  Let $\alpha: G\curvearrowright X$ be a minimal free continuous action of an
infinite countable amenable group on an infinite compact metrizable space. In
this paper, under the hypothesis that the invariant ergodic probability Borel
measure space $E_G(X)$ is compact and zero-dimensional, we show that the action
$\alpha$ has the small boundary property. This partially answers an open
problem in dynamical systems that asks whether a minimal free action of an
amenable group has the small boundary property if its space $M_G(X)$ of
invariant Borel probability measures forms a Bauer simplex. In addition, under
the same hypothesis, we show that dynamical comparison implies almost
finiteness, which was shown by Kerr to imply that the crossed product is
$\mathcal{Z}$-stable. Finally, we discuss some rank properties and provide two
classifiability results for crossed products, one of which is based on the work
of Elliott and Niu.
