Weighted topological entropy and intersecting random translates of Bedford--McMullen carpets

We establish a relativised variational principle for the Feng--Huang weighted topological entropy associated with a factor map between dynamical systems. Combined with a recent theorem of Yin, this yields an almost-everywhere equivalence between the Feng--Huang entropy and its combinatorial version on fibers. As an application, we compute the Hausdorff dimension of the intersection of random translates of two Bedford--McMullen carpets. The resulting formula extends the Kenyon--Peres formula from the self-similar to the self-affine setting, and also points to a new problem concerning random matrix products.