On the integral simplicial volume of cyclic covers of mapping tori

In this paper, we investigate the asymptotic behavior of the integral simplicial volume of cyclic covers of manifolds that fiber over the circle with fiber given by an $n$-dimensional torus. By studying the integral filling volume -- an invariant introduced by Frigerio and the first author -- for the monodromy, we establish both lower and upper bounds for the limit of the integral simplicial volume of these covers, normalized by the degree of the covering. These bounds are expressed in terms of the action of the monodromy on the real homology of the fiber. As applications, we establish a close connection between the topological entropy and the integral filling volume of self-homeomorphisms of $n$-dimensional tori, we find new examples for which the Delta-complexity and the integral simplicial volume are not equivalent, and we prove the nonvanishing of the filling volume for Anosov self-diffeomorphisms of infranilmanifolds.