Attracting currents and equilibrium measures for quasi-attractors of $\mathbb {P}^k$
Résumé
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 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.