2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/158759A homeomorphism f of a manifold M is called H_1-transitive if there is a transitive lift of an iterate of f to the universal Abelian cover \tM. Roughly speaking, this means that f has orbits which repeatedly and densely explore all elements of H_1(M). For a rel pseudo-Anosov map ϕof a compact surface M we show that the following are equivalent: (a) ϕis H_1-transitive, (b) the action of ϕon H_1(M) has spectral radius one, and (c) the lifts of the invariant foliations of ϕto \tM have dense leaves. The proof relies on a characterization of transitivity for twisted \Z^d-extensions of a transitive subshift of finite type.Dynamical Systems37E30 (Primary); 37E45, 57M10 (Secondary)text