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

TL;DR

This paper characterizes intervals in the unit interval where all or finitely many cycles of the doubling map avoid intersecting the interval, providing sharp bounds for these properties.

Contribution

It completely describes the sets of intervals with finitely many or no cycles intersecting them for the doubling map, establishing sharp bounds.

Findings

Intervals with length less than 1/6 have finitely many intersecting cycles.

Intervals with length at most 2/15 have no intersecting cycles.

The bounds 1/6 and 2/15 are proven to be sharp.

Abstract

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.