In this short note, for countably infinite amenable group actions, we provide topological proofs for the following results: Bowen topological entropy (dimensional entropy) of the whole space equals the usual topological entropy along tempered Følner sequences; the Hausdorff dimension of an amenable subshift (for certain metric associated to some Følner sequence) equals its topological entropy. This answers questions by Zheng and Chen  and Simpson .
"A note on dimensional entropy for amenable group actions." Topol. Methods Nonlinear Anal. 51 (2) 599 - 608, 2018. https://doi.org/10.12775/TMNA.2017.056