Feng and Huang in 2016 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:
(1) Inspired from the well-known classical entropy theory, we define various weighted topological (measure-theoretic) entropies and investigate their relationships.
(2) The classical entropy formula of subsets and their transformations by factor maps is generalized to the weighted version.
(3) A formula which comes from the Brin-Katok theorem of weighted conditional entropy is established.
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