Errata for “cubic Polynomial Maps with Periodic Critical Orbit, Part Ii: Escape Regions”

In this note we fill in some essential details which were missing from our paper. In the case of an escape region Eh with non-trivial kneading sequence, we prove that the canonical parameter t can be expressed as a holomorphic function of the local parameter η = a−1/μ (where a is the periodic critical point). Furthermore, we prove that for any escape region Eh of grid period n ≥ 2, the winding number ν of Eh over the t-plane is greater or equal than the multiplicity μ of Eh. A result which can be stated as follows is claimed in §6 of the paper Cubic Polynomial Maps with Periodic Critical Orbit, Part II: Escape Regions, Conformal Geometry and Dynamics 14 (2010), 68–112 (referred to below as [BKM]). Assertion A. For any escape region Eh, the residue ∮ dt/2πi at the ideal point ∞h is zero. Furthermore, whenever the kneading sequence of Eh is non-trivial, the indefinite integral t = ∫ dt can be expressed as a holomorphic function of the local parameter η = ξ = a−1/μ. This assertion is true; however, there is a gap in our proof when the kneading sequence is non-trivial. In this case, [BKM, Lemma 5.19 and Theorem 6.2] do show that the quotient dt/da can be expressed as a locally holomorphic function of η, vanishing at η = 0. However, this is not enough to prove the assertion. Since a = η−μ, we have dt dη = dt da da dη = −μ dt da η−μ−1 . Thus we must show that dt/da is divisible by η in order to complete the proof. In fact, we will prove a slightly sharper statement. The necessary details follow. Lemma B. Consider a Branner-Hubbard marked grid of period n ≥ 2, denoting its finite column heights by L1, . . . , Ln−1. If Ln−1 > 0, then Lj = Ln−1 − j for 1 ≤ j ≤ Ln−1 . Received by the editors April 2, 2010. 2010 Mathematics Subject Classification. Primary 37F10, 30C10, and 30D05. The first author was partially supported by the Simons Foundation. The second author was supported by Research Network on Low Dimensional Dynamics PBCT/CONICYT, Chile. 1Our mistake was to ignore the ξ2 in the denominator of [BKM, Equation (6.3)]. 2The period p of the critical orbit can be any multiple of the grid period n; but we will work only with the grid. Note that n ≥ 2 if and only if the kneading sequence is non-trivial. c ©2010 American Mathematical Society Reverts to public domain 28 years from publication