Attracting Currents and Equilibrium Measures for Quasi-attractors of ${\mathbb{P}}^k$

Abstract

Let $f$ be a holomorphic endomorphism of ${\mathbb{P}}^k$ of degree $d.$ For each quasi-attractor of $f$ we construct a finite set of currents with attractive behaviors. To every such an attracting current is associated an equilibrium measure which allows for a systematic ergodic theoretical approach in the study of quasi-attractors of ${\mathbb{P}}^k.$ As a consequence, we deduce that there exist at most countably many quasi-attractors, each one with topological entropy equal to a multiple of $\log d.$ We also show that the study of these analytic objects can initiate a bifurcation theory for attracting sets.