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