  We develop a new thermodynamic formalism to investigate the transient
behaviour of maps on the real line which are skew-periodic
$\mathbb{Z}$-extensions of expanding interval maps. Our main focus lies in the
dimensional analysis of the recurrent and transient sets as well as in
determining the whole dimension spectrum with respect to $\alpha$-escaping
sets. Our results provide a one-dimensional model for the phenomenon of a
dimension gap occurring for limit sets of Kleinian groups. In particular, we
show that a dimension gap occurs if and only if we have non-zero drift and we
are able to precisely quantify its width as an application of our new
formalism.
