On cycles for the doubling map which are disjoint from an interval

Let (T:[0,1]rightarrow [0,1]) be the doubling map and let (0 . We say that an integer (nge 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: $$begin{aligned} D_2={(a,b) : B(a,b),text {is finite}} end{aligned}$$ and $$begin{aligned} D_3={(a,b) : B(a,b)=varnothing }. end{aligned}$$ In particular, we show that if (b-a , then ((a,b)in D_2) , and if (b-ale frac{2}{15}) , then ((a,b)in D_3) , both constants being sharp.