Absence of line fields and Mane's theorem for non-recurrent transcendental functions

Abstract

Let f be a transcendental meromorphic function. Suppose that the finite part of the postsingular set of f is bounded, that f has no recurrent critical points or wandering domains, and that the degree of pre-poles of f is uniformly bounded. Then we show that f supports no invariant line fields on its Julia set. We prove this by generalizing two results about rational functions to the transcendental setting: a theorem of Mane about the branching of iterated preimages of disks, and a theorem of McMullen regarding absence of invariant line fields for "measurably transitive" functions. Both our theorems extend results previously obtained by Graczyk, Kotus and Swiatek.

28 pages; V3. Proof of Theorem 7.4 corrected, as well as some other minor corrections

References in corpus (3)

• On the Hausdorff dimension of the escaping set of certain meromorphic functions
• A note on hyperbolic leaves and wild laminations of rational functions
• Linearization problem on structurally finite entire functions

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• On invariance of order and the area property for finite-type entire functions
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• Quasisymmetric rigidity in one-dimensional dynamics
• Density of hyperbolicity for classes of real transcendental entire functions and circle maps
• A note on hyperbolic leaves and wild laminations of rational functions
• Complex box bounds for real maps
• Rigidity and absence of line fields for meromorphic and Ahlfors islands maps
• The escaping set in transcendental dynamics
• No entire function with real multipliers in class S
• Speiser meets Misiurewicz