Tangencies between holomorphic maps and holomorphic laminations
نویسندگان
چکیده
We prove that the set of leaves of a holomorphic lamination of codimension one that are tangent to a germ of a holomorphic map is discrete. Let F be a holomorphic lamination of codimension one in an open set V in a complex Banach space B. In this paper, this means that V = W × C, where W is a neighborhood of the origin in some Banach space, and the leaves Lλ of the lamination are disjoint graphs of holomorphic functions w 7→ f(λ,w), W → C. For holomorphic functions in a Banach space we refer to [5]. Here λ is a parameter and we assume that the dependence of f on λ is continuous. A natural choice of this parameter is such that λ = f(λ, 0), in which case the continuity with respect to λ follows from the so-called λlemma of Mane-Sullivan-Sad and Lyubich, see, for example [5]. With this choice of the parameter, our definition of a lamination coincides with that of a holomorphic motion of C parametrized by W . Let γ : U → V be a holomorphic map, U ⊂ C. We say that γ is tangent to the lamination at a point z0 ∈ U if the image of the derivative γ(z0) is contained in the tangent space TL(γ(z0)), where L is the leaf passing through γ(z0). A leaf for which this holds is called a tangent leaf to γ. Theorem. Let K be a compact subset of U . Then the set of leaves tangent to γ at the points of K is finite. ∗Supported by NSF grant DMS-0555279. †Supported by the NSF.
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