Measurable dynamics of the exponential map
Consider the exponential map f: \(z\mapsto \exp z\) of the complex plane and denote by \(f^ n\) its n-th iterate. The purpose of this paper is to consider the characteristic Lebesgue measure properties of the dynamical system, determined by the trajectories of \(\{f^ n\}\). The main results are: (1) The exponential transformation of the complex plane is not ergodic. Every ergodic component has Lebesgue measure zero. (This answers a question from \textit{D. S. Sullivan}'s lecture notes [Seminar on conformal and hyperbolic geometry, Bures-sur-Yvette, IHES, 1982, preprint, p. 45].); (2) The exponential transformation of the complex plane has wandering domains of positive Lebesgue measure; (3) The exponential transformation of the complex plane doesn't have absolutely continuous invariant measure, finite on compacta. (This gives a partial answer to a question of \textit{E. Ghys}, \textit{L. R. Goldberg}, and \textit{D. P. Sullivan} [Ergodic Theory Dyn. Syst. 5, 329-335 (1985; Zbl 0616.58006)].) The author also gives a detailed description of the behaviour of a typical trajectory of the exponential transformation, in particular, he shows that it is not recurrent.
scientific article; zbMATH DE number 4092410
scientific article