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A Spectral Gap for the Transfer Operator on Complex Projective Spaces

Adventurous Berkeley Complex Dynamics May 02, 2022 - May 06, 2022

May 05, 2022 (02:00 PM PDT - 03:00 PM PDT)
Speaker(s): Fabrizio Bianchi (Université de Lille)
Location: SLMath: Eisenbud Auditorium, Online/Virtual
Video

A Spectral Gap For The Transfer Operator On Complex Projective Spaces

Abstract

We study the transfer (Perron-Frobenius) operator on Pk(C) induced by a generic holomorphic endomorphism 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 such as the equidistribution of points, speed of convergence, K-mixing, mixing of all orders, exponential mixing, central limit theorem, Berry-Esseen theorem, local central limit theorem, almost sure invariant principle, law of iterated logarithms, almost sure central limit theorem and the large deviation principle. Most of the 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.

Supplements
93181?type=thumb A Spectral Gap for the Transfer Operator on Complex Projective Spaces 492 KB application/pdf Download
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A Spectral Gap For The Transfer Operator On Complex Projective Spaces

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