eu ON THE DYNAMICS OF A FAMILY OF RATIONAL FUNCTIONS WHICH CONTAINS KOEBE FUNCTION

We study the dynamics of a parameterized family of quadratic rational functions z 7→ λz/(1− z), λ ∈ (0,∞). We have investigated the dynamical and geometric properties of Julia sets and the nature of the variance of these properties according to the parameter. We have shown the Julia sets are connected when λ > 1 and are dynamically defined cantor sets when λ < 1. We also have shown the continuity of geometric properties specially in terms of Hausdorff dimensions of these Julia sets. AMS Subject Classification: 37F


Introduction
In this paper we consider the family When λ=1, we get the famous Koebe function.Koebe function is most celebrated function in geometric function theory.It is instrumental in proving many important results.However from dynamical point of view the function is not explored much.Stienmetz [11] has discussed the Koebe function in unsolved exercises of his book.Here we describe the complete dynamics of a larger family of rational maps containing Koebe function as a member and show the continuity of the Julia sets and their Hausdorff dimensions.
Two rational maps R and S are said to be conjugate if there is a mobius transformation g such that S = gRg −1 .It is a well known fact that two conjugate maps are dynamically equivalent to each other.It can easily be seen that the results obtained for the familyK λ are also valid for the families containing the maps z → z/(λ − z) 2 and z → z/(1 − λz) 2 whereλ is taken in (0, ∞).
A lot of work has been done on the dynamics of quadratic rational maps.Milnor [7,8] and Rees [9] have investigated the moduli space of quadratic rational maps in general.Particular slices of this moduli space have also been studied by Hawkins [4] and Milnor [7].
The concept of conformal measures is one of the main tools to understand fractal properties of a Julia set, i.e. its dimensions and measures.The concept of conformal measures was first employed to the case of rational functions by Sullivan in [12].Basic results concerning conformal measures, Hausdorff dimension and fractal and ergodic properties of Julia sets of rational functions of the Riemann sphere can be found in a survey done by Urbanski [13].

Preliminaries
The Fatou set F (f ) of a function f is defined as the set of those points z ∈ C that has an open neighborhood U such that the family of iterates {f n (U )} n∈N is normal with respect to some spherical metric on C. The Julia set J(f ) is then defined as C − F (f ).A remarkable Theorem of Montel, which paved the way for Fatou and Julia to develop a comprehensive global theory of iterates of rational functions, is as follows: Theorem 1.Let F be a family of meromorphic functions defined on a domain U .Suppose there exist points a, b, c in C such that f ∈F f (U ) ∩ {a, b, c} = φ .Then F is a normal family on U .
The following basic properties of Julia sets may be found in [1] and [6]: 3. Repelling periodic points of f are dense in J(f ).
Given a nonnegative real number t, t-dimensional Housdorff measure H t (A) of a set A is defined as where the infrimum is taken over all countable covers {A i : i 1} of A by arbitrary sets whose diameters do not exceed ǫ.The Hausdorff dimension HD(A) of A is defined as, Along with the Hausdorff dimension there are some other notions which have been frequently employed to visualize the fractal structures of Julia sets.
Here we recall some of them which we shall use in the proofs 1.The Poincaré series is defined as, The critical exponent at z by for every Borel set A ⊂ J(f ) such that f | A is injective map.Moreover minimal possible exponent for which a conformal measure exists for f will be denoted by δ(f ).
It is well known that for a geometrically finite map f the Hausdorff dimension of the Julia set HD(J(f )) is equal to δ(f ).We define Comp( C) as the space of all non empty compact sets in C with the Hausdorff topology.We recall from [5] that in the Hausdorff topology, If {X n } is a sequence of subsets of a space S. The set of all points x in S such that every open set containing x intersects all but a finite number of the sets X n is called the limit inferior of the sequence {X n } and is abbreviated "lim inf X n "; the set of all points y in S such that every open set containing y intersects infinitely many sets X n is called the limit superior of {X n } and is abbreviated "lim sup X n ."If these two sets coincide so that lim inf X n = L = lim sup X n , we say that {X n } is a convergent sequence of sets and that L is the limit of {X n }, which is abbreviated "L = limX n ."

Dynamics of the maps
Theorem 2. The Julia set J(K λ ) = J λ of the map K λ is the set of all nonnegative real numbers with infinity if λ 1.
∈ F then z / ∈ F , For let K λ (z) = r where r is nonnegative real number then z satisfies, Thus z is a nonnegative real number so K λ (F ) = F .Using this fact with Montel Theorem the family Next 0∈J λ .For if 0∈F λ then there exist an open neighborhood of 0, F 0 which is contained in the Fatou set F λ .let us take, λ [0, ε) using the fact that Fatou sets of rational maps are completely invariant and K λ [0, 1] = [0, ∞], we have [0, ∞] ⊂ F ′ ⊂ F λ and thus F λ = C which is not possible since Julia set of a rational map of degree greater then 2 is nonempty.
open set there exist ε > 0 such that (0, ε) ⊂ F ′ which leads to ∞ ∈ F λ as before, a contradiction.So d(0, F ′ ) = 0 and therefore there exist δ > 0 such that, (0, δ) ∩ F ′ = φ =⇒ (0, δ) ∩ F λ = φ =⇒ (0, δ) ⊂ J λ .Since J λ is completely invariant, K n λ (0, δ) ⊂ J λ for every n ∈ Z.We have noted above that there exist n ∈ Z such that [0, ∞] ⊂ K n λ (0, δ), so [0, ∞] ⊂ J λ this with the fact Proof.As in Theorem 2 we can see that the Fatou set F λ contains F = C − [0, ∞] this implies Fatou set is connected.From this and the fact 0 is attractive fixed point we can deduce We now begin our construction of Julia set.first we observe that Now we define R 1 , R 2 : (0, ∞) → (0, 1) such that, We consider the images of J under the elements of semi group containing maps generated by R 1 and R 2 .So for every sequence of integers i n in {1, 2}, we define: J(i 1 , ..., i m ) = R i 1 ...R im (J) It can be easily seen that J(i 1 , ..., i m , i m+1 ) ⊂ J(i 1 , ..., i m ).Since the maps R 1 and R 2 are homeomorphisms onto two disjoint sets, the 2 m sets of the form J(i 1 , ..., i m ) are pairwise disjoint compact sets, We claim that the Julia set, For, by (2) and complete invariance of J λ we get, By construction it is clear that J λ is a compact set.To prove J λ is cantor set we only have to show that it is totally disconnected.If J ′ is any connected subset of J λ , then for some n it is contained in one and only one of the sets of the form J(i 1 , ..., i n ), so we have a unique sequence (k n ) such that n , where σ is chordal metric on Riemann sphere.Chordal and Euclidean metrics are comparable on compact set J, this together with the fact Corollary 4. The map K λ restricted to its Julia set J λ is topologically conjugate to shift map σ in {1, 2} N .
Proof.From the proof of the theorem it is clear that for a given x in J λ and for each n ∈ N there exist a unique i n ∈ {1, 2} such that x ∈ J(i 1 , ..., i n ).To every x ∈ J we can associate a sequence θ x = i 1 , i 2 , ..., i n , ..., then θ K λ (x) = i 2 , i 3 , ..., i n , ... We define f : J λ → {1, 2} N , such that f (x) = θ x .f is required conjugacy.Now, we investigate the continuity of Julia sets with respect to the parameter λ.It is well-known that the Julia set may vary discontinuously at some points.Parabolic implosions for quadratic polynomials is studied in [13].Parabolic implosion refers to the phenomenon of discontinuity that sometimes appear at parabolic bifurcations.So it is particularly useful to investigate the continuity of Julia set at λ = 1, since it is a parabolic bifurcation point.Here we demonstrate with relatively simple arguments that Julia set does depend continuously on the parameter even at the parabolic bifurcation in the case of family taken by us.We use some arguments of Douady [3] in the first part of the proof of the following theorem Theorem 5. Julia sets J λ continuously depend on λ Proof.We shall prove that the function f : [0, ∞] → Comp( C) such that f (λ) = J λ is a continuous function or equivalently J λn → J λ whenever λ n → λ.
It is obvious when λ 1.So let us take a sequence (λ n ) in (0,1) such that λ n → λ ∈ (0, 1].First we shall show that J λ ⊂ lim inf J λn .For let x ∈ J λ and U be some neighborhood of x then since Julia set is closure of the set of repelling periodic points there exist a repelling point y ∈ U .This y is a simple solution of the equation K λ k (z) − z = 0 for some k ∈ N. By the implicit function theorem there exist m > 0 such that K λn has a repelling periodic point of order k in U whenever n ≥ m implying that J λn ∩ U = φ whenever n ≥ m.Thus J λ ⊂ lim inf J λn .
To prove J λn → J λ , we only have to show lim sup J λn ⊂ J λ .Now let us assume x ∈ F λ = C − J λ .We have to show that there exist a neighborhood of x which is contained in all but finitely many F λn .Since Fatou set is completely invariant, it is sufficient to show it for any iterate K λ i (x) in place of x.We have already seen that J λn ⊂ [1 − √ λ n , 1/(1 − √ λ n )].So there exist m ∈ N and a neighborhood U of 0 such that U ⊂ F λn for all n ≥ m.But since 0 is attractive fixed point when λ ∈ (0, 1) and x ∈ F λ , we have K λ i (x) ∈ U .So lim sup J λn ⊂ J λ .When λ = 1 it is obvious since F 1 = C − [0, ∞] ⊂ F λ for every λ.Hence J λn → J λ .