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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.

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