Mean dimension of the dynamical system of Brody curves

TL;DR

This paper determines the exact mean dimension of the dynamical system of Brody curves, linking it to their energy density, and employs novel metric mean dimension techniques.

Contribution

It provides the first exact formula for the mean dimension of Brody curves' dynamical system, connecting it to energy density.

Findings

Exact mean dimension formula derived

Mean dimension expressed via energy density

Innovative application of metric mean dimension theory

Abstract

Mean dimension measures the size of an infinite dimensional dynamical system. Brody curves are one-Lipschitz entire holomorphic curves in the projective space, and they form a topological dynamical system. Gromov started the problem of estimating its mean dimension in the paper of 1999. We solve this problem. Namely we prove the exact mean dimension formula of the dynamical system of Brody curves. Our formula expresses the mean dimension by the energy density of Brody curves. The proof is based on a novel application of the metric mean dimension theory of Lindenstrauss and Weiss.