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On typical behavior of the trajectories of a rational mapping of the sphere. (English. Russian original) Zbl 0595.30034

Sov. Math., Dokl. 27, 22-25 (1983); translation from Dokl. Akad. Nauk SSSR 268, 29-32 (1983).
Let f be a rational map of the Riemann sphere and let \(\Omega\) (f) be its nonwandering set. A point z is regular for f if the family \(\{f^ m\}_{m\geq 0}\) is normal in some neighbourhood of z. Then: (a) Any invariant measure with positive characteristic exponents is singular with respect to Lebesgue measure. (b) If the set of regular points is nonempty then any invariant ergodic measure of positive entropy is singular with respect to Lebesgue measure. (c) For polynomials of degree two, the Lebesgue measure of \(\Omega\) (f) is generically zero.
The proofs (given in outline only) use Fatou’s classification of points of the sphere [P. Fatou, Bull, Soc. Math. France 47, 161-271 (1919); ibid. 48, 33-94 (1920); ibid. 48, 208-314 (1920)] and the principle that the behaviour of almost all points whose orbits never enter Int \(\Omega\) (f) (so-called points of the first kind) is governed by the behaviour of the critical points. Some of the same ideas are to be found in a paper by the author [Ergodic Theory, Dyn. Syst. (to appear)]. (Note that the reference given by the author is incorrect.)

MSC:

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37A99 Ergodic theory
28D99 Measure-theoretic ergodic theory

Keywords:

rational map