Geometry and Dynamics of Automorphisms of $\mathbb{P}^{2}_\mathbb{C}$

Abstract

In this chapter we study and describe the geometry, dynamics and algebraic classification of the elements in PSL(3, $\mathbb{C}$), extending Goldman’s classification for the elements in PU(2, 1) ⊂ PSL(3,$\mathbb{C}$). Just as in that case, and more generally for the isometries of manifolds of negative curvature, the automorphisms of $\mathbb{P}^{2}_\mathbb{C}$ can also be classified into the three types of elliptic, parabolic and loxodromic (or hyperbolic) elements, according to their geometry and dynamics. This classification can be also done algebraically, in terms of their trace.