Rabinowitz-Floer homology is the semi-infinite dimensional Morse-Bott homology in the sense of Floer of the Rabinowitz action functional introduced by Kai Cieliebak and Urs Frauenfelder in 2009. In our work, we consider a generalisation of this theory to a Rabinowitz-Floer homology of a Liouville automorphism. As an application we show the existence of noncontractible periodic Reeb orbits on bases of universal coverings. In particular, our theory applies to lens spaces.

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We show that in general the gradient of the twisted Rabinowitz action functional does not exist without further modifications.

We show that the limit set of a family of parametrised periodic orbits on a homotopy of stable regular energy surfaces is not empty.