Examples of codification of the dynamics of a rational function into a topological tree

Abstract. In 1736 L. Euler gave solution to the famous Seven Bridges of Königsberg problem, considerin a graph consisting of nodes representing the landmasses and arcs representing the bridges. This problem is a referent of how to codify the information given of a problem into a simpler and richer structure. In the case of the Dynamics of rational functions, Shishikura in [5] explores this idea in the context of rational functions, and he stated a connection between a certain kind of topological tree with a p-cycle of Herman rings associated to a rational function. In this work we develop some examples of realizable configurations for rational functions, two of them sketched in [5], and an example of a non realizable configuration which we modify in order to be realizable.


Introduction
In this paper we will deal with the class of automorphisms on the Riemann sphere, denoted by R, where R = f :Ĉ →Ĉ : f (z) = p(z) q(z) where p and q are complex coprime polynomials .
P. Fatou and G. Julia in [2] and [4] studied the iteration of functions with degree at least two in the class R, that is, the convergence on compacts of sequence of iterates of the function f, {f n } n∈N , where f n is the composition of f with itself n times. In particular, when n = 0 and n = 1, f 0 = Id and f 1 = f, respectively. Observe that the class of rational functions R is a semigroup with the composition of functions, i.e., if f ∈ R, then f n ∈ R. They consider the iteration of all the points z ∈Ĉ under f ∈ R and obtain two disjoint sets, namely, the Fatou set, denoted by F (f ), which is the set of point z ∈Ĉ such that the sequence of iterates of f forms a normal family in a neighbourhood of z and the Julia set, which is the complement of the Fatou set and denoted by J (f ) =Ĉ \ F (f ).
We recall some well known properties of the Fatou and Julia sets for rational functions: is perfect and non-empty; (iii) the sets J (f ) and F (f ) are completely invariant under f ; and finally (iv) the repelling periodic points are dense in J (f ) (see [1] for a proof).
D. Sullivan in [8] classifies the connected components U of the Fatou set of a rational function f as follows: (i) attracting, (ii) parabolic and (iii) linearizable rotation domains (Siegel disk and Herman ring).
In this paper we are particularly interested in Herman rings, whose existence was proved by Herman in [3], and later on Shishikura in [5] constructed them using quasiconformal surgery. A significant characteristic of these kind of components is that their existence is not related to the existence of periodic points like the others (see [1] for a proof).
In [6] Shishikura studies the realization problem, which is an inverse construction to the existence theorem; that is, he studied the conditions under which there exists a rational f such that given a finite collection with p elements of topological annuli that will correspond later on to a p-cycle of Herman rings for some function f. In order to do that, he constructed three objects: a topological tree T associated to the collection of topological annuli, a linear metric d onĈ and a continuous function F onto T , which conform a Shishikura's triple.
In what follows, we shall explain the construction of the elements that conform a Shishikura's triple, say (T, d, F ), and we give the procedure for obtain the orbit model on (T, d, F ). Finally, we enunciate the Realization theorem for rational functions.
Organization of the paper. This paper contains two sections. In Section 2 we explain the necessary elements to guarantee the realization of a configuration of a p-cycle of Herman rings for a rational function through a construction of a Shishikura's triple given in [6] and [7], and Section 3 contains examples of some realizable configurations and the procedure to obtain them.

Shishikura's triple and the Realization theorem
We recall that a component U in the Fatou set is called: (i) A p-cyclic Siegel disc if there exits an analytic homeomorphism ϕ : U → D, where D is the unit disc, such that ϕ conjugates f p with e 2πα z for some α ∈ R \ Q, where α is called the rotation number of the disc.
(ii) A p-cyclic Herman ring if f p is conformally conjugate to an irrational rotation on a concentric annulus, that is, there exist 0 < r 0 < 1, θ ∈ (0, 1) \ Q and a conformal change where R θ is the rigid rotation of angle 2πθ, and H r0,1 is the standard annulus of inner radius r 0 and outer radius 1.
A configuration for a p-cycle of Herman rings is defined as follows: fix r such that r 0 < r < 1 and define γ 0 = h −1 H ({z ∈ C :| z |= r}) with a determined orientation and γ j = f j (γ 0 ); then f induces equivariant orientations of γ j , and {γ j } defines a configuration, which is independent of the choice of r and the orientation of γ 0 . Shishikura in [6] studies the conditions under which a finite arrangement of topological annulli D := {A 0 , ..., A p1 } corresponds to a configuration of cyclic Herman rings (the elements in D satisfies that for some i, j, A i ⊂ A j or, for every i, j, A i ∩ A j = ∅ or the combination of the two cases). In order to do that, on D constructs three objects: a topological tree T , a linear metric d onĈ and a continuous function F onto T , which conform a Shishikura's triple.
In the same paper, he defines an orbit model on (T, d, F ) in order to obtain f ∈ R and a collection of Herman rings .., p1}. When these conditions are satisfied, it is said that f realizes the configuration D.
It should be mentioned that initially we obtain a rational function with a p-cycle of Siegel discs, which via quasi-conformal surgery is transformed into Herman rings (see [5] for a proof).

Abstract tree
Consider D as in Section 2 and assume that there exists a non-empty closed set B such that B consists of a finite number of connected components B 1 , ..., B n , where none of them is a point and each A ∈ D is disjoint with B and separates B.
In this part, we will construct an abstract tree associated to the configuration D. For that, consider onĈ an equivalence relation ∼ d as follows: for x, y ∈Ĉ, x ∼ d y if, and only if, d(x, y) = 0, where d is a metric given by d(x, y) = A∈D mod(A(x, y)), which is continuous onĈ ×Ĉ and d is a pseudo-metric onĈ. Thus, the quotient space  of specific arcs with determined end points and satisfies that the projections π(A) are dense in T (D) and are isometric to open intervals. For a proof see [6].
A. Algorithm for obtaining the tree associated to a configuration D.
Intuitively it is possible to construct T as it shall follow the following algorithm, which is exemplified in each step.
(i) Consider a collection C of topological annuli with the conditions mentioned in Section 2.1 (see Figure 1(a)).
(ii) Now, in C consider a graph Γ whose vertices are the connected components of the complement of the annuli (see Figure 1(b)).
(iii) Two vertices are connected by an edge if, and only if, the corresponding two vertices have a common annulus whose boundary is contained in them (see Figure 1(c)). (iv) Thus, we obtain a tree as follows. [Revista Integración, temas de matemáticas

Shishikura's tree associated to a configuration given by a p-cycle of Herman rings
With base to the previous definition of an abstract tree we are able to define a Shishikura's tree for a configuration of a cycle of Herman rings (see [6], [7]). Let f ∈ R and f n the n-iterate of f . Take f such that it has a p-cycle of Herman rings {H 0 , ..., H p1 }, and define the following sets: (iv) B = ∪∂H i , where ∂H i denotes the boundaries of the Herman ring H i , and The sets A 0 , A ′ and A are collections of disjoint annuli. Taking A as D and B as B in the previous section, T (A) is a metric space and d can be projected to a linear metric. T (A) is called a Shishikura's tree which is denoted by T.

Continuous function on T
Let T be as Shishikura's tree and F : T → T be defined by where π is the canonical projection and ∂(π −1 (x)) is the boundary of π −1 (x) inĈ; this function is well-defined and continuous.
It is observable that f induces a map on T and therefore a permutation on the elements of T. Moreover, it satisfies that if J ⊂ T is an arc that does not contain any branch points or the image under π of a critical point of f , then R = π −1 (J) is an annulus and f : R → f (R) is a convering map degree of degree k for some k ∈ N and d(F (x), F (y)) = kd(x, y), so that DF (z) := k denotes an unbranched point z ∈ (x, y).
It follows from the construction of T that F is a piece wise linear function that acts as a permutation on the elements of T, and the arcs I j can be oriented in such way that f respects the orientations, that is, considering ∂ − I j and ∂ + I j the beginning and the end of the arc I j , then ∂I j = {∂ − I j , ∂ + I j } and F (∂ − I j ) = ∂ − I j+1 , F (∂ + I j ) = ∂ + I j+1 .
Thus, in order to represent T we need a finite set of arrowed segments, denoted by x 1 , ..., x n , and their respective lengths as well as a periodic arc, x n+1 , with its end points.

Local models
Now, we will explain how Shishikura in [7] gets a rational function f that realizes a configuration of Herman rings, for which the concept of local model for singular orbits is needed. Since each complementary component of A is collapsed to a point, we lose the dynamics, which can be recovered by defining local models via surgery. The plan behind this concept is as follows.
First, consider a tree associate to a configuration of annulli (see Figure 3).
Second, thicken all the segments of the tree to tubes with a small common radius, blowing up all the singular points to balls. Gluing the balls and the tubes, a topological sphere is obtained (see Figure 4).  Finally, define a function on the tubes so that if x, F (x) ∈ Sing(T, F ), it is a covering of degree DF (x) from the circle corresponding to x to that corresponding to F (x). Thus, we define a rational function with the same degree as the the local degree of F. We obtain a cyclic function G : mĈ → mĈ, where mĈ := (Z\Z) ×Ĉ (see Figure 5).

The Realization Theorem
Shishikura proved in [6] the following theorem related with a Shishikura's triple and its local model.
[Revista Integración, temas de matemáticas Summarizing the ideas exposed so far, in order to determinate the realization of a configuration of annuli, we need: 2. Determinate the configuration of the configuration D.
3. Construct the abstract tree T (D) associated to D, as in Section 2.1.A.
4. Construct the narrowed abstract tree associated to T (D) taking into consideration the permutation given by the configuration.
5. Determinate the length of the edges of the narrowed abstract tree. 6. Determinate the local model of singular orbits on the end points of the narrowed abstract tree associated to T (D).

Remark 2.2.
As a result of step 5 we obtain a linear system of n × n + 1 equations, where one of the variables must be the length of the cyclic arc x n+1 . Thus, we will obtain that all the solutions will depend on x n+1 .
In the following section, we will give example of realizable configuration, keeping on the steps above.

Examples of realizable configurations for rational functions
In what follows we shall construct some examples of specific configurations of Herman rings realizable by a rational function by using the results from Section 2.
Example 3.1. Consider the configuration C 1 of the cyclic annuli given in Figure 6.
This configuration was proposed by Shishikura in [7], and it is the simplest one. The abstract tree associated to the configuration C 1 is given in Figure 7.   As we mention in Subsection 2.3, it is necessary to construct an associated narrowed tree (see Figure 8).
By [7], the configuration C 1 is realizable by a rational function.
Example 3.2. In the following example, consider the configuration C 2 given in Figure 9.
The associated abstract tree to his configuration is given in Figure 10.  Figure 10. Abstract tree associated to the configuration C 2 .
As we know, to construct the Shishikura's tree T and define the function F for the configuration given we know that it is only necessary to determinate the length of the edges of T in term of the periodic arc x 5 , for which we will consider the following narrowed tree (see Figure 11).
In this case let us determine the lengths of the segments that form the abstract tree; in order to do it, consider the following system of equations: (1) [Revista Integración, temas de matemáticas Figure 11. Arrowed tree associated to the configuration C 2 Solving the above system of equations, we obtain that x 1 = x 2 = x 5 , x 3 = 4x 5 , x 4 = 5x 5 with x 5 > 0. Therefore, is possible to assign finite lengths to each segment and construct a finite topological tree. Now, let us construct a local model for singular points in the given tree. In Table 1, we present the function g at each point of X, where the expression x → ϕ with x ∈ X, means that p β = ϕ, for the branch β at x containing the segment x. Considering that the degree of each one must be 2 for some functions given that DF = 2 for some segments, and that each function has a Siegel disk, and by using quasi-conformal surgery, we have a p-cycle of Herman rings.
We recall two important facts: (i) α is a Diophantine number if |αq| < 1 q µ has infinitely many solutions in integers q for µ < v for some real number v and at most finitely many for µ < v; (ii) If the eigenvalue of a periodic orbit of rational function f is of the form e 2πα with α a Diophantine number, then f has a p-cycle of Siegel discs, and therefore a cycle of Herman rings.  Following the algorithm described in the Section 2.1.A, we obtain the tree associated to the configuration C 3 (see Figure 13).