  It is well-known that any $\mathbb{Z}$ subshift with the specification
property has the property that every factor is intrinsically ergodic, i.e.,
every factor has a unique factor of maximal entropy. In recent work, other
$\mathbb{Z}$ subshifts have been shown to possess this property as well,
including $\beta$-shifts and a class of $S$-gap shifts. We give two results
that show that the situation for $\mathbb{Z}^d$ subshifts with $d >1 $ is quite
different. First, for any $d>1$, we show that any $\mathbb{Z}^d$ subshift
possessing a certain mixing property must have a factor with positive entropy
which is not intrinsically ergodic. In particular, this shows that for $d>1$,
$\mathbb{Z}^d$ subshifts with specification cannot have all factors
intrinsically ergodic. We also give an example of a $\mathbb{Z}^2$ shift of
finite type, introduced by Hochman, which is not even topologically mixing, but
for which every positive entropy factor is intrinsically ergodic.

Dynamical Systems