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Let $f$ be a holomorphic endomorphism of $\mathbb P^k$ of degree $d.$ Foreach quasi-attractor of $f$ we construct a finite set of currents withattractive behaviors. To every such an attracting current is associated anequilibrium measure which allows for a systematic ergodic theoretical approachin the study of quasi-attractors of $\mathbb P^k.$ As a consequence, we deducethat there exist at most countably many quasi-attractors, each one withtopological entropy equal to a multiple of $\log d.$ We also show that thestudy of these analytic objects can initiate a bifurcation theory forattracting sets.

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