Antipode preserving cubic maps: the Fjord theorem

Abstract

This note will study a family of cubic rational maps which carry antipodal points of the Riemann sphere to antipodal points. It focuses particularly on the fjords, which are part of the central hyperbolic component but stretch out to infinity. These serve to decompose the parameter plane into subsets, each of which is characterized by a corresponding rotation number.

Invisible tricorns in real slices of rational maps

“…For the family A 3 , we prove the following result which confirms a conjecture of Buff, Bonifant, and Milnor [3,Remark 6.12].…”

“…Part 2 of the paper deals with the family A 3 . This family has been extensively studied in [4,3], and the proofs of most of the basic results about this family that we will have need for can be found in their work. We briefly recall the various types of hyperbolic components of A 3 in Section 8.…”

“…It follows that the family A 3 := {f q : q ∈ C} is bicritical. The family A 3 was introduced by Bonifant, Buff, and Milnor in [4].…”

“…Figure 1 shows a Tricorn component (in black). 4 The following proposition follows from the general theory of hyperbolic components of rational maps [20,Theorem 7.13 Let us take a more careful look at the dynamics of the maps in a Tricorn component, which are the main objects of study of this article. Let a belong to a Tricorn component H of period 2n.…”

Parabolic arcs of the multicorns: Real-analyticity of Hausdorff dimension, and singularities of $\mathrm{Per}_n(1)$ curves

“…Proof. (1) =⇒ (2). The critical points of any P ∈ F d are 0 and the d roots of the equation z d + a = 0.…”