and we continue this convention. Note that a regular tree is simply a Galton-Watson tree with $Z \equiv b$.

Let \[gamma-def\] $\gamma^T := (\rtt, \gamma^T_1, \gamma^T_2, \ldots)$ be the random sequence whose $n$th element is the unique $v$ with $|v|=n$ such that $v \leftrightarrow \infty$ via a downward path in the invasion cluster $I^T$. Let $\nu_T$\[nu\_T-def\] denote the law of $\gamma^T$ given $T$. Let $\nu_\TT$ denote the random measure on the random space $(\TT , \partial \TT)$ induced by the $\gamma^\TT$. In other words, for measurable $A \subseteq \partial \ulam$, $\nu_\TT (A , \omega) = \P [\gamma^T \in A]$ evaluated at $T = \TT(\omega)$. By Corollary \[cor:path\], this is a well defined probability measure for almost every $\omega$.

Preliminary comparison of limit-uniform and invasion measures
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Our main goal is to see whether $\nu_\TT$ is almost surely absolutely continuous with respect to $\mu_\TT$. The following easy first result shows that $\nu_\TT$ has the same entropy as $\mu_\TT$.

Let $T$ be a rooted tree with branching number $\mu$ and $S \subset \partial T$. For each $n$, define $S_n$ to be the set of vertices in paths in $S$ at height $n$. If there exists an $\ee > 0$ so that $|S_n| 