Foliation Structure for Generalized Henon Mappings.

Complex Dynamics;Foliation;Hyperbolicity;Brody Curve;Mathematics;Science;Mathematics

Ahn, TaeyongYoung, Virginia R. ;

University of Michigan

关键词: Complex Dynamics;    Foliation;    Hyperbolicity;    Brody Curve;    Mathematics;    Science;    Mathematics;   

Others  :  https://deepblue.lib.umich.edu/bitstream/handle/2027.42/93830/taeyonga_1.pdf?sequence=1&isAllowed=y

瑞士|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

We consider a generalized Henon mapping f from the two dimensional complex plane to itself defined by f(z, w)=(p(z)-aw, z) where p(z) is a monic polynomial of degree greater than or equal to 2 and a is not equal to 0, and its Green function g of the two dimensional complex plane to the set of all real numbers. We show that there is no non-trivial holomorphic curve in the two dimensional projective space passing through I_+ and supported in the closure of {g=c} for c>0, where I_+ is a point of indeterminancy when f is projectivised. Next, we consider the foliation structure of a level set of the form {g=c} for c>0. It is known that each of them is foliated by conformal images of the set of all complex numbers and each leaf is dense. We prove each leaf is actually an injective Brody curve. More precisely, for any biholomorphic parametrization of any leaf, the Fubini-Study metric of its derivative is uniformly bounded.