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The \(M\)-set of \(\lambda \exp(z)/z\) has infinite area. (English) Zbl 1362.37094

The paper under review deals with the \(M\)-set of the family \(\lambda \exp(z)/z\), that is the set of all parameters \(\lambda\) so that the Fatou set of \(\lambda \exp(z)/z\) is empty.
It was shown by M. Misiurewicz [Ergodic Theory Dyn. Syst. 1, 103–106 (1981; Zbl 0466.30019)] that the Fatou set of the function \(z\mapsto \exp(z)\) is empty. This leads to consideration of the \(M\)-set of the family \(\lambda\exp(z)\). It was shown by W. Qiu [Acta Math. Sin., New Ser. 10, No. 4, 362–368 (1994; Zbl 0817.30011)] that the \(M\)-set for this family has Hausdorff dimension \(2\). The question on whether the \(M\)-set for this family has positive area is still open.
The authors consider here the \(M\)-set for the related family \(z\mapsto \lambda\exp(z)/z\). Their main result consists in proving that the \(M\)-set for this family has infinite area, and in particular it has Hausdorff dimension 2. They also show that the complement of this \(M\)-set has positive area.
The main techniques used are the Kœbe distortion theorem and a criterion developed by C. McMullen for constructing a nested intersection of dynamically defined sets with positive area [Trans. Am. Math. Soc. 300, 329–342 (1987; Zbl 0618.30027)].

MSC:

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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References:

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