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Non-existence of absolutely continuous invariant probabilities for exponential maps. (English) Zbl 1167.37024
The author considers the problem of existence of invariant measures for entire maps, in particular for exponential maps. The author shows that for entire maps of the form \(z \mapsto \lambda \exp(z)\) such that its Julia set is the whole plane, the forward orbit of \(0\) is bounded and Lebesgue almost very point is transitive, no absolutely continuous invariant probability measure can exist. The proof uses an important method of J. Rivera-Letelier’s construction of nice sets for rational dynamics [see Ergodic Theory Dyn. Syst. 27, No. 2, 595–636 (2007; Zbl 1110.37037)], which the author adapts to the setting of entire maps. The author first constructs a nice set on which the density of a hypothetical measure must be bounded away from zero, and then shows that the return time to the nice set is not integrable.

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37A10 Dynamical systems involving one-parameter continuous families of measure-preserving transformations
37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
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