Stable Manifolds of Holomorphic Diffeomorphisms
نویسنده
چکیده
Problem: Determine the complex structure of the stable manifolds of f . It is not hard to see, using f(W s p ) = W s fp, that W s p is a monotone union of balls, and this in turn implies [Br] that it is diffeomorphic to real Euclidean space. Moreover, by the contracting nature of the dynamics, one sees that the Kobayashi pseudometric of W s p vanishes identically. However, when dim(W s p ) ≥ 3, it is not possible to deduce only from these properties that W s p is biholomorphic to Euclidean space. Indeed, there exist monotone unions of balls which are not Stein [F]. (The question of Steinness of monotone unions of balls in complex dimension 2 is open.) Of course, when dim(W s p ) = 1, the Uniformization Theorem implies that W s p is biholomorphic to C. The main results of this paper are proved in the non-uniform setting, i.e., with respect to compactly supported invariant measures. More precisely, we say that a subset A ⊂ M is invariant if fA = A, and that it has total measure if μ(A) = 1 for every compactly supported invariant probability measure μ. Our main objective in this paper is to prove the following theorem.
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