This talk is a short presentation of my PhD project.

The class of \(\mathbb{Z}^d\)-periodic lifts can be seen as a generalisation of lifts of circle maps of degree one. Assuming that the underlying map is Markovian with finitely many conformal branches we consider transient dynamics of escaping points which deviate from their asymptotic drift \(\alpha\) quasi-uniformly. These constitute the \(\alpha\)-escaping sets. Applying methods from thermodynamic formalism and convex analysis, the Hausdorff dimension of any non-empty \(\alpha\)-escaping set can be shown to be given as the zero of a certain pressure function: a fibre induced pressure function. Moreover, we find that a \(\mathbb{Z}^d\)-periodic lift in our class is recurrent with full dimension if and only if it does not exhibit an essential drift.