A Spectral Gap for the Transfer Operator on Complex Projective Spaces - Lecture 4

Explore the transfer operator on complex projective spaces in this advanced mathematics lecture. Delve into the study of the Perron-Frobenius operator on P^k(C) induced by a generic holomorphic endomorphism and a continuous weight. Learn about the existence of a unique equilibrium state and the introduction of new invariant functional spaces, including a dynamical Sobolev space. Discover how these spaces allow for a spectral gap in the action of the endomorphism, leading to various statistical properties of equilibrium states. Gain insights into the application of pluripotential theory and interpolation between Banach spaces in constructing invariant functional spaces. This 1 hour 36 minute lecture, part of the Simons Semester on Dynamics, presents joint work with Tien-Cuong Dinh, offering novel results even in one-dimensional cases and for constant weight functions.

Fabrizio Bianchi (CNRS Universite Lille) lecture 4