Nagoya Mathematical Journal

Moduli space of Brody curves, energy and mean dimension

Masaki Tsukamoto

Full-text: Open access

Abstract

A Brody curve is a holomorphic map from the complex plane $\mathbb{C}$ to a Hermitian manifold with bounded derivative. In this paper we study the value distribution of Brody curves from the viewpoint of moduli theory. The moduli space of Brody curves becomes infinite dimensional in general, and we study its "mean dimension". We introduce the notion of "mean energy" and show that this can be used to estimate the mean dimension.

Article information

Source
Nagoya Math. J., Volume 192 (2008), 27-58.

Dates
First available in Project Euclid: 22 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1229955904

Mathematical Reviews number (MathSciNet)
MR2477610

Zentralblatt MATH identifier
1168.32016

Subjects
Primary: 32H30: Value distribution theory in higher dimensions {For function- theoretic properties, see 32A22}

Keywords
moduli space of Brody curves mean dimension mean energy the Nevanlinna theory

Citation

Tsukamoto, Masaki. Moduli space of Brody curves, energy and mean dimension. Nagoya Math. J. 192 (2008), 27--58. https://projecteuclid.org/euclid.nmj/1229955904


Export citation

References

  • F. Berteloot and J. Duval, Sur l'hyperbolicité de certains complémentaires, Enseign. Math., 47 (2001), 253--267.
  • R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401--414.
  • R. Brody, Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc., 235 (1978), 213--219.
  • J. Clunie and W. K. Hayman, The spherical derivative of integral and meromorphic functions, Comment. Math. Helv., 40 (1966), 117--148.
  • A. Eremenko, Normal holomorphic curves from parabolic regions to projective spaces, preprint, Purdue University (1998), arXiv:0710.1281.
  • H. Grauert and R. Remmert, Coherent analytic sheaves, Springer-Verlag, Berlin, 1984.
  • M. L. Green, Holomorphic maps to complex tori, Amer. J. Math., 100 (1978), 615--620.
  • M. Gromov, Topological invariants of dynamical systems and spaces of holomorphic maps: I, Math. Phys. Anal. Geom., 2 (1999), 323--415.
  • E. Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Inst. Hautes Études Sci. Publ. Math., 89 (1999), 227--262.
  • E. Lindenstrauss and B. Weiss, Mean topological dimension, Israel J. Math., 115 (2000), 1--24.
  • D. Minda, Yosida functions, Lectures on complex analysis (Chi-Tai Chuang, ed.), World Sci. Publishing, Singapore, 1988, pp. 197--213.
  • R. Nevanlinna, Analytic functions, Translated from the second German edition by Phillip Emig. Die Grundlehren der mathematischen Wissenschaften, Band 162, Springer-Verlag, New York, Berlin, 1970.
  • C. Robinson, Dynamical systems: stability, symbolic dynamics, and chaos, Second edition, CRC Press, Boca Raton, 1999.
  • M. Tsukamoto, A packing problem for holomorphic curves, preprint, arXiv: math.CV/0605353, to appear in Nagoya Math. J\.
  • M. Tsukamoto, On holomorphic curves in algebraic torus, J. Math. Kyoto Univ., 47-4 (2007), 881--892.
  • J. Winkelmann, On Brody and entire curves, Bull. Soc. math. France, 135 (1) (2007), 25--46.