The Kobayashi distance in holomorphic dynamics and operator theory

TL;DR

This paper explores how the Kobayashi distance, a geometric tool, can be used to analyze and encode properties of holomorphic functions and maps, with applications in complex dynamics and operator theory.

Contribution

It provides a comprehensive overview of the Kobayashi distance's properties and demonstrates its applications in holomorphic dynamics and operator theory on complex manifolds.

Findings

Kobayashi distance characterizes complex manifold geometry.

Applications to holomorphic dynamics in various domains.

Connections to operator theory in Bergman spaces.

Abstract

These are the notes of a short course I gave in the school "Aspects m\'etriques et dynamiques en analyse complete", Lille, May 2015. The aim of this notes is to describe how to use a geometric structure (namely, the Kobayashi distance) to explore and encode analytic properties of holomorphic functions and maps defined on complex manifolds. We shall first describe the main properties of the Kobayashi distance, and then we shall present applications to holomorphic dynamics in taut manifolds, strongly pseudo convex domains and convex domains, and to operator theory in Bergman spaces (Carleson measures and Toeplitz operators).