We study the transfer (Perron-Frobenius) operator on $Pk(C)$ induced by a generic holomorphic endomorphism $f$ and a suitable continuous weight. We prove the existence of a unique equilibrium state and we introduce various new invariant functional spaces, including a dynamical Sobolev space, on which the action of $f$ admits a spectral gap. This is one of the most desired properties in dynamics. It allows us to obtain a list of statistical properties for the equilibrium states. Most of our results are new even in dimension $1$ and in the case of constant weight function, i.e., for the operator $f_*$. Our construction of the invariant functional spaces uses ideas from pluripotential theory and interpolation between Banach spaces. This is a joint work with Tien-Cuong Dinh.
Meeting ID: 892 9535 2196 Passcode: 392668