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

Let $T:[0,1]\to[0,1]$ be the doubling map and let $0<a<b<1$. We say that an integer $n\ge3$ 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: $D_2=\{(a,b) : B(a,b)\,\text{is finite}\}$ and $D_3=\{(a,b) : B(a,b)=\varnothing\}.$ In particular, we show that if $b-a<\frac16$, then $(a,b)\in D_2$, and if $b-a\le\frac2{15}$, then $(a,b)\in D_3$, both constants being sharp.

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