Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-27T08:07:39.384Z Has data issue: false hasContentIssue false
  • English
  • Français

Deformation of Brody curves and mean dimension

Published online by Cambridge University Press:  03 February 2009

MASAKI TSUKAMOTO
MASAKI TSUKAMOTO*
Affiliation:
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan (email: tukamoto@math.kyoto-u.ac.jp)
Get access Rights & Permissions [Opens in a new window]

Abstract

The main purpose of this paper is to show that ideas of deformation theory can be applied to ‘infinite-dimensional geometry’. We develop the deformation theory of Brody curves. A Brody curve is a kind of holomorphic map from the complex plane to the projective space. Since the complex plane is not compact, the parameter space of the deformation can be infinite-dimensional. As an application we prove a lower bound on the mean dimension of the space of Brody curves.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 5 , October 2009 , pp. 1641 - 1657
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bonk, M. and Eremenko, A.. Covering properties of meromorphic functions, negative curvature and spherical geometry. Ann. of Math. 152 (2000), 551–592.CrossRefGoogle Scholar
[2]Brody, R.. Compact manifolds and hyperbolicity. Trans. Amer. Math. Soc. 235 (1978), 213–219.Google Scholar
[3]Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition(Classics in Mathematics). Springer, Berlin, 2001.CrossRefGoogle Scholar
[4]Gromov, M.. Filling Riemannian manifolds. J. Differential Geom. 18 (1983), 1–147.CrossRefGoogle Scholar
[5]Gromov, M.. Topological invariants of dynamical systems and spaces of holomorphic maps: I. Math. Phys. Anal. Geom. 2 (1999), 323–415.CrossRefGoogle Scholar
[6]Kollár, J. and Mori, S.. Birational Geometry of Algebraic Varieties (Cambridge Tracts in Mathematics, 134). Cambridge University Press, Cambridge, 1998 (with the collaboration of C.H. Clemens and A. Corti).CrossRefGoogle Scholar
[7]Lindenstrauss, E.. Mean dimension, small entropy factors and an embedding theorem. Publ. Math. Inst. Hautes Études Sci. 89 (1999), 227–262.CrossRefGoogle Scholar
[8]Lindenstrauss, E. and Weiss, B.. Mean topological dimension. Israel J. Math. 115 (2000), 1–24.CrossRefGoogle Scholar
[9]McDuff, D. and Salamon, D.. J-holomorphic Curves and Quantum Cohomology (University Lecture Series, 6). American Mathematical Society, Providence, RI, 1994.CrossRefGoogle Scholar
[10]Tsukamoto, M.. A packing problem for holomorphic curves. Nagoya Math. J. to appear, arXiv: math.CV/0605353.Google Scholar
[11]Tsukamoto, M.. Macroscopic dimension of the ℓ p-ball with respect to the ℓ p-norm. J. Math. Kyoto Univ. 48(2) (2008), 445–454.Google Scholar
[12]Tsukamoto, M.. Moduli space of Brody curves, energy and mean dimension. Nagoya Math. J. 192, to appear, arXiv:0706.2981.CrossRefGoogle Scholar