Martingales for quasisymmetric systems and complex manifold structures

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 L 1 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 Teichmuller metric on the fiber of the forgetful map at the basepoint. We prove that these two metrics are not equivalent.