Regularizing quadrangles in the M¨obius plane

. For any given cross ratio δ ∈ C we deﬁne a non-linear M¨obius-invariant procedure creating vertices of a new quadrangle from a given quadrangle. Reiterating the process we discover a remarkable regularizing property. The values δ ∈ C for which this phenomenon of regularization develops are examined. The regularizing eﬀect also depends upon the shape of the starting quadrangle. The investigation of such shapes leads into the ﬁeld of complex dynamics in one variable. Methods of discrete dynamics are employed to explore the regularizing behaviour of the procedure.


Introduction
Euclidean and affine iterative procedures creating sequences of polygons have been studied by several authors (e.g. [5,8,11,12] and [6]). At times, these procedures show some regularizing effect. This paper is devoted to the study of a procedure based on Möbius geometry (see [3]). This seems to be the first attempt to prove such a regularizing effect on quadrangles created by a nonlinear procedure. α2 z+α3 with complex numbers α 0 , α 1 , α 2 , α 3 and α 0 α 3 = α 1 α 2 are called Möbius transformations. Along with this group P GL(C) of Möbius transformations C * is called the Möbius plane.

New generations of quadrangles
In this section we define a procedure generating a new quadrangle from a given quadrangle.
Definition 2.1. We select a complex number δ ∈ C \ {0, 1}. For a given quadrangle (p 0 , p 1 , (2.1) Here and in the following paragraphs the index j is to be interpreted mod 4. The complex number δ is referred to as the construction cross ratio.
We can also offer an explicit formula for the vertices of the next generation quadrangle: Vol. 109 (2018) Regularizing quadrangles in the Möbius plane Page 3 of 11 47 Figure 1 The initial quadrangle Q 0 = (p 0 , p 1 , p 2 , p 3 ) and some of the following generations (scaled and shifted) for three different procedures defined by the construction cross ratios This procedure can be applied iteratively to a starting quadrangle (see Fig. 1). The shape z n of the next generation quadrangle Q n = (p * 0 , p * 1 , p * 2 , p * 3 ) is fully determined by the chosen construction cross ratio δ and the shape z n−1 of the quadrangle Q n−1 = (p 0 , p 1 , p 2 , p 3 ). So we can compute the characteristic cross ratio z n of Q n straight from the characteristic cross ratio z n−1 of Q n−1 : For pairwise distinct points p 0 , p 1 , p 2 Lemma 2.2 tells us that we are entitled to replace the quadrangle Q n−1 = (p 0 , p 1 , p 2 , p 3 ) by some prototype (0, ∞, 1, z n−1 ) with the same characteristic cross ratio z n−1 . For this prototype the vertices of the next generation quadrangle Q n are: Its corresponding characteristic cross ratio f δ (z n−1 ) := z n is .
The same result can be derived if p 0 , p 1 , p 2 are not pairwise distinct. This formula can also be viewed as an iteration rule enabling us to compute z n = f n δ (z 0 ) from the shape z 0 of a starting quadrangle Q 0 . According to our construction it is clear that the following functional equations apply:

Regular quadrangles
Obviously, this yields two kinds of regular quadrangles: z * = 1 2 and z * = ∞. For the cross ratio z * = 1 2 we can think of the example (1, i, −1, −i) which reminds us of a square. This first kind of regular quadrangles is sometimes referred to as harmonic quadrangle 3 (see [4, p. 63]). As for z * = ∞ (second kind), we can think of the example (1, −1, 1, −1). The next generation quadrangle Q n of a regular quadrangle Q n−1 is again regular of the same kind.  .2) is said to be regularizing if the sequence (2.4) of the corresponding shapes z 0 , z 1 , . . . converges to 1 2 The explicit formula (2.2) enables us to implement the procedure and to apply it to particular examples of quadrangles. Mind that the image of a regular quadrangle will not necessarily appear regular in terms of Euclidean geometry in R 2 ; after all, the setting is Möbius Geometry. Apart from this, Regularization (see Definition 3.1) does not mean that the quadrangle Q n tends towards some regular limit quadrangle. It may well be that the sequence Q 0 , Q 1 , . . . of quadrangles shrinks towards a degenerate quadrangle (p, p, p, p) with p ∈ C * . Even in such a case the regularizing property in terms of Definition 3.1 might still apply 4 ; see also Fig. 1.
In Fig. 1 the initial quadrangle Q 0 = (p 0 , p 1 , p 2 , p 3 ) has the shape z 0 ≈ 1.168−0.345i. The three illustrated procedures are defined by the construction cross ratios δ 1 = 2.5+i, δ 2 = 2.5 and δ 3 = 0.75+0.25 i. They behave differently: • For δ 1 we get the shape z 25 ≈ 0.526 − 0.007 i for the highlighted quadrangle Q 25 ; the procedure regularizes towards the shape z * = 1 2 . The quadrangles, while getting more and more regular of first kind, keep shrinking from generation to generation towards a point.
• For δ 2 the quadrangle Q 100 is highlighted; its shape is z 100 ≈ 0.5 + 1.87 × 10 −11 i. The procedure again regularizes towards the shape z * = 1 2 . The circumcircles of any three distinct points of the quadrangles tend towards Vol. 109 (2018) Regularizing quadrangles in the Möbius plane Page 5 of 11 47 one limit circle c. The points of the quadrangles asymptotically approach that Möbius circle c, but still there does not exist any limit quadrangle. • For δ 3 the procedure regularizes towards z * = ∞. The quadrangle Q 10 is highlighted; its shape is z 10 ≈ −715.48 + 637.32 i.
Whether or not the process (2.2) regularizes may depend on the construction cross ratio δ as well as on the characteristic cross ratio (shape) z 0 of the starting quadrangle. In order to examine the behaviour we need not compute the points of the sequence of quadrangles via (2.2) explicitly. According to Definition 3.1 we instead study the function (2.4); this way we zero in on the behaviour of the shape.

The rational iteration and its fixed points
Along with the emergence of f δ (z) (see Equation (2.4)) we have arrived in the realm of complex-valued rational iteration (see also [2] and [7]). The functional equations (2.5) inspire the substitutions In terms of ζ and γ the iteration (2.4) now reads: Properties of the discrete dynamical system defined by R γ have a direct impact on the geometry of our iterative procedure defined in (2.1) and (2.2). This is why we focus on the behaviour of this system in the next few sections.
Every limit value ζ of a rational iteration (4.2) has to be a fixed point. So we first go for the set of fixed points of R γ : t 0 and t 1 are the values relating to the shape-values ∞ and 1/2 of regular quadrangles. The two fixed points t 2,3 , however, are different from t 0 , t 1 due to γ = −1; they also characterize quadrangles Q 0 that keep their shape during all subsequent steps of iteration. For any attracting fixed point t there exists a neighbourhood U ⊂ C * of t such that for all ζ ∈ U we have lim n→∞ R n γ (ζ) = t. In order to determine the local behaviour of R γ (ζ) at the fixed points we consider the derivatives From (4.4) we infer that for all γ ∈ C \ {±1} the two fixed points t 2,3 are repelling. t 0 is attracting if and only if γ ∈ C 0 , t 1 is attracting if and only if γ ∈ C 1 with: The two domains C 0 and C 1 are disjoint.
It can also happen that an iteratively generated sequence of points is heading towards a periodic orbit; these are cycles of length n ≥ 2 relating to fixed points of R n γ (z) that are not fixed points of R m γ (z) for any m ∈ {1, . . . , n − 1}. Such a cycle is called attracting (repelling) if and only if it contains an attracting (a repelling) fixed point of R n γ . Due to R γ (±1) = ∓1 the pair (−1, 1) ∈ C * 2 is one example of a cycle of length 2. For this cycle we have Hence, this cycle is attracting if and only if γ ∈C := {γ ∈ C Re(γ) < −1}; this domainC is disjoint from both, C 0 and C 1 . Additionally, we have (R n γ ) (±1) = (R γ (±1)) n which will be of importance in Sect. 6. In the following sections we lock a value γ in one of the domains C 0 or C 1 . Then, either t 0 = ∞ or t 1 = 0, is an attracting fixed point of the rational function R γ given by (4.2). The other fixed points are repelling, though there might well exist further attracting cycles of R γ . For all γ = ±1 the points c 0 , . . . , c 3 are the four critical points for R γ (counted with multiplicity). Any basin of attraction for attracting cycles or fixed points has to contain at least one critical point of R γ (see [2, p. 194]  According to [2, p. 194] we can be sure that in the cases 1 and 2 there are no further attracting cycles of any length. Thus, we have

Critical points and basins of attraction
For γ ∈ C 1 ⊂ C the only point of attraction is t 1 = 0. In any of these cases there exist no further attracting fixed points or cycles of any length.
In each of the cases 1 and 2 there is only one basin of attraction of the corresponding fixed point. This does not mean, though, that any point ζ ∈ C * converges to t 0 or t 1 ; the exceptions will be described further on.

The sets of Fatou and Julia
We follow [2, p. 50] and define: The family of functions {R n γ | n ∈ N} is called equicontinuous atz if and only if The Fatou set F γ of R γ is the maximal open subset of C * (see [2, p. 50] or [7, p. 40]) on which the sequence R n γ is equicontinuous. The Julia set J γ is the complementary set J γ = C * \ F γ .
Obviously, the sequence R n γ is not equicontinuous at repelling fixed points or repelling cycles: Such points are contained in the Julia set J γ .

Figure 2
The Julia set J γ for γ = 0.96 (cos 3π 4 + i sin 3π 4 ). Here t 0 = ∞ is the attracting fixed point, the three marked points −1, 0 and 1 belong to J γ . The diamond-shaped symbols indicate the critical points c 0 , . . . , c 3 , the squares mark the two finite preimages of ∞. According to Theorem 6.1 J γ has a diameter ≤ 2 × 4.6 If γ lies in C 0 the set {−1, 0, 1} is contained in the Julia set J γ and F γ = A ∞ .
If γ lies in C 1 the set {−1, 1, ∞} is contained in the Julia set J γ and F γ = A 0 .
One property of the Fatou set F γ is that, for γ ∈ C 0 , it contains the subset {z ∈ C * |z| > M γ } -the respective Julia set J γ is bounded. The Julia set J γ (see [2, p. 71]) is the closure of the backward orbit O − (z) := {w ∈ C * | ∃n > 0 with R n γ (w) = z} for any z ∈ J γ . This property can be used to visualize the Julia set J γ in any of the two aforementioned cases γ ∈ C 0 and γ ∈ C 1 . Figures 2 and 3 refer to γ ∈ C 0 and γ ∈ C 1 , respectively; the Julia set J γ is displayed in black. 5 A brief look at the Figs. 2 and 3 implores the question: Are the Julia sets J γ connected, disconnected or do we encounter the phenomenon of a Cantor Set (compare [2, p. 227])? Due to numerical observations for particular values of γ we can imply that the preconditions for [9, Theorem 2.2] are fulfilled: For the considered values of γ the Julia set J γ is connected. Without being able