Entropy of real rational surface automorphisms: actions on the fundamental groups

This article develops a method to compute the action induced on the fundamental group by an automorphism of a real rational surface. Then, it uses this method to compute the induced actions for a certain family of basic quadratic real automorphisms. By utilizing an invariant set in the fundamental group, we introduce a method to estimate a lower bound of the action’s growth rate on the fundamental group. The growth rate of this induced action provides a lower bound for the entropy of the real surface automorphism. These calculations are carried out for an important family of real surface automorphisms, and new lower bounds are obtained for the entropy of these automorphisms.