  Given a countable group $G$ and a $G$-flow $X$, a measure $\mu\in P(X)$ is
called characteristic if it is $\mathrm{Aut}(X, G)$-invariant. Frisch and Tamuz
asked about the existence of a minimal $G$-flow, for any group $G$, which does
not admit a characteristic measure. We construct for every countable group $G$
such a minimal flow. Along the way, we are motivated to consider a family of
questions we refer to as minimal subdynamics: Given a countable group $G$ and a
collection of infinite subgroups $\{\Delta_i: i\in I\}$, when is there a
faithful $G$-flow for which every $\Delta_i$ acts minimally?
