Monotonicity of entropy for real quadratic rational maps

On Entropy and Monotonicity for Real Cubic Maps

Topological Models for Some Quadratic Rational Maps

Consider a quadratic rational self-map of the Riemann sphere such that one critical point is periodic of period 2, and the other critical point lies on the boundary of its immediate basin of attraction. We will give explicit topological models for all such maps. We also discuss the corresponding parameter picture. Stony Brook IMS Preprint #2007/1 February 2007

M ay 2 00 9 Monotonicity of entropy for real multimodal maps ∗

In [16], Milnor posed the Monotonicity Conjecture that the set of parameters within a family of real multimodal polynomial interval maps, for which the topological entropy is constant, is connected. This conjecture was proved for quadratic by Milnor & Thurston [17] and for cubic maps by Milnor & Tresser, see [18] and also [5]. In this paper we will prove the general case.

The Moduli Space of Quadratic Rational Maps

Let M2 be the space of quadratic rational maps f : P 1 → P , modulo the action by conjugation of the group of Möbius transformations. In this paper a compactification X of M2 is defined, as a modification of Milnor’s M 2 ' CP , by choosing representatives of a conjugacy class [f ] ∈ M2 such that the measure of maximal entropy of f has conformal barycenter at the origin in R, and taking the clos...

On the geometry of bifurcation currents for quadratic rational maps

We describe the behaviour at infinity of the bifurcation current in the moduli space of quadratic rational maps. To this purpose, we extend it to some closed, positive (1, 1)-current on a two-dimensional complex projective space and then compute the Lelong numbers and the self-intersection of the extended current.