Erratum to ‘Semi-hyperbolic fibered rational maps and rational semigroups’ (Ergodic Theory and Dynamical Systems 26 (2006), 893-922)

Hiroki Sumi Department of Mathematics, Graduate School of Science, Osaka University, 1-1, Machikaneyama, Toyonaka, Osaka, 560-0043, Japan E-mail: sumi@math.sci.osaka-u.ac.jp

http://www.math.sci.osaka-u.ac.jp/∼sumi/welcomeou-e.html

January 22, 2008

Abstract We give a correction to the assumption of Theorem 1.12 and Theorem 2.6 in the paper [H. Sumi. Semi-hyperbolic fibered rational maps and rational semigroups. Ergod. Th. & Dynam. Sys. 26 (2006), 893-922].

1 Correction to Theorem 1.12 in [S1]

We use the same notation as that in [S1].

Let (π, Y = X × \overline{ℂ}, X) be a trivial \overline{ℂ}-bundle. Let f : Y → Y be a fibered polynomial map over g : X → X (see [S1, Definition 1.11]). For each x ∈ X, we set A_{x}(f) := \{y ∈ Y_{x} | π\overline{ℂ}(f_{x}^{n}(y)) → ∞\} and K_{x}(f) := Y_{x} \ A_{x}(f). Moreover, we denote by intK_{x}(f) the set of all interior points of K_{x}(f) with respect to the topology in Y_{x}.

Theorem 1.12 in [S1] should be replaced by the following form.

Theorem 1.1. Let (π, Y = X × \overline{ℂ}, X) be a trivial \overline{ℂ}-bundle. Let f : Y → Y be a semi-hyperbolic fibered polynomial map over g : X → X such that d(x) ≥ 2, for any x ∈ X. Assume that either:

(1) J_{x}(f) is connected for each x ∈ X; or

(2) the map x ↦ f_{x} from X to the space of polynomials is locally constant (here, we identify f_{x} with a polynomial on \overline{ℂ}), inf\{d(a, b) | a ∈ π\overline{ℂ}(J_{x}(f)), b ∈ π\overline{ℂ}(A_{x}(f) ∩ C(f)), x ∈ X\} > 0, and for each z ∈ Y with z ∈ intK_{π(z)}(f), there exists no sequence {z_{j}}_{j=1}^{∞} in Y with z_{j} ∈ A_{π(z_{j})}(f) (∀j) such that z_{j} → z as j → ∞.

Then, there exists a positive constant c such that, for any x ∈ X, the basin of infinity A_{x}(f) is a c-John domain.

Remark 1. We have added an extra assumption in condition (2).

Remark 2. In the proof of Theorem 1.1, we need the following Claim (∗), which does not hold in general when we have condition (2) in [S1, the original Theorem 1.12] instead of the condition (2) of Theorem 1.1 :

Claim (∗): Under the assumption of Theorem 1.1, for each (x, y) ∈ X × ℂ let G_{x}((x, y)) := lim_{n→∞}(1/d_{n}(x)) log^{+} |π\overline{ℂ}(f_{x}^{n}((x, y)))| (see [S1, p. 909]). Moreover, for each x ∈ X and z ∈ Y_{x}, let δ_{x}(z) := inf_{w∈J_{x}(f)}\{|π\overline{ℂ}(z) − π\overline{ℂ}(w)|\}. Let a_{1} > 0 be a number. Then, there exists a positive number a_{2} such that if (x, y) ∈ X × ℂ, (x, y) ∈ A_{x}(f), and δ_{x}((x, y)) > a_{1} then G_{x}((x, y)) > a_{2}.

proof of Claim (∗): Under the assumption of Theorem 1.1, suppose that there exists a sequence {z_{j}}_{j=1}^{∞} in Y such that for each j, z_{j} ∈ A_{π(z_{j})}(f) and δ_{π(z_{j})}(z_{j}) > a_{1}, and such that G_{π(z_{j})}(z_{j}) → 0 as j → ∞. We will deduce a contradiction. We may assume that there exists a point z ∈ X × ℂ such that z_{j} → z. Since (x, y) ↦ G_{x}((x, y)) is continuous in X × ℂ ([S1, p. 910]) and G_{x} in A_{x}(f) \ \{∞\} is harmonic (under the canonical identification Y_{x} ≅ \overline{ℂ}), it follows that G_{π(z)} is identically zero in a neighborhood of z in Y_{π(z)}. Hence z ∈ intK_{π(z)}(f). From the assumption of Theorem 1.1, it follows that for each x ∈ X, J_{x}(f) is connected. Since z ∈ intK_{π(z)}(f) ⊂ F_{π(z)}(f), [S2, Theorem 2.14] implies that there exists a neighborhood U of z in Y_{z} such that

diamf_{π(z)}^{n}(U) → 0 as n → ∞  (1)

and

\overline{∪_{n∈ℕ}f_{π(z)}^{n}(U)} ⊂ F̃(f).  (2)

By (1), we obtain

d(CV(f_{π(z)}^{n}), f_{π(z)}^{n}(z)) → 0 as n → ∞,  (3)

where CV(·) denotes the set of all finite critical values. Combining (2) and (3), we get that there exists an n ∈ ℕ such that the point f_{π(z)}^{n}(z) and a point v ∈ CV(f_{π(z)}^{n}) belong to a component V of F_{g^{n}(π(z))}(f). Since x ↦ J_{x}(f) is continuous (see [S2, Theorem 2.14]), it follows that there exists an m ∈ ℕ such that the point f_{π(z_{m})}^{n}(z_{m}) and a point v_{m} ∈ CV(f_{π(z_{m})}^{n}) belong to a component V_{m} of F_{g^{n}(π(z_{m}))}(f). However, since f_{π(z_{m})}^{n}(z_{m}) ∈ A_{g^{n}(π(z_{m}))}(f) and J_{x}(f) is connected for each x ∈ X, it causes a contradiction. Thus we have proved Claim (∗).

Example 1.2. Let c ∈ ℂ, |c| > 6. Let h_{1}(z) := z^{2} + c and h_{2}(z) := z^{2} − c. Let f : Σ_{2} × \overline{ℂ} → Σ_{2} × \overline{ℂ} be the fibered rational map associated with the generator system {h_{1}, h_{2}} (see [S1, Definition 1.4]). Then, f is a hyperbolic fibered polynomial map over the shift map σ : Σ_{2} → Σ_{2} and f satisfies condition (2) in Theorem 1.1. In fact, for each x ∈ Σ_{2}, intK_{x}(f) = ∅.

2 Correction to Theorem 2.6 in [S1]

In [S1, Theorem 2.6], the sequence (n_{j}) of ℕ should be strictly increasing.

References

[S1] H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups, Ergod. Th. & Dynam. Sys. 26 (2006), 893-922

[S2] H. Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products, Ergod. Th. & Dynam. Sys. 21 (2001), 563-603.