  We consider the problem of when a symbolic dynamical system supports a Borel
probability measure that is invariant under every element of its automorphism
group. It follows readily from a classical result of Parry that the full shift
on finitely many symbols, and more generally any mixing subshift of finite
type, supports such a measure. Frisch and Tamuz recently dubbed such measures
characteristic, and further showed that every zero entropy subshift has a
characteristic measure. While it remains open if every subshift over a finite
alphabet has a characteristic measure, we define a new class of shifts, which
we call language stable subshifts, and show that these shifts have
characteristic measures. This is a large class that is generic in several
senses and contains numerous positive entropy examples.
