On the iteration behaviour of meromorphic functions on the Julia set (Q2762223)

Let \(f\) be a rational function of degree at least \(2\) or a transcendental meromorphic function in the plane, and denote by \(f^n\) the \(n\)-th iterate of \(f\). The Fatou set \(\mathcal{F}(f)\) is the maximal open set in which all iterates \(f^n\) are defined and form a normal family in the sense of Montel, while the complement of \(\mathcal{F}(f)\) in \(\widehat{\mathbb C}\) is the Julia set \(\mathcal{J}(f)\). Because of the normality of the sequence \((f^n)\) in components of \(\mathcal{F}(f)\) the dynamics of \((f^n)\) in \(\mathcal{F}(f)\) is well understood. On the other hand, the behaviour of a point in \(\mathcal{J}(f)\) under iteration is more complicated and it may change significantly under small perturbations. For example, every neighbourhood of a point in \(\mathcal{J}(f)\) contains both periodic points and points whose forward orbit is dense in \(\mathcal{J}(f)\). NEWLINENEWLINENEWLINENevertheless, in many cases the forward orbits of points in \(\mathcal{J}(f)\) have a typical behaviour in the sense that the set of points whose forward orbit has a different behaviour has measure zero. In order to state the main result of this thesis some notations are needed. The closure of the forward orbit of the set of singularities of \(f^{-1}\) is called the postsingular set and is denoted by \(P(f)\). If \(z\) is a point which belongs to the domain of definition of each iterate \(f^n\), the set of all cluster points of the sequence \((f^n(z))\) is denoted by \(\omega(z)\). For a set \(A \subset \widehat{\mathbb C}\) and a point \(z \in \widehat{\mathbb C}\), let \(\text{dist}_\chi{(z,A)}\) be the distance of \(z\) to \(A\) in the chordal metric \(\chi\). Then the main theorem reads as follows. NEWLINENEWLINENEWLINETheorem. Let \(f\) be a rational function of degree at least \(2\) or a transcendental meromorphic function. Then at least one of the following statements hold: (1) \(\lim_{n\to\infty}{\text{dist}_\chi{(f^n(z),P(f))}}=0\) for almost all \(z \in \mathcal{J}(f)\). (2) \(\mathcal{J}(f)=\widehat{\mathbb C}\), and for every set \(A \subset \widehat{\mathbb C}\) of positive measure the set \(\{ n \in \mathbb N : f^n(z) \in A \}\) is infinite for almost all \(z \in \widehat{\mathbb C}\). From this result it follows that \(\omega(z) \subset P(f)\) for almost all \(z \in \mathcal{J}(f)\) or \(\omega(z) = \widehat{\mathbb C}\) for almost all \(z \in \widehat{\mathbb C}\). NEWLINENEWLINENEWLINEIn general, it may be difficult to decide which of these two possibilities applies to a given function \(f\). From a result of \textit{M.~Yu.~Lyubich} [Sov. Math. Dokl. 27, 22-25 (1983; Zbl 0595.30034)] it follows that if \(f\) is a rational function of degree at least \(2\) whose Fatou set is not empty, then statement (1) holds. It may also occur that (1) holds for functions whose Julia set is \(\widehat{\mathbb C}\). This is the case for \(f=\exp\) as \textit{M.~Yu.~Lyubich} [Russ. Math. Surv. 41, No. 2, 207-208 (1986; Zbl 0617.30032)] and \textit{M.~Rees} [Math. Z. 191, 593-598 (1986; Zbl 0595.30033)] have shown independently. NEWLINENEWLINENEWLINEThe author gives further conditions which help to decide whether (1) or (2) holds. If there is some \(n \in \mathbb N\) such that the set \(I_n(f):=\{ z \in \mathbb C : \lim_{m\to\infty}{f^{mn}(z)}=\infty\}\) has positive measure then (1) holds. The author also gives a sufficient condition for \(I_n(f)\) to have positive measure. This allows to treat the example \(f(z) = \tan{(\pi iz/2)}\), where \(\text{meas}{I_1(f)} = 0\), \(\text{meas}{I_2(f)} > 0\) and \(\omega(z) \subset \{i,-i,\infty\}\) for almost all \(z \in \mathbb C\). NEWLINENEWLINENEWLINEOn the other hand, if the set of singularities of \(f^{-1}\) is finite, and if each singularity is preperiodic but not periodic, then condition (2) holds. For example, this is the case for the functions \(f(z)=(z-2)^2/z^2\) and \(f(z)=\tan{\pi iz}\). Note that a transcendental entire function cannot satisfy the assumptions of the latter criterion.