Weighted topological and measure-theoretic entropy (Q2423648)

The present article is devoted to dynamical systems and ergodic theory. In particular, the main attention is given to the entropy theory and topological dynamical systems. Certain known results are considered. The authors state that the aim of the present article is to generalize some investigations in this area, to define some notions, as well as to study their relationships (including their relationships with similar known notions). Specifically, their investigations concern the following entropies: weighted Bowen topological entropy; weighted upper capacity topological entropy; weighted classical topological entropy; weighted packing topological entropy; weighted measure-theoretic entropy; weighted conditional entropy. One open problem is formulated. One can note the authors' abstract: ``\textit{D. Feng} and \textit{W. Huang} in [J. Math. Pures Appl. (9) 106, No. 3, 411--452 (2016; Zbl 1360.37080)] defined a new notion called weighted topological entropy (pressure) and obtained the corresponding variational principle for compact dynamical systems. In this paper, it was our hope to carry out a further study from the following three aspects: inspired from the well-known classical entropy theory, we define various weighted topological (measure-theoretic) entropies and investigate their relationships; the classical entropy formula of subsets and their transformations by factor maps is generalized to the weighted version; a formula which comes from the Brin-Katok theorem of weighted conditional entropy is established.''