Tangencies between holomorphic maps and holomorphic laminations

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 grant DMS-0801050. 1 For the case of holomorphic curves (n = 1) this result is contained in [1, Lemma 9.1] where it is credited to Douady. Artur Avila, in a conversation with the authors, proposed to extend this result to arbitrary holomorphic maps. According to Avila, this generalization has several applications to holomorphic dynamics [1]. Proof. We assume without loss of generality that f(0,w) ≡ 0, and that L0 is tangent to γ at z0 = 0. We have to show that other tangent leaves cannot accumulate to L0. Suppose the contrary, that is suppose that there is a sequence λk → 0 such that Lλk are tangent to γ, and let Lλk be the graphs of the functions fk(w) = f(λk,w). We may assume that tangency points zk → 0. We make several preliminary reductions. 1. Let γ(z) = (φ(z), ψ(z)) ∈W × C. Consider the new lamination in U×C whose leaves are the graphs of f(λ, z) = f(λ, φ(z)) and the new map γ(z) = (z, ψ(z)). Then γ is tangent to a leaf L if and only if γ is tangent to L. This reduces our problem to the case that W is an open set in C and the map γ is a graph of a function ψ of the same variable as the functions fk. From now on we assume that U =W and γ(w) = (w, ψ(w)). 2. Now we reduce the problem to the case that ψ is a monomial. For this we use the desingularization theorem of Hironaka [4, 2, 3, 6]. Let X be a complex analytic manifold, and ψ an analytic function on X. Then there exists a complex analytic manifold M and a proper surjective map π : M → X such that the restriction of π onto the complement of the π-preimage of the set {ψ = 0, ψ = 0} is injective and for each point z0 ∈ π ({ψ = 0}) there is a local coordinate system with the origin at z0 such that ψ ◦ π is a monomial z1 1 . . . z mn n . We choose such M corresponding to X =W . Let Y = W × C, and let S ⊂ Y be the set of points (w, t) with t 6= 0 such that the graph of t = ψ(w) is tangent to the lamination. In our proof by contradiction, we assume that the origin belongs to the closure of S. Let N = M × C, and let ρ : N → Y be the map defined by ρ(z, t) = (π(z), t).