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It is well-known that any $\mathbb{Z}$ subshift with the specificationproperty 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 resultsthat show that the situation for $\mathbb{Z}^d$ subshifts with $d >1 $ is quitedifferent. First, for any $d>1$, we show that any $\mathbb{Z}^d$ subshiftpossessing a certain mixing property must have a factor with positive entropywhich is not intrinsically ergodic. In particular, this shows that for $d>1$,$\mathbb{Z}^d$ subshifts with specification cannot have all factorsintrinsically ergodic. We also give an example of a $\mathbb{Z}^2$ shift offinite type, introduced by Hochman, which is not even topologically mixing, butfor which every positive entropy factor is intrinsically ergodic.

On intrinsic ergodicity of factors of $\mathbb{Z}^d$ subshifts