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

Johan Taflin (1)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]

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.

Domaines: Systèmes dynamiques [math.DS]; Variables complexes [math.CV]

Johan Taflin. Attracting currents and equilibrium measures for quasi-attractors of $\mathbb {P}^k$. Inventiones Mathematicae, 2018, 213 (1), pp.83-137. ⟨10.1007/s00222-018-0786-0⟩. ⟨hal-01899994⟩