The branched virtual fibering theorem by Makoto Sakuma states that every closed orientable 3-manifold M with a Heegaard surface of genus g has a branched double cover which is a genus g surface bundle over the circle.
It is proved by Brooks and Montesinos that such surface bundle can be chosen to be hyperbolic. i.e, the monodromy map of such surface bundle can be chosen to be pseudo-Anosov.
So it makes sense to talk about the topological entropies of hyperbolic surface bundles over the circle as branched double covers of M.
I discuss some properties of entropies of those hyperbolic surface bundles.
In joint work with Susumu Hirose, we prove that when M is the 3-sphere S^3, the minimal entropy over all hyperbolic, genus g surface bundles as branched double covers of S^3 behaves like 1/g.
If time permits, I will introduce some questions related to the branched virtual fibering theorem.