe even interesting! :P - example. Let $C=\{$pointed perfect sets$\}$ (where a pointed perfect set is a nonempty closed subset $K$ of $\omega^\omega$ with no isolated points, such that every element of $K$ computes the tree representation of $K$ - that is, the set of strings $\sigma\in\omega^{<\omega}$ such that $\exists f\in K(\sigma\prec f)$). Then $C$ spreads onto $A$ (where $A$ is the set of dominating families as above). This argument hinges on two observations:

There is a definable way to assign to each perfect $K\subseteq\omega^\omega$ a surjection $s_K: K\rightarrow \omega^\omega$. (Look at the tree representations.)

Every real computes only countably many tree representations of perfect sets.

We combine these facts as follows. Given a real $f\in\omega^\omega$, let $T_i$ ($i\in\omega$) be the perfect sets whose tree representation is computable in $f$ (these come ordered by index of Turing reduction), and let $g_i=s_{T_i}(f)$. Then let $F$ be defined as $$F(f)(n)=1+\sum_{i=0}^ns_{T_i}(f)(n).$$ That is, $F$ takes in $f$, and spits out a real dominating each of the countably many reals which $f$ could correspond to in any of the perfect sets $f$ computes.

Now suppose $T$ is any pointed perfect set; I want to argue that $F(T)$ is a dominating family. Since $s_T$ is surjective, it's enough to show that for each $f\in T$, there is some $g\in F(T)$ such that $g$ dominates $s_T(f)$. But since $T$ is perfect, $f$ computes the tree representation of $T$; so we may take $g=F(f)$, and problem solved.

However, each of the two results above has a natural follow-up question:

Question 1.Does $B$ (escaping) spread onto $A$ (dominating)?

Of course the answer is consistently "no," since it is consistent that the dominating number is strictly greater than the bounding number. However, I would like a ZF answer. Less valuable, but still very interesting, would be answers in either ZFC or ZF+AD+etc (which don't decide whether $\mathfrak{b}<\mathfrak{d}$ and which prove a weak CH, respectively - so the question is not trivial in either case).

Question 2.Can we drop "pointed" in the second example? That is, does the family ofmerely perfectsets spread onto $A$?

I don't see how to do this even in ZFC; we seem to need some way to assign to each real $f$ a family of "few" (specifically, smaller than the escaping number) reals which it needs to dominate. Via pointedness, we were able to cheat - every real is in only countably many pointed perfect sets. However, without this I don't see how to get the argument to go through. (Note that there are perfect sets whose tree representations aren't computed by *any* of their elements!)

*Actually, I'm beginning to think that the right approach for this type of question is to restrict attention to "pointed subsets of $\omega^\omega$" - that is, dominating/escaping/perfect/whatever sets, every element of which computes "the right representation" of the set in question. However, I still don't know quite what I mean by that. Any thoughts on the matter, while not the main focus of this question, would be welcome!*