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

摘要

In a previous paper [Pavlov, A characterization of topologically completely positive entropy for shifts of finite type. Ergod. Th. & Dynam. Sys. 34 (2014), 2054-2065], the author gave a characterization for when a Z(d)-shift of finite type has no non-trivial subshift factors with zero entropy, a property which we here call zero-dimensional topologically completely positive entropy. In this work, we study the difference between this notion and the more classical topologically completely positive entropy of Blanchard. We show that there are one-dimensional subshifts and two-dimensional shifts of finite type which have zero-dimensional topologically completely positive entropy but not topologically completely positive entropy. In addition, we show that strengthening the hypotheses of the main result of Pavlov [A characterization of topologically completely positive entropy for shifts of finite type. Ergod. Th. & Dynam. Sys. 34 (2014), 2054-2065] yields a sufficient condition for a Z(d)-shift of finite type to have topologically completely positive entropy.