Rational Misiurewicz Maps Are Rare Ii

The notion of Misiurewicz maps has its origin from the paper [10] from 1981 by M. Misiurewicz. The (real) maps studied in this paper have, among other things, no sinks and the omega limit set ω(c) of every critical point c does not contain any critical point. In particular, in the quadratic family fa(x) = 1 − ax 2, where a ∈ (0, 2), a Misiurewicz map is a non-hyperbolic map where the critical point 0 is non-recurrent. D. Sands showed in 1998 [14] that these maps has Lebesgue measure zero, answering a question posed by Misiurewicz in [10]. In this paper we state a corresponding theorem for rational maps on the Riemann sphere. From another viewpoint, in 2003 [19], S. Zakeri showed that the Hausdorff dimension of the set of Misiurewicz maps is full, i.e. equal to one in the quadratic family. Conjecturally, a similar statement holds in higher dimensions too. In the complex case, there have been some variations on the definition of Misiurewicz maps. (The sometimes used definition of being a postcritically finite map is far too narrow to adopt here.) In [17], S. van Strien studies Misiurewicz maps with a definition similar to the definition in [10], (allowing super-attracting cycles but no sinks). In [6] by J. Graczyk, G. Światek, and J. Kotus, a Misiurewicz map is roughly a map for which every critical point c has the property that ω(c) does not contain any critical point, (allowing sinks but not super-attracting cycles). In this paper we allow attracting cycles, and only care about critical points on the Julia set (suggested by J. Graczyk), hence generalising the earlier version [1]. Parabolic cycles are ruled out for technical reasons. Let f(z) be a rational function of a given degree d ≥ 2 on the Riemann sphere Ĉ. Let Crit(f) be the set of critical points of f , J(f) the Julia set of f and F (f) the Fatou set of f .