Structure of Julia sets of polynomial semigroups with bounded finite postcritical set
Introduction
A rational semigroup is a semigroup generated by non-constant rational maps on the Riemann sphere with the semigroup operation being the composition of maps. We denote by 〈hλ: λ ∈ Λ〉 the rational semigroup generated by the family of maps {hλ: λ ∈ Λ}. A polynomial semigroup is a semigroup generated by non-constant polynomial maps. Research on the dynamics of rational semigroups was initiated by Hinkkanen and Martin in [2], who were interested in the role of the dynamics of polynomial semigroups while studying various one-complex-dimensional moduli spaces for discrete groups. Also, Ren, Gong, and Zhou studied such semigroups from the perspective of random dynamical systems (see [18], [1]).
The polynomial maps fc(z) = z2 + c for c in the Mandelbrot set are such that the orbit of the sole critical point is bounded, which in turn leads to many important dynamic and structural properties. It is then natural to look at the more general situation of polynomial semigroups with bounded postcritical set. We discuss the dynamics of such polynomial semigroups as well the structure of their Julia sets. For some properties of polynomial semigroups with bounded finite postcritical set, also see [12], [14].
Note that the research of polynomial semigroups is deeply related to the research of random dynamics of polynomials (see [16]). Definition 1 Let G be a rational semigroup. We setWe call F(G) the Fatou set of G and J(G) the Julia set of G. The Fatou set and Julia set of the semigroup generated by a single map g is denoted by F(g) and J(g), respectively.
From the definition, we get that F(G) is forward invariant under each element of G, i.e., g(F(G)) ⊂ F(G) for all g ∈ G, and thus J(G) is backward invariant under each element of G, i.e., g−1(J(G)) ⊂ J(G) for all g ∈ G (see [2, p. 360]). The sets F(G) and J(G) are, however, not necessarily completely invariant under the elements of G. This is in contrast to the case of single function dynamics, i.e., the dynamics of semigroups generated by a single rational function. For a treatment of alternatively defined completely invariant Julia sets of rational semigroups the reader is referred to [4], [5], [6], [7].
Note that J(G) contains the Julia set of each element of G. Moreover, the following result due to Hinkkanen and Martin holds. Theorem 2 [2, Corollary 3.1] For rational semigroups G with ♯(J(G)) ⩾ 3, we have
The backward orbit of z is given by G−1(z) = ∪g∈Gg−1({z}) and the forward orbit of z is given by G(z) = ∪g∈Gg({z}). For any subset A of , we set G−1(A) = ∪g∈G g−1(A).
For any polynomial g, we set which is known as the filled in Julia set of g. We note that J(g) = ∂K(g) and that K(g) is the polynomial hull of J(g). The appropriate extension (to our situation with polynomial semigroups) of the concept of filled in Julia set is as follows. Definition 3 For a polynomial semigroup G, we set Remark 4 We note that for all g ∈ G, we have and . Definition 5 The postcritical set of a rational semigroup G is defined byWe say that G is hyperbolic if P(G) ⊂ F(G) and we say that G is subhyperbolic if both #{P(G) ∩ J(G)} < +∞ and P(G) ∩ F(G) is a compact set.
For research on (semi-)hyperbolicity and Hausdorff dimension of Julia sets of rational semigroups see [8], [9], [10], [15], [11]. Definition 6 The finite postcritical set (or, the planar postcritical set) of a polynomial semigroup G is defined by Definition 7 Let be the set of all polynomial semigroups G with the following properties: Each element of G is of degree at least two, and P∗(G) is bounded in .
Moreover, we set .
Remark 8
Since P(G) is forward invariant under G, we see that implies , and thus P∗(G) ⊂ K(g) for all g ∈ G.
Remark 9
For a polynomial g of degree two or more, it is well known that implies J(g) is connected. (Hence, for any , we have that J(g) is connected.) We note, however, that the analogous result for polynomial semigroups does not hold as there are many examples where , but J(G) is not connected (see [17]). See also [13] for an analysis of the number of connected components of J(G) involving the inverse limit of connected components of the realizations of the nerves of finite coverings of J(G), where consists of backward images of J(G) under maps in G.
The aim of this paper is to investigate what can be said about the structure of the Julia sets and the dynamics of semigroups ? We begin by examining the structure of the Julia set and note that a natural order (that is respected by the backward action of the maps in G) can be placed on the components of J(G), which then leads to implications on the connectedness of Fatou components. Definition 10 For a polynomial semigroup , we denote by the set of all connected components of J(G) which do not include ∞. Definition 11 We place a partial order on the space of all non-empty compact connected sets in as follows. For any connected compact sets K1 and K2 in , “K1 ⩽ K2” indicates that K1 = K2 or K1 is included in a bounded component of . Also, “K1 < K2” indicates K1 ⩽ K2 and K1 ≠ K2. We call ⩽ the surrounding order and read K1 < K2 as “K1 is surrounded by K2”. Theorem 12 [12] Let (possibly infinitely generated). Then is totally ordered. Each connected component of F(G) is either simply or doubly connected. For any g ∈ G and any connected component J of J(G), we have that g−1(J) is connected. Let g∗(J) be the connected component of J(G) containing g−1(J). If , then . If and J1 ⩽ J2, then both g−1(J1) ⩽ g−1(J2) and g∗(J1) ⩽ g∗(J2).
With this order and the following notation we will then be able to state our main results.
Let h1, … , hm be rational functions on . Let be the one-sided shift space and let σ: Σm → Σm be the shift map, i.e., σ(x1, x2, …) = (x2, x3, …). Let be the map defined by , where x = (x1, x2, …) ∈ Σm. This is called the skew product map associated with {h1, … , hm}. Let and be the natural projections. We set and we denote by the set of points y ∈ π−1x which has a neighborhood U in π−1x such that is normal. Furthermore, we set . Remark 13 Note that is equal to the set of points where the sequence of rational functions is not normal. This is sometimes called the Julia set along the trajectory (sequence) x ∈ Σm. Theorem 14 Let and let A and B be disjoint subsets of . Suppose that we have one of the following conditions: A and B are doubly connected components of F(G). A is a doubly connected component of F(G) and B is a connected of F(G) with ∞ ∈ B. and B is a doubly connected component of F(G). There exists an open set U in with and H = 〈α1, α2〉 is hyperbolic. Let be the skew product map associated with {α1, α2} Then (disjoint union), for any component J of J(H), there exists an x ∈ Σ2 with and there exists a constant K ⩾ 1 such that any component J of J(H) is a K-quasicircle.
Then ∂A ∩ ∂B = ∅. Furthermore, and are separated by a Cantor family of quasicircles with uniform dilatation which all lie in J(G). More precisely, there exist two elements α1, α2 ∈ G satisfying all of the following.
is totally ordered with ⩽, consisting of mutually disjoint subsets of J(H). Furthermore, for each x ∈ Σ2, the set separates and .
Remark 15
It should be noted that in the above theorem, the quasicircles are all disjoint components of J(H), but may all lie in the same component of J(G).
Example 16
We give an example of a semigroup such that J(G) is a Cantor set of round circles. Let f1(z) = azk and f2(z) = bzj for some positive integers k and j. Then J(f1) and J(f2) are both circles centered at the origin. Let A denote the closed annulus between J(f1) and J(f2). For positive integers m1 and m2 large enough, we see that the iterates and will yield where . Now iteratively define and note that for G = 〈g1, g2〉 we have that , since J(G) is the smallest closed backward invariant (under each element of G) set which contains three or more points.
The next results concern the (semi-)hyperbolicity of polynomial semigroups in , and in particular show how one can build larger (semi-)hyperbolic polynomial semigroups in from smaller ones by including maps with certain properties. For this result we need to note the existence of a minimal element in and state a few of its properties. Theorem 17 [12] Let G be a polynomial semigroup in . Then there is a unique element such that Jmin meets (and therefore contains) . Furthermore, we have the following: Jmin ⩽ J for all . P∗(G) is contained in the polynomial hull of Jmin.
Definition 18
A rational semigroup H is semi-hyperbolic if for each z ∈ J(H) there exists a neighborhood U of z and a number such that for each g ∈ H we have deg(g: V → U) ⩽ N for each connected component V of g−1(U).
Theorem 19
Let and let G = 〈H, h1, … , hn〉 be a polynomial semigroup generated by H and h1, … , hn. Suppose
- (1)
,
- (2)
J(hj) ∩ Jmin(G) = ∅ for each j = 1, … , n, and
- (3)
H is semi-hyperbolic.
Remark 20
Theorem 19 would not hold if we were to replace both instances of the word semi-hyperbolic with the word hyperbolic (see Example 37). However, with an additional hypothesis we do get the following:
Theorem 21
Let and let G = 〈H, h1, … , hn〉 be a polynomial semigroup generated by H and h1, … , hn. Suppose
- (1)
,
- (2)
J(hj) ∩ Jmin(G) = ∅ for each j = 1, … , n,
- (3)
P∗(H) ∩ J(H) = ∅, and
- (4)
for each j = 1, … , n, the critical values of hj do not meet Jmin(G). Then, P∗(G) ∩ J(G) = ∅.
Remark 22
We note that if in Theorem 21 we replace hypothesis (3) with the hypothesis “H is hyperbolic”, then the conclusion becomes “G is hyperbolic”. This follows immediately as one can show that ∞ ∈ F(H) implies ∞ ∈ F(G) (see the Proof of Theorem 19 for more details).
The rest of this paper is organized as follows. In Section 2, we give the necessary background and tools required, in Section 3 we give the Proof of Theorem 14, and in Section 4 we give the proofs of Proof of Theorem 19, Proof of Theorem 21 along with Example 37.
Section snippets
Background and tools
Although not all connected compact sets in are comparable in the surrounding order, we do have the following lemma whose proof we leave to the reader. Lemma 23 Given two connected compact sets A and B in we must have exactly one of the following: A < B, B < A, A ∩ B ≠ ∅. A and B are outside of each other, i.e., A is a subset of the unbounded component of and B is a subset of the unbounded component of .
For various sets of interest in this paper the last case listed above is not possible (see Corollary 33
Proof of Theorem 14
Definition 34 For compact connected sets K1 and K2 in such that K1 < K2 we define Ann(K1, K2) = U ∩ V where U is the bounded component of which contains K1, and V is the unbounded component of . Thus Ann(K1, K2) is the open doubly connected region “between” K1 and K2. Remark 35 For any connected compact set A ⊂ Ann(K1, K2) we immediately see that A < K2 and, by Lemma 23, either K1 and A are outside of each other or K1 < A. Lemma 36 Let be such that J(f) and J(g) lie in different components of J(G) with J(f) < J(g). Then for
Proof of Theorems 19 and 21
Example 37 Let f1(z) = z2 + c where c > 0 is small (thus J(f1) is a quasi-circle). Let denoted the finite attracting fixed point of f1. Note that increases to z0. Choose and note that J(f2) = C(z0,∣c − z0∣). For large and each map B(z0,∣c − z0∣) into itself and J(G) is disconnected for G = 〈h1, h2〉. Note that and so . We have H = 〈h2〉 is hyperbolic, but since f1(0) = c ∈ J(h2) ⊂ J(G), G is not hyperbolic. By conjugating h2 by a rotation we may
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