Unlike most publications devoted to the application of the self-consistent method of the nonlinear Vlasov-Poisson system to the study of beam-beam interaction, in this article an alternative strategy using the approach of the Perron-Frobenius operator for symplectic twist maps was developed. A detailed analysis of the establishment of an equilibrium density distribution in phase space and the behaviour of the perturbed distribution function with respect to the coherent stability of the two beams was carried out.

Using the Renormalisation Group technique for the reduction of the Perron-Frobenius operator, the case where the unperturbed rotation frequency (unperturbed betatron tune) of the map is far from any structural resonance driven by the beam-beam kick perturbation was analysed in detail. It was shown that up to second order in the beam-beam parameter, the renormalised map propagator with nonlinear stabilisation describes a random walk of the angle variable, implying that there is an equilibrium distribution function depending only on the action variable.

The linearised Perron-Frobenius operators for each beam imply a discrete form of the linearised Vlasov equations, which essentially comprises a new method for calculating coherent beam-beam instabilities using a matrix mapping technique. In the special case of an isolated coherent beam-beam resonance, a stability criterion for coherent beam-beam resonances was found in closed form.