Annales Academić Scientiarum Fennicć
Mathematica
Volumen 38, 2013, 115-140

MARTINGALES FOR QUASISYMMETRIC SYSTEMS AND COMPLEX MANIFOLD STRUCTURES

Yunchun Hu, Yunping Jiang and Zhe Wang

CUNY Graduate Center, Department of Mathematics
New York, NY 10016, U.S.A.; yhu 'at' ccny.cuny.edu

CUNY Queens College and Graduate Center, Department of Mathematics
Flushing, NY 11367, U.S.A.; Yunping.Jiang 'at' qc.cuny.edu

Bronx Community College, Department of Mathematics and Computer Science
New York, NY 10453, U.S.A.; wangzhecuny 'at' gmail.com

Abstract. We construct an infinite martingale sequence on the dual symbolic space from a uniformly quasisymmetric circle endomorphism preserving the Lebesgue measure. This infinite martingale sequence is uniformly bounded. Thus from the martingale convergence theorem, there is a limiting martingale which is the unique L¹ limit of this uniformly bounded infinite martingale sequence. Moreover, we prove that the classical Hilbert transform gives an almost complex structure on the space of all uniformly quasisymmetric circle endomorphisms preserving the Lebesgue measure. Furthermore, we discuss the complex manifold structure which is the integration of the almost complex structure. We further discuss the comparison between the global Kobayashi's metric and the global Teichmüller metric on the fiber of the forgetful map at the basepoint. We prove that these two metrics are not equivalent.

2010 Mathematics Subject Classification: Primary 37E10, 37F15, 37F30, 30F60.

Key words: Martingale, quasisymmetric homeomorphism, uniformly quasisymmetric circle endomorphism, Hilbert transform.

Reference to this article: Y. Hu, Y. Jiang and Z. Wang: Martingales for quasisymmetric systems and complex manifold structures. Ann. Acad. Sci. Fenn. Math. 38 (2013), 115-140.

Full document as PDF file

doi:10.5186/aasfm.2013.3812