geometrically finite rational maps

A rational map is called *geometrically finite* if every critical point in the Julia set is eventually periodic. In my paper
*
"Semiconjugacies between the Julia sets of geometrically finite rational maps"
* (in Erg. Th. and Dyn. Sys. **23**(2003)), the following is shown:
*For suitable perturbations of a geometrically finite rational map, there exist conjugacies or "alomost injective" semiconjugacies between the dynamics on the perturbed Julia sets and the original one.*

Here is an example in polynomial case: First, let us consider a geometrically finite polynomial,

f(z)=z(1+z)(1+z)(1-3z+1.75zz).

This polynomial has the following properties:

1. z=0 is a parabolic fixed point with one petal.

2. z=-1 is one of two critical points in the Julia set, and f(-1)=0.

3. z=1 is another critical point in the Julia set, and f(1)=-1, thus f(f(1))=0.

4. z=-0.30888.. is an attracting fixed point.

The following picture shows the filled Julia set of f:

The region outside (colored in gradation from white to light gray) is the immediate basin of infinity. The gray region is the attracting basin of z=-0.30888... The region with cauliflower-like part (colored in gradation from black to dark gray) is the parabolic basin of z=0. The Julia set J(f) of f is the boundary between these three regions.

Next, let us consider the family of polynomials parameterized by a complex number C,

F(z,C):=z(1+z)(1+z)(C-(2C+1)z+(3+4C)zz/4).

Then for any C, F(z,C) has the following properties:

1. z=0 is a fixed point with multiplier C.

2. z=-1 is a critical point and F(-1,C)=0.

3. z=1 is another critical point, and F(1,C)=-1, thus F(F(1,C),C)=0.

Thus f(z) is the special case of C=1, that is,

f(z)=F(z,1).

So changing C of F(z,C) near C=1 gives a perturbation of f(z).

Set f1(z):=F(z,1.05). Then the Julia set of f1(z) is as following:

In this case, z=0 is perturbed into a repelling fixed point with multiplier 1.05. The original parabolic point splits into a pair of repelling and attracting fixed points. Here are the pictures near 0, centered at 0, and with critical orbits drawn in.

J(f) near z=0 (original)
J(f1) near z=0 (perturbed)

However, the topology ("shape") of the Julia set is not changed. By the main theorem in the paper, there exists a topological conjugacy h1: J(f1) -> J(f); that is, the dynamics on the both Julia sets are essentially the same.

Set f2(z):=F(z,0.95). Then the Julia set of f2(z) is as following:

In this case, z=0 is perturbed into an attracting fixed point with multiplier 0.95. The original parabolic point again splits into a pair of repelling and attracting fixed points.

J(f2) near z=0, centered at 0.

Now the topology of the Julia set is slightly changed. Indeed, J(f) and J(f1) contain z=-1 and z=1 (critical points) but J(f2) does not. z=-1 and z=1 land on z=0, which is now perturbed into an attracting point.

However, by the main theorem in the paper, there exists a semiconjugacy h2: J(f2) -> J(f). Moreover, except at the preimages (backward orbits) of z=0 of f, h2 gives a topological conjugacy. That is, the dynamics on the both Julia sets are "almost" the same.

Set f3(z):=F(z,1.02+0.05i). Then the Julia set of f3(z) is as following:

In this case, as in Case 1, z=0 is perturbed into a repelling fixed point (with multiplier 1.02+0.05i). The original parabolic point also splits into a pair of repelling and attracting fixed points. Note that the attractive one is not on the real axis.

J(f3) near z=0, and a critical orbit.

Though the original parabolic basin is changed to be swirling near z=-1, 0, 1 and so on, the topology of the Julia set is not changed as in Case 1.

J(f3) near z=1

Actually, there exists a topological conjugacy h3: J(f3) -> J(f) by the theorem.

For more examples, see
*Tiles 2 *
.

Next let us go to the
*parameter planes*.