A New Characterization of the $F$ Set of a Rational Function

• Published in: Transactions of the American Mathematical Society 1972-04, Vol.166, p.297-308, Article 297
• Main Authors: Oba, Marilyn K., Pitcher, Tom S.
• Format: Article
• Language: English
• Subjects: Borel sets; Continuous functions; Ergodic theory; Harmonic functions; Log integral function; Mathematical theorems; Polynomials; Rational functions; Riemann manifold; Uniqueness
• ISSN: 0002-9947; 1088-6850
• DOI: 10.2307/1996050

In the early part of this century G. Julia and P. Fatou extensively studied the iteration of functions on the complex plane. More recently Hans Brolin reopened the investigation. In this paper, we are interested in the $F$ set which is the set of points at which the family of iterates of a given rational function $R$ is not normal and in a measure which is in some sense naturally imposed on the $F$ set by the iterates of $R$. We construct a sequence of probability measures via the inverse functions of the iterates of $R$ and almost any starting point. The measure of primary interest is the weak limit of such sequences. These weak limits are supported by $F$ and have certain invariance properties. We establish that this weak limit measure is unique and is ergodic with respect to the transformation $R$ on the $F$ set for a large class of rational functions. In the course of the proof of uniqueness we develop expressions for the logarithmic potential function and for the energy integral of $F$. We also establish inequalities for the capacity of the $F$ set which become equalities for the polynomial case.