Mean dimension of the dynamical system of Brody curves

Abstract

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. Mean dimension formulaThis is the local Lipschitz constant. For a unit tangent vector u ∈ T z C the Fubini-Study length of df (u) ∈ T f (z) CP N is equal to the spherical derivative |df |(z) Recently new waves have begun, and Brody curves attract new interests of several authors [1,3,4,5,6,12,14,15,19]. One source of new interests is the work of Gromov [8]. He introduced a new topological invariant of dynamical systems called mean dimension, and proposed a program to study many infinite dimensional dynamical systems in geometric analysis from the viewpoint of this invariant. The dynamical system consisting of Brody curves is the simplest example in this program. The purpose of our paper is.

Metric mean dimension and analog compression

“…Metric mean dimension was introduced in [23] and proved to be, when calculated with respect to any compatible metric, an upper bound for the topological mean dimension. It was recently successfully used in [52] and [53] to obtain formulas for mean dimension of dynamical systems arising from geometric analysis.…”

“…This implies that f is injective on B n , hence #f (B n ) = #B n . Moreover, by (52), elements of the…”

Double variational principle for mean dimension

“…Although our investigation in this direction has just started, the result in §6 seems to suggest a high potential of this research direction. It is desirable to study geometric examples in [Gro99,T18a,T18b] from the viewpoint of the double variational principle.…”

Around the variational principle for metric mean dimension

“…Metric mean dimension turned out to be a useful tool in calculating the topological mean dimension of certain systems (see e.g. [Tsu18a] and [Tsu18b] for applications to dynamical systems arising from geometric analysis) as well as a meaningful quantity from the point of view of analog compression (see [GŚ19,GŚ20]).…”