Counting hyperbolic components

Abstract

We give formulae for the numbers of type II and type IV hyperbolic components in the space of quadratic rational maps, for all fixed periods of attractive cycles.

Persistent Markov partitions for rational maps

“…The family of such maps, quotiented by Möbius conjugation, is of complex dimension one, and is well known to have no finite singular points. (See, for example, Theorem 2.5 of [9].) So V , or a natural quotient of it, is a Riemann surface, with some punctures at ∞, where the degree of the map degenerates.…”

“…We use certain facts about the ends of V . These appear in Stimson's thesis [17] and in various other papers, for example [9]. Choosing suitable representatives of g k up to Möbius conjugation,chosen, in particular, so that c 1 (g k ) = 1 for all k, g k converges to a periodic Möbius transformation g(z) = e 2πir/q z for some integer q ≥ 2 and some r ≥ 1 which is coprime to q, and the set {g i k (v 1 (g k )) : i ≥ 0}∪{v 2 (g k )} = Z 1 (g k ) converges Z 1 (g) = {e 2πij/q : 0 ≤ j ≤ q−1}.…”

On the geometry of bifurcation currents for quadratic rational maps

“…where ν q (n) ∼ 2 n−1 /(2 q − 1) and ν q (n) ≥ 2 n−1 /(2 q − 1) − 1/2 (see [KR,Theorem 1.1]). A standard transversality statement asserts that this number actually coincides with Let us now proceed by contradiction, assuming that µ ∞ = 0.…”