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On the Hausdorff dimension of the Julia set of a regularly growing entire function. (English) Zbl 1198.30027

Let \(f\) be an entire function and let \(f_{1} = f, f_{2} = f \circ f,\) and \(f_{n} = f \circ f_{n-1}\). The Julia set \(J(f)\) of an entire function \(f\) is the set of points at which the family \(\{f_{n}: 1 \leq n < \infty \}\) is not a normal family. The escaping set \(I(f)\) for \(f\) is the set \(\{z: f_{n}(z) \to \infty\}\). Let \(M(r,f) = \sup{\{|f(z)|: |z| \leq r\}}\). The function \(f\) is said to satisfy the condition \((*)\) if there exist constants \(A, B, C\), and \(r_{0} > 1\) such that \[ A \log{M(r,f)} \leq \log{M(Cr,f)} \leq B \log{M(r,f)} \;\;\;\;\;\text{for } r \geq r_{0}. \] The authors prove that if \(f\) is an entire function satisfying condition \((*)\), then the Hausdorff dimension of the set \( I(f) \cap J(f)\) is 2. For non-negative constants \(\alpha_{1}, \alpha_{2}, q,\lambda\), and an entire function \(f\), let \(T(\alpha_{1}, \alpha_{2}, q, \lambda, f)\) be the set of all \(z\) such that all of the conditions \[ \alpha_{1} \log{M(|z|,f)} \leq \bigg|\frac{z f'(z)} {f(z)}\bigg| \leq \alpha_{2} \log{M(|z|,f)}, \]
\[ |f(z)| \geq |z|^{q}, \] and \[ \bigg|\frac {\zeta f'(\zeta)} {f(\zeta)}\bigg| \leq \alpha_{2} \log{M(|\zeta|, f)} \;\;\text{ for } \;|\zeta - z| < \lambda \frac {|z|}{\log{M(|z|, f)}} \] are satisfied. For measurable sets \(X\) and \(Y\), define the density of \(X\) in \(Y\) as \[ \text{dens}(X,Y) = \frac {\text{area}(X \cap Y)} {\text{area}(Y)} \;. \] Also, for \(R > 0\), let \(A(R) = \{z: R \leq |z| \leq 2R\}\). The authors prove that if \(f\) is an entire function satisfying condition \((*)\), then there exist positive constants \(\alpha_{1}, \alpha_{2}\), and \(\eta\) such that if \(q\) and \(\lambda\) are positive numbers, then dens\((T(\alpha_{1}, \alpha_{2}, q, \lambda, f), A(R)) > \eta\) for sufficiently large \(R\). These results are related to results of K. Baranski [Math. Proc. Camb. Philos. Soc. 145, No. 3, 719–737 (2008; Zbl 1162.30013)] and H. Schubert [Dissertation, University of Kiel, (2007)].

MSC:

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems

Citations:

Zbl 1162.30013
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References:

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