On Cycles for the Doubling Map Which Are Disjoint from an Interval

Let T : [0, 1] → [0, 1] be the doubling map and let 0 < a < b < 1. We say that an integer n ≥ 3 is bad for (a, b) if all n-cycles for T intersect (a, b). Let B(a, b) denote the set of all n which are bad for (a, b). In this paper we completely describe the sets: D2 = {(a, b) : B(a, b) is finite} and D3 = {(a, b) : B(a, b) = ∅}. In particular, we show that if b − a < 1 6 , then (a, b) ∈ D2, and if b − a ≤ 2 15 , then (a, b) ∈ D3, both constants being sharp.