Dimensional entropy of amenable group actions over stable sets and fibres
This paper is devoted to the study of Bowen's dimensional entropy on subsets for actions of amenable groups. We prove three main results. (1) First, topological conditional entropy is characterized by the dimensional entropy of stable sets (Theorem 1.1), answering a question of Dou, Wang and the second author of the present paper raised in [Fund. Math., 2025]. We remark that our Theorem 1.1 is the first characterization of topological conditional entropy via Bowen's dimensional entropy of stable sets even for $\mathbb{Z}$-actions. (2) Second, we establish a dimensional entropy inequality for factor maps (Theorem 1.2). It relates dimensional entropy of a set to that of its image and topological entropy of fibres, and may be viewed as the dimensional-entropy counterpart of the factor-map inequality for packing topological entropy due to Dou, Zheng, and Zhou proved as Theorem 1.4 in [Ergodic Theory Dynam. Systems, 2023]. (3) Third, the relative topological entropy of a factor map is determined by the dimensional entropy of the fibres (Theorem 1.3). Notably, our proof of this formula (Theorem 1.3) is purely topological, in contrast to the recent measure-theoretic approach of Dou, Wang and Zhou based on relative Shannon--McMillan--Breiman theorems. These results (Theorem 1.2 and 1.3) not only generalize the work of Oprocha and the second author of the present paper [Nonlinearity, 2011] from single transformations to amenable group actions, but also provide a purely topological and self-contained proof of a fibre entropy characterization recently obtained through measure-theoretic arguments.