Recent zbMATH articles in MSC 30D05https://www.zbmath.org/atom/cc/30D052022-05-16T20:40:13.078697ZWerkzeugEscaping Fatou components of transcendental self-maps of the punctured planehttps://www.zbmath.org/1483.370542022-05-16T20:40:13.078697Z"Martí-Pete, David"https://www.zbmath.org/authors/?q=ai:marti-pete.davidHolomorphic self-maps of the Riemann sphere are the most well-studied families of systems in complex dynamics (rational dynamics), followed by self-maps of the once punctured plane (transcendental dynamics). This paper concerns the study of holomorphic self-maps of the twice punctured Riemann sphere, a subject still in its early days.
In transcendental dynamics, an important role is played by the escaping set, which consists of those points which iterate to the essential singularity of the function. In the present setting, however, there are two essential singularities. The main result is that any possible way of escaping is possible for a wandering Fatou component. The author's main technique is approximation theory.
Reviewer: Kirill Lazebnik (New Haven)Exponential polynomials with Fatou and non-escaping sets of finite Lebesgue measurehttps://www.zbmath.org/1483.370552022-05-16T20:40:13.078697Z"Wolff, Mareike"https://www.zbmath.org/authors/?q=ai:wolff.mareikeSummary: We give conditions ensuring that the Fatou set and the complement of the fast escaping set of an exponential polynomial \(f\) both have finite Lebesgue measure. Essentially, these conditions are designed such that \(|f(z)|\geq \exp (|z|^\alpha)\) for some \(\alpha>0\) and all \(z\) outside a set of finite Lebesgue measure.Asymptotic upper bound for tangential speed of parabolic semigroups of holomorphic self-maps in the unit dischttps://www.zbmath.org/1483.370572022-05-16T20:40:13.078697Z"Cordella, Davide"https://www.zbmath.org/authors/?q=ai:cordella.davideThe author studies continuous semigroups of holomorphic maps in the unit disc \((\Phi_t)_{t\ge 0}\). For a non-elliptic semigroup, \textit{F. Bracci} [Ann. Univ. Mariae Curie-Skłodowska, Sect. A 73, No. 2, 21--43 (2019; Zbl 1436.30007)] introduced and studied three kinds of speeds: the total speed, the orthogonal speed, and the tangential speed. The tangential speed \(v^T(t)\) is related to the slope of convergence of orbits to the Denjoy-Wolff point of the semigroup. In the present paper, the author proves a conjecture in [loc. cit.] claiming that \(\limsup_{t\to\infty}\left(v^T(t)-\frac 12\log t\right)<\infty\) holds for parabolic semigroups.
Reviewer: Barbara Drinovec Drnovsek (Ljubljana)