Untitled

Abstract

SEBASTIAN VAN STRIEN AND EDSON VARGAS i.e., the points where f is zero. Throughout this paper we will assume that f is C k outside these critical points, and that f is non-flat at these critical point, i.e., that for i = 1,. .. , d and x near c i we can write f (x) = ±|φ i (x)| βi + f (c i), where φ i is C k , φ i (c i) = 0 and β i > 1. Here k = 2 is enough for Theorems A, B(1) and C(1), and k = 3 is enough for the remaining theorems. In fact, for Theorems A, B(1) and C(1) it suffices to take k = 1 + Zygmund; see [17]. We denote the class of such maps by A k and note that A 3 ⊂ A 2 ⊂ A 1+Zygmund. If B is a Borelean set, we will denote its Lebesgue measure by |B|. We will also use the following two definitions: Definition 1. Let U, V be bounded intervals such that the closure of U is contained in the interior of V. We say that V is an α-scaled neighbourhood of U if |U + | ≥ α|U | and |U − | ≥ α|U | , where U + and U − are the connected components of V \ U. We also sometimes say that U is α-well-inside V. Definition 2. An open interval I ⊂ [−1, 1] is called a nice interval if the forward orbit of its boundary does not intersect I; that is, I ∩ f i (∂I) = ∅ for each i ≥ 0.