On the Hausdorff dimension of the escaping set of certain meromorphic functions

Abstract

The escaping set I(f) of a transcendental meromorphic function f consists of all points which tend to infinity under iteration. The Eremenko-Lyubich class B consists of all transcendental meromorphic functions for which the set of finite critical and asymptotic values of f is bounded. It is shown that if f is in B and of finite order of growth, if infinity is not an asymptotic value of f and if the multiplicities of the poles of f are uniformly bounded, then the Hausdorff dimension of I(f) is strictly smaller than 2. In fact, we give a sharp bound for the Hausdorff dimension of I(f) in terms of the order of f and the bound for the multiplicities of the poles. If f satisfies the above hypotheses but is of infinite order, then the area of I(f) is zero. This result does not hold without a restriction on the multiplicities of the poles.

23 pages

References in corpus (1)

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

Cited by in corpus (13)

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• Absence of line fields and Mane's theorem for non-recurrent transcendental functions
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• The escaping set in transcendental dynamics
• Dynamics of meromorphic functions outside a countable set of essential singularities
• The Hausdorff dimension of escaping sets of meromorphic functions in the Speiser class
• Hausdorff dimension of escaping sets of meromorphic functions
• Dynamics of Generalized Nevanlinna Functions
• Hausdorff measure of escaping sets on certain meromorphic functions
• Dimensions of non-autonomous meromorphic functions of finite order
• Hausdorff dimension of escaping sets of Nevanlinna functions
• The exact value of Hausdorff dimension of escaping sets of class B meromorphic functions