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In a previous paper ("A characterization of topologically completely positiveentropy for shifts of finite type"), the author gave a characterization forwhen a $\mathbb{Z}^d$-shift of finite type (SFT) has no nontrivial subshiftfactors with zero entropy, a property which we here call zero-dimensionaltopologically completely positive entropy (ZTCPE). In this work, we study thedifference between this notion and the more classical topologically completelypositive entropy (TCPE) of Blanchard. We show that there are one-dimensionalsubshifts and two-dimensional SFTs which have ZTCPE but not TCPE. In addition,we show that strengthening the hypotheses of the main result of theaforementioned paper yields a sufficient condition for a $\mathbb{Z}^d$-SFT tohave TCPE.

Topologically completely positive entropy and zero-dimensional topologically completely positive entropy