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Topologically completely positive entropy and zero-dimensional topologically completely positive entropy. (English) Zbl 1396.37010

Summary: In a previous paper [R. Pavlov, Ergodic Theory Dyn. Syst. 34, No. 6, 2054–2065 (2014; Zbl 1351.37068)], the author gave a characterization for when a \(\mathbb{Z}^{d}\)-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 [loc. cit.] yields a sufficient condition for a \(\mathbb{Z}^{d}\)-shift of finite type to have topologically completely positive entropy.

MSC:

37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010)
37B40 Topological entropy
28D20 Entropy and other invariants

Citations:

Zbl 1351.37068
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References:

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