For $\mathbb{A},\mathbb{B}\subseteq\mathcal{P}(\omega^\omega)$, say $\mathbb{A}$ **spreads onto** $\mathbb{B}$ iff there is some $F:\omega^\omega\rightarrow\omega^\omega$ such that for all $X\in\mathbb{A}$ the set $F[X]=\{F(r):r\in X\}$ is in $\mathbb{B}$.


Let $\mathbb{D},\mathbb{E}$ be the sets of dominating, escaping families of reals respectively. Trivially $\mathbb{D}$ spreads onto $\mathbb{E}$ via the identity function, since every dominating family is escaping. The converse however is more complicated, even in $\mathsf{ZFC}$: $\mathbb{E}$ obviously can't spread onto $\mathbb{D}$ if $\mathfrak{b}<\mathfrak{d}$, but $\mathsf{CH}$ implies that $\mathbb{E}$ does spread onto $\mathbb{D}$ per vzoltan's answer to [an old question of mine](https://mathoverflow.net/questions/249418/reducing-largeness-notions-uniformly).


I'm curious what happens in $\mathsf{ZF+DC+AD}$:



> 
> Under $\mathsf{ZF+DC+AD}$, does $\mathbb{E}$ spread onto $\mathbb{D}$?
> 
> 
> 



Do escaping sets "uniformly" cover dominating sets under determinacy?
Here's something that seems to work. I can add more details if needed.


Suppose that every set of reals has the property of Baire. Then every function from $\omega^{\omega} \to \omega^{\omega}$ is continuous on a comeager set. Note that every comeager set is unbounded. Fix $F \colon \omega^{\omega} \to \omega^{\omega}$ and let $C$ be a comeager set on which $F$ is continuous. We will find a dense open sets $D\_{i}$ ($i \in \omega$) and $y \in \omega^{\omega}$ such that $F(x)$ fails to dominate $y$ whenever $x \in C \cap \bigcap\_{i \in \omega}D\_{i}$. Let $\langle (i\_{n}, \sigma\_{n}) : n \in \omega \rangle$ be an enumeration of $\omega \times \omega^{<\omega}$. In the $n$th stage of the construction choose $y \upharpoonright k\_{n}$ for some $k\_{n} \in \omega$ and also some $\tau\_{n} \in \omega^{<\omega}$ extending $\sigma\_{n}$ such that $F(x) \upharpoonright k\_{n}$ is the same sequence $\rho\_{n}$ for all $x \in C$ extending $\tau\_{n}$, and $\{ j < k\_{n} : \rho\_{n}(j) < y(j)\}$ has size at least $i\_n$. Then we can let $D\_{i}$ be the set of $x \in \omega^{\omega}$ extending any of the sets $\tau\_{n}$ where $i\_{n} = i$.


The $\mathbb{P}\_{\mathrm{max}}$ claim has to be modified somewhat, since $\mathbb{P}\_{\mathrm{max}}$ needs AD$^{+}$ to work, which is not known to follow from AD + DC. Forcing over a model of AD$^{+}$, $\mathbb{P}\_{\mathrm{max}}$ produces a model of ZFC which is $\Pi\_{2}$ maximal for the powerset of $\omega\_{1}$ by (very roughly speaking) forming a generic direct limit of all countable models, subject to agreement about stationarity for subsets of $\omega\_{1}$. There is a natural $\mathbb{P}\_{\mathrm{max}}$ variation for each $\Sigma\_{2}$ sentence, where each model fixes a witness to the sentence, and a model in the direct limit passes its witness on to stronger conditions. This doesn't always succeed, but for the statement $\mathfrak{b} = \aleph\_{1}$ it does succeed, producing (without adding reals) a model of $\mathfrak{b} = \aleph\_{1}$ in which every $\Pi\_{2}$ sentence for $\mathcal{P}(\omega\_{1})$ holds which can be provably forced to hold along with $\mathfrak{b} = \aleph\_{1}$. For instance, the statement $\mathfrak{d} > \aleph\_{1}$. This is discussed at a general level in the last section of [my article from the Handbook of Set Theory](https://link.springer.com/chapter/10.1007/978-1-4020-5764-9_25), and in more detail in the Shelah-Zapletal paper ["Canonical models for $\aleph\_{1}$-combinatorics"](https://arxiv.org/abs/math/9806166).


Having produced such a model, one can then take an unbounded family $B$ of size $\aleph\_{1}$, find a real $y$ which is not dominated by $F[B]$, and then find the unbounded set of $x$ such that $F(x)$ does not dominate $y$ back in the ground model.


There is an intermediate argument one could run using the fact that, under AD$^{+}$, true $\Sigma^{2}\_{1}$ sentences have Suslin, co-Suslin witnesses. This is at the heart of the $\mathbb{P}\_{\mathrm{max}}$ argument. For the current problem, continuity on a comeager set works as a substitute for being Suslin and co-Suslin.


Morally, anything that is consistent with large cardinals should be forceable over a determinacy model without adding reals.


