Rational maps with half symmetries, Julia sets, and Multibrot sets in parameter planes

Dynamics of regularly ramified rational maps: Ⅰ. Julia sets of maps in one-parameter families

“…Julia sets of regularly ramified maps A • R Gj with 2 ≤ j ≤ 5 are first explored in [6] through computer-generated pictures. Each of those maps A • R Gj satisfies the following two assumptions:…”

“…Without loss of generality, one can arrange p at the origin and q at infinity. Such a map A • R Gj is called a normalized projection map in [6], which is now called in this paper a normalized regularly ramified rational map and is briefly denoted by A • R Gj .…”

“…Section 4 is the last section. The first half considers two one-parameter families of maps of form A • R G5 or A • R G5 • φ explored in [6] and states, without proofs, three main results: nonexistence of Herman rings, a tetrachotomy of the Julia sets of the maps in one family, and a trichotomy of the Julia sets of the maps in another family. The second half constructs two one-parameter families of maps of form A • R G3 or A • R G3 • φ satisfying the two assumptions given in this section.…”

“…Our computer-generated pictures of Julia sets indicate that the family f λ (2,ν) (z) behaves very similarly to F ν c (z) when ν > 2 and when ν = 2. It is given in [6]…”

Symmetries of Julia sets for analytic endomorphisms of the Riemann sphere

“…This sudden, quickly reversed change at |λ| = 1 when m = d can, in fact, be understood as the transversal movement of half-symmetries of R λ through I( C) in PSL(2, C). In [HJM12], the half-symmetries of a rational map R were defined as the Möbius transformations γ ∈ PSL(2, C) satisfying Rγ = R -i.e., they permute the fibres of R. It is readily seen that these form a group G(R) ⊂ PSL(2, C), and that γ[J(R)] = J(R) for any γ ∈ G(R). It was proved by Hu and his co-workers that G(R) is always a finite group, and that it is conjugate to a group of isometries (see Theorem 4.1.1).…”