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      A sharp growth condition for a fast escaping spider's web

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          Abstract

          We show that the fast escaping set \(A(f)\) of a transcendental entire function \(f\) has a structure known as a spider's web whenever the maximum modulus of \(f\) grows below a certain rate. We give examples of entire functions for which the fast escaping set is not a spider's web which show that this growth rate is best possible. By our earlier results, these are the first examples for which the escaping set has a spider's web structure but the fast escaping set does not. These results give new insight into a conjecture of Baker and a conjecture of Eremenko.

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          Iteration of meromorphic functions

          Walter Bergweiler (1993)
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            Dynamics of meromorphic functions with direct or logarithmic singularities

            , , (2008)
            We show that if a meromorphic function has a direct singularity over infinity, then the escaping set has an unbounded component and the intersection of the escaping set with the Julia set contains continua. This intersection has an unbounded component if and only if the function has no Baker wandering domains. We also give estimates of the Hausdorff dimension and the upper box dimension of the Julia set of a meromorphic function with a logarithmic singularity over infinity. The above theorems are deduced from more general results concerning functions which have "direct or logarithmic tracts", but which need not be meromorphic in the plane. These results are obtained by using a generalization of Wiman-Valiron theory. The method is also applied to complex differential equations.
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              On a question of Eremenko concerning escaping components of entire functions

              Lasse Rempe (2006)
              Let f be an entire function with a bounded set of singular values, and suppose furthermore that the postsingular set of f is bounded. We show that every component of the escaping set I(f) is unbounded. This provides a partial answer to a question of Eremenko.
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                Author and article information

                Journal
                Publication date Created: 16 August 2012
                Article
                ArXiV ID: 1208.3371
                SO-VID: 5eca0021-c519-4a9a-bf25-15fbd2f5a76a
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                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                ACM classes 30D05, 37F10
                Categories math.DS math.CV

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