Non-ergodic maps in the tangent family (Q1433186)

For a meromorphic function \(f\) denote by \(J(f)\) the Julia set and by \(P(f)\) the closure of the set of forward orbits of the singularities of the inverse of \(f.\) It was shown by \textit{H. Bock} [Über das Iterationsverhalten meromorpher Funktionen auf der Juliamenge, PhD Thesis, Aachen (1998; Zbl 0990.30016)] that at least one of the following two statements holds: (i) \(\lim_{n\rightarrow\infty} \text{dist} (f^n(z), P(f))=0\) for almost all \(z\in J(f);\) (ii) \(J(f)=\overline{\mathbb{C}}\) and for all sets \(A\) of positive measure the set \(\{n\in \mathbb{N}: \; f^n(z)\in A\}\) is infinite for almost all \(z\in J(f).\) Here dist\((\cdot , \cdot)\) denotes spherical distance. In case (ii) the function\(f\) is recurrent and ergodic, and \(\omega(z)=\overline{\mathbb{C}}\) for almost all \(z\in \mathbb{C},\) where \(\omega(z)\) is the \(\omega\)-limit set. The present paper is concerned with the functions \(f_\lambda (z)=\lambda \tan z\) where the parameter \(\lambda\) is chosen such that \(f_\lambda^p(\pm \lambda i)=\infty\) for some \(p\in \mathbb{N}.\) Parameters \(\lambda\) with this property play an important role in the description of the parameter space of the tangent family given by \textit{L. Keen} and \textit{L. Kotus} [Conform. Geom. Dyn. 1, 28--57 (1997; Zbl 0884.30019)]. For such parameters \(\lambda\) it follows that \(J(f_\lambda)=\overline{\mathbb{C}}.\) It follows from further results of H. Bock [loc. cit.] that for such \(\lambda\) statement (i) of the above theorem holds and hence that \(\omega(z)\subset P(f_\lambda)\) for almost all \(z\in \mathbb{C}.\) In the paper under review this result is strengthened by showing that in fact \(\omega(z)=P(f_\lambda)\) for almost all \(z\in \mathbb{C}.\) Moreover, it is proved that \(f\) is not ergodic and that there does not exist an \(f_\lambda-\)invariant measure which is absolutely continuous with respect to Lebesgue measure and which is finite on compact subsets of \(\mathbb{C}.\)