An entropy formula over the dynamical balls and its applications

Wang, Tao

doi:10.1090/proc/15756

An entropy formula over the dynamical balls and its applications
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Proc. Amer. Math. Soc. 150 (2022), 4743-4755 Request permission

Abstract:

The Bowen topological entropy was introduced by Bowen in way which resembles the Hausdorff dimension. It is well-known that the Bowen topological entropy and the topological entropy of subsets are not equal in general. We show in the paper that the supremum of the Bowen topological entropy of the dynamical balls is the same as that of the topological entropy of the dynamical balls, which naturally implies that every positively countably expansive map is also positively entropy expansive. This answers a question posed by Artigue, Carvalho, Cordeiro, and Vieitez [Proc. Amer. Math. Soc. 150 (2022), pp. 3369–3378]. Besides, our main result is applied to find some other interesting phenomena.

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