  Given a dynamical system $(X, \Gamma)$, the corresponding crossed product
$C^*$-algebra $C(X)\rtimes_{r}\Gamma$ is called reflecting, when every
intermediate $C^*$-algebra $C^*_r(\Gamma)<\mathcal{A} < C(X)\rtimes_{r}\Gamma$
is of the form $\mathcal{A}=C(Y)\rtimes_{r}\Gamma$, corresponding to a
dynamical factor $X \rightarrow Y$. It is called almost reflecting if
$\mathbb{E}(\mathcal{A}) \subset \mathcal{A}$ for every such $\mathcal{A}$.
These two notions coincide for groups admitting the approximation property
(AP). Let $\Gamma$ be a non-elementary convergence group or a lattice in
$\text{SL}_d(\mathbb{R})$ for some $d \ge 2$. We show that any uniformly rigid
system $(X,\Gamma)$ is almost reflecting. In particular, this holds for any
equicontinuous action. In the von Neumann setting, for the same groups $\Gamma$
and any uniformly rigid system $(X,\mathcal{B},\mu, \Gamma)$ the crossed
product algebra $L^{\infty}(X,\mu)\rtimes\Gamma$ is reflecting. An inclusion of
algebras $\mathcal{A}\subset\mathcal{B}$ is called $\textit{minimal ambient}$
if there are no intermediate algebras. As a demonstration of our methods, we
construct examples of minimal ambient inclusions with various interesting
properties in the $C^*$ and the von Neumann settings.
