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On intrinsic ergodicity of factors of
$\mathbb{Z}^{d}$ subshifts
Published online by Cambridge University Press: 06 October 2015
Abstract
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
$\unicode[STIX]{x1D6FD}$-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 subshift factor is intrinsically ergodic.
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- © Cambridge University Press, 2015