Brody curves and mean dimension

Abstract

We study the mean dimensions of the spaces of Brody curves. In particular we give the formula of the mean dimension of the space of Brody curves in the Riemann sphere. A key notion is a non-degeneracy of Brody curves introduced by Yosida (1934). We develop a deformation theory of non-degenerate Brody curves and apply it to the calculation of the mean dimension. Moreover we show that there are sufficiently many non-degenerate Brody curves. ∂ 2 ∂y 2 . |df |(z) is classically called a spherical derivative. It evaluates the dilatation of the map f with respect to the Euclidean metric on C and the Fubini-Study metric on CP N . (See the equation (6) in Section 4.2.) A holomorphic map f : C → CP N is called a Brody curve ([3]) if it satisfies |df |(z) ≤ 1 for all z ∈ C. Let M(CP N ) be the space of Brody curves in CP N . It is endowed with the compact-open topology (the topology of uniform convergence on compact subsets): A sequence of Brody curves {f n } ⊂ M(CP N ) converges to f ∈ M(CP N ) if and only if for any compact subset K ⊂ C we have sup z∈K d(f n (z), f (z)) → 0 as n → ∞. ( d(·, ·) is the distance on CP N with respect to the Fubini-Study metric.) M(CP N ) is an infinite dimensional compact metrizable space, and it admits the following continuous C-action.

Mean dimension of the dynamical system of Brody curves

“…In a series of works [14,16,12], we have been sharpening Gromov's estimate. In the present paper we finally reach the answer to the problem.…”

“…The original motivation for the study of Brody curves comes from this connection to the hyperbolicity. 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].…”

“…Namely we can decompose a global problem into local ones which are more suitable for detailed analysis. This enables us to apply the analytic machinery developed in [16,12] to the problem of proving the upper bound (1.5). The techniques in [16,12] were originally introduced for the lower bound (1.4).…”

“…This is equivalent to the condition that the closure of the C-orbit of f in M(CP 1 ) does not contain a constant function. This idea of Yosida played a quite important role in [12] for the proof of the lower bound dim(M(CP…”

Brody curves in complicated sets

“…Recently, the space of Brody curves has been much studied. In particular, for the projective spaces, see [6], [7], [12], [15], [16], [17] and [18]. However, it is not easy to find interesting examples (cf.…”