The impetus for Ergodic Theory came from Boltzmann's hypothesis that a particle in an equilibriated system (say, for instance, a gas) would visit each possible state in the level set $H^{-1}(e)$ of the energy as time asymptotically tended toward infinity. Walters has written that the hypothesis as stated was false, and the well-known ergodic theorem due to Brikhoff instead equates temporal averages of paths with spatial averages of the system.
My question is are there any conditions under which Boltzmann's original claim is true? Specifically, Suppose $(X,\mathcal{B},m)$ is a measure space and $T$ is a transformation on $X$. Suppose we call a subset $U$ of the state space universally reachable if for every $x \in X$ there is some index $k$ such that $T^k(x) \in U$. Does knowledge of ergodicity help us to draw any conclusions about the existence and number of such sets? Under what conditions are ergodicity and universal reachability equivalent for some given subset of the state space?