Non-persistently recurrent points, qc-surgery and instability of rational maps with totally disconnected Julia sets

Let $ R $ be a rational map with totally disconnected Julia set $ J(R). $ If the postcritical set on $ J(R) $ contains a non-persistently recurrent (or conical) point, then we show that the map $ R $ can not be a structurally stable map.


Introduction and Statements
Fatou's problem of the density of hyperbolic maps in the space of rational maps is one of the principal problems in the field of holomorphic dynamics. Due to Mané, Sad and Sullivan [MSS] we can reformulate this problem in the following way: If the Julia set J(R) contains a critical point, then the rational map R is a structurally unstable map.
For convenience we give the definition of the structural stability of a rational map. For other basic notations and definitions refer to the book of Milnor [M].
Definition 1. Let Rat d be the space of all rational maps of degree d with the topology of coefficient convergence. A map R ∈ Rat d is called structurally stable if there exists a neighborhood U ⊂ Rat d of R such that: For any map R 1 ∈ U there exists a quasiconformal map f : C → C conjugating R to R 1 .
We give a condition "Assumption G" (see below) on the rational map with totally disconnected Julia set and with a critical point on J(R) to be unstable. In a pioneer paper [BH], Branner and Hubbard prove that the Lebesgue measure of the Julia set is zero if there exists only one critical point on J(R). Our result (see theorem A below) restricted on the Branner-Hubbard case is weaker, but it can be applied for maps with two or more critical points on J(R). Let R be a rational map with totally disconnected Julia set. Let us normalize R so that the point z = ∞ becomes the attractive fixed point. Let P c(R) be postcritical set of the map R and P (R) = P c(R) ∩ J(R) be postcritical set on the Julia set. Let S = C\∪ n R −n (P c(R)), then R : S → S is an unbranched autocovering.

Typeset by A M S-T E X
Definition 2. We call a closed simple geodesic γ ⊂ S linked with P (R) if the interior I(γ) of γ intersects P (R).
Assumptions "G". Let R be a rational map with totally disconnected Julia set. Assume there exists a simple closed geodesic γ ∈ S such that: (1) there exists an infinite subsequence of simple closed geodesics γ i ∈ ∪ n R −n (γ) linked with postcritical set P (R), and (2) for all i = 1, ... the lengthes L(γ i ) are bounded uniformly away from ∞, in the hyperbolic metric on S.
The aim of this paper is to prove the following theorem.
Theorem A. Let R be a rational map with totally disconnected Julia set satisfying Assumption "G". Then the map R is not a structurally stable map (that is to say, is an unstable map.) Apriori it is not clear when the Assumption G holds. We give a series of sufficient conditions on R that imply Assumption "G". The next proposition translates Assumption "G" into the language of "non-persistently recurrent points" on P(R).
Definition 3. A point x ∈ P (R) is called persistently recurrent if any backward orbit U 0 , U −1 , ... of any neighborhood U 0 of x along P (R) hits a critical point infinitely many times.
Lemma (Sufficient condition). Let R be a rational map with totally disconnected Julia set and assume there exists a non-persistently recurrent point x ∈ P (R). Then R satisfies Assumption "G".
Prof. Follows immediately from Definition 3.
Another sufficient condition is connected with the conical points of P (R).
Definition 4. let R be a rational map, and denote by U (x 0 , R k , δ) the component of has degree no more than d.
Several other notions of conical point appear in the literature. One can see that the definition of conical point is somehow in the spirit of the notion of conical set of Lyubich and Minsky [LM]. Definition 4 above appears in [P], where Przytycki compares different notions of conical points. McMullen [MM] and independently Urbanski [DMNU] call a point conical if the mappings in Definition 4 can be chosen to be conformal.
Theorem 1. Let R be a rational map with totally disconnected Julia set and assume there exists a conical point x ∈ P (R). Then R is an unstable map.
The following two results are immediate corollaries of the theorem 1.
Corollary 1. Let R be a rational map with totally disconnected Julia set and assume that the postcritical set P (R) contains a periodic point x. Then R is unstable map.
Proof. By assumption, the periodic point x ∈ J(R). Hence x is either parabolic or repelling. Now assume that R is a structurally stable map, then x should be repelling and hence conical. Applying Theorem 1 we are done.
Corollary 2. Let R be a rational map with totally disconnected Julia set. Assume J(R) = P (R), then R is an unstable map.
Proof. In this case P (R) contains all repelling periodic points and by Corollary 1 we are done.

Proof of Theorem A
To prove Theorem A, we use a kind of quasiconformal surgery in the spirit of Shishikura [Sh].
Let ∆(r) be a disk of the radius r centered at z = 0 and ∆ = ∆(1). Let A(p, q) = {z : 2 ) ⊂ A(p) be a point. Now we define a quasiconformal homeomorphism f p : ∆ → ∆ as follows: (1) f p (z) = z+a 1+az on ∆(p) and (2) f p is a quasiconformal mollifier on A(p), that is i)f p = id on ∂∆ and ii)f p = z+a 1+az on other boundary component of A(p) and iii) the L ∞ − norm of the dilatation µ = ∂f p ∂f p is minimal among all dilatations of the quasiconformal homeomorphisms satisfying i)-ii).
Remark 1. Note that the L ∞ − norm of µ depends only on the modulus of the ring A(p), or in other words, if p i ∈ ∆ converges to p 0 ∈ ∆, then the L ∞ − norm of the dilatations µ i are uniformly bounded away from 1.
According to results of D. Sullivan ([S]) and C. McMullen, D. Sullivan ([MS]) the space of full orbits of the points on S forms a Riemann surface S(R) which is (conformally) the torus with a finite number of punctures: the punctures correspond to the full orbits of the critical points belonging to F (R). Hence there exists a fundamental domain F ⊂ F (R) for the action R on F (R). We can choose the fundamental domain as follows: (1) F ⊂ F (R) is a closed topological ring, (2) the boundary components α 1 ∪ α 2 = ∂F ⊂ S are smooth closed Jordan curves (3) R(α 1 ) = α 2 and restrictions R n |F are univalent for all n. Let O(F ) be the full orbit of the fundamental domain F. Let α ⊂ S be any geodesic, then α intersects a finite number, say n(α), of elements of O(F ), say F 1 (α), ..., F n(α) (α).
Remark 2. By the properties of the fundamental domain we can always assume that there exists i 0 so that the forward orbit O + (F i 0 (α)) = ∪ i≥1 R i (F i 0 (α)) never intersects the interior of the geodesic α. For convenience we redefine F 1 (α) = F i 0 (α).
be an annulus containing α as a non-trivial curve with modulus m(α) of B(α) as large as possible. Note that B(α) is not unique. Now let β ⊂ S be an iterated preimage of α (that is there exists an integer k such that R k (β) = α). If d(β) is the degree of the covering R k : β → α, then the hyperbolic length l(β) = d(β)l(α), and m(β) ≥ m(α) d(β) , as well as n(β) ≤ d(β)n(α). Let us start with any closed simple geodesic γ ∈ S linked with P (R). Now we associate a qc-homeomorphism f (γ, p) : C → C as follows: Let I(γ) be the interior of γ and the point b be the first hit in I(γ) of the forward orbit of a critical point c / ∈ I(γ. Let h : I(γ) → ∆ be the Riemann map with h(b) = 0, h ′ (b) = 1. Now let p > 0 be an number so that A(p) ⊂ h(B(γ)). Adjusting h by a rotation we can construct a conformal map φ(γ) : I(γ) → ∆ so that the point a = 1+3p 4 ∈ φ(F 1 (γ)). Then we set Hence for any simple closed geodesic γ ⊂ S and a suitable number 0 < p < 1 we can define a quasi-regular map P (γ, p) = f (γ, p) • R : C → C.
Lemma 1. Let γ ⊂ S be closed simple geodesic linked with P (R) and 1 > p > 0 be a suitable number. Then (1) there exists an invariant conformal structure σ on C so that P (γ, p) : (C, σ) → (C, σ is a holomorphic map, (2) the norm of the dilatation of σ (that is L ∞ −norm of the corresponding Beltrami differential) depends only on the numbers p and n(γ).
Proof. Follows immediately from the definition of f (γ, p).
By the Riemann Mapping Theorem there exists a quasi-conformal homeomorphism f σ fixing the points 0, 1 and ∞ so that R(γ, p) = f σ • R • f −1 σ is a rational map. Let s(R) be the number of critical points whose forward orbits converge to ∞.
Corollary 1. Let γ and p be as in Lemma 1 above. Assume that γ has a sufficiently small spherical diameter. Then s(R(γ, p) ≥ s(R) + 1.
Proof. Let the spherical diameter of γ be so small that the interior I(γ) does not contain any critical point of the Fatou set F (R). Hence if the critical point c ∈ F (R), then P n (γ, p)(c) = R n (c) → ∞. Now let c ∈ J(R) be the critical point coming from the definition of f (γ, p). Then again by the construction, P n (γ, p)(c) → ∞. Now we are ready to prove Theorem A. Let γ i ∈ ∪ n R −n (γ) be coming geodesics from the assumption. Let us redefine: Let ν i be the Beltrami differentials of the structures of σ i respectively. Now our aim is to show that there exists a subsequence {ν i j } with norms uniformly bounded away from 1. By Remark 1 and Lemma 1 it is enough to show that we can choose a subsequence p i j uniformly bounded away from 1.
Let k i be integers so that R k i (γ i ) = γ. Let B i be a component of R −k i (B(γ)) containing the geodesic γ i . Then by our assumptions there exists a constant C so that the moduli There exists a subsequence {i j } and a number p < 1 so that A(p) ⊂ A i j for any j.
Proof. The argument is simple. Let A i 0 be any annulus of minimal modulus. Then there exist conformal injections h i : A i 0 → A i such that h i (∂∆) = ∂∆ and h i (1) = 1. The family {h i } is normal so let {h i j } be a convergent subsequence. Then the limit map h ∞ = const. Now let q < 1 be so that A(q) ⊂ A i 0 , then by the reflection principle {h i j } converges to h ∞ uniformly on A( q+1 2 ). Hence h ∞ (A( q+1 2 )) ⊂ h i j (A i 0 ) for all large enough j. Let p < 1 be an integer so that A(p) ⊂ h ∞ (A( q+1 2 )); then by the discussion above for all large j. The lemma is thus proved.
By Remark 1, Lemma 1 and Lemma 2 we have that the family of quasiconformal homeomorphisms {f i j } is normal, and after passing to a subsequence we can assume that {f i j } converges to a quasi-conformal homeomorphism f ∞ . The Julia set J(R) is a Cantor set hence the spherical diameter diam(γ i ) → 0. Then the homeomorphisms f (γ i j , p) converge to the identity uniformly on C, and hence P i j → R.
Again after passing to a subsequence we can assume that lim j→∞ R i j = R ∞ , where R ∞ is a rational map of degree smaller or equal to the degree of R. Then we can pass to the limit in the following equality: Now to obtain a contradiction assume that R is a structurally stable map. Then R ∞ is structurally stable ( a being a qc-deformation of R) and s(R) = s(R ∞ ). By construction R ∞ = lim j→∞ R i j and by Corollary 1 s(R i j ) ≥ s(R) + 1 = s(R ∞ ) + 1 which contradicts the structural stability of R ∞ .

Proof of Theorem 1
Here we show that the existence of a conical point x ∈ P (R) implies Assumption "G" .
Lemma 3. Assume R satisfies the assumptions of the Theorem 1. Then R satisfies Assumption "G".
Proof. Let x 0 ∈ P (R) be a conical point. Let integer d and a sequence {k j } be as in the definition of the conical point. Let U (x 0 , R k j , δ) be the component of R −k (D(R k j (x 0 ), δ)) that contains x 0 , where D(R k (x 0 ), δ) is the disk centered in R k (x 0 ) with the radius δ.
The lemma is proved.
An application of Theorem A completes the proof of Theorem 1.