  Motivated by fractal geometry of self-affine carpets and sponges, Feng--Huang
(2016) introduced weighted topological entropy and pressure for factor maps
between dynamical systems, and proved variational principles for them. We
introduce a new approach to this theory. Our new definitions of weighted
topological entropy and pressure are very different from the original
definitions of Feng--Huang. The equivalence of the two definitions seems highly
nontrivial. Their equivalence can be seen as a generalization of the dimension
formula for the Bedford--McMullen carpet in purely topological terms.
