  Masures are generalizations of Bruhat--Tits buildings and the main examples
are associated with almost split Kac--Moody groups G over non-Archimedean local
fields. In this case, G acts strongly transitively on its corresponding masure
$\Delta$ as well as on the building at infinity of $\Delta$, which is the twin
building associated with G. The aim of this article is twofold: firstly, to
introduce and study the cone topology on the twin building at infinity of a
masure. It turns out that this topology has various favorable properties that
are required in the literature as axioms for a topological twin building.
Secondly, by making use of the cone topology, we study strongly transitive
actions of a group G on a masure $\Delta$. Under some hypotheses, with respect
to the masure and the group action of G, we prove that G acts strongly
transitively on $\Delta$ if and only if it acts strongly transitively on the
twin building at infinity $\partial$$\Delta$. Along the way a criterion for
strong transitivity is given and the existence and good dynamical properties of
strongly regular hyperbolic automorphisms of the masure are proven.
