Mean dimension of the dynamical system of Brody curves

Abstract

Mean dimension is a topological invariant of dynamical systems counting the number of parameters averaged by dynamics. Brody curves are Lipschitz holomorphic maps $$f:\mathbb {C}\rightarrow \mathbb {C}P^N$$ , and they form an infinite dimensional dynamical system. Gromov started the problem of estimating its mean dimension in 1999. We solve this problem by proving the exact mean dimension formula. Our formula expresses the mean dimension by the energy density of Brody curves. As a key novel ingredient, we use an information theoretic approach to mean dimension introduced by Lindenstrauss and Weiss.